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# MTH302BusinessMathematicsStatistics.pdf MATH 223: Vector Calculus

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This page Class Notes was uploaded by zeee Notetaker on Friday November 20, 2015. The Class Notes belongs to MATH 223: Vector Calculus at Central Washington University taught by Aaron Ekstrom in Fall 2015. Since its upload, it has received 10 views. For similar materials see Vector Calculus in Mathematics (M) at Central Washington University.

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Business Mathematics amp Statistics MTH 302 Lesson 22 Lesson 23 Lesson 24 Lesson 25 Lesson 26 Lesson 27 Lesson 28 Lesson 29 Lesson 30 Lesson 31 Lesson 32 Lesson 33 Lesson 34 Lesson 35 Lesson 36 Business Mathematics amp Statistics MTH 302 VU TABLE OF CONTENTS Lesson 1 COURSE OVERVIEW 3 Lesson 2 APPLICATION OF BASIC MATHEMATICS 12 Lesson 3 APPLICATION OF BASIC MATHEMATICS 22 Lesson 4 APPLICATION OF BASIC MATHEMATICS 29 Lesson 5 APPLICATION OF BASIC MATHEMATICS 399 Lesson 6 APPLICATION OF BASIC MATHEMATICS Error Bookmark not defined8 Lesson 7 APPLICATION OF BASIC MATHEMATICS Error Bookmark not defined9 Lesson 8 COMPOUND INTEREST 709 Lesson 9 COMPOUND INTEREST 776 Lesson 10MATRCES 809 Lesson 11 MATRICES 854 Lesson 12 RATIO AND PROPORTION 94 Lesson 13 MATHEMATICS OF MERCHANDISING 1009 Lesson 14 MATHEMATICS OF MERCHANDISING 105 Lesson 15 MATHEMATICS OF MERCHANDISING 11211 Lesson 16 MATHEMATICS OF MERCHANDISING 12120 Lesson 17 MATHEMATICS FINANCIAL MATHEMATICS 12524 Lesson 18 MATHEMATICS FINANCIAL MATHEMATICS 13029 Lesson 19 PERFORM BREAKEVEN ANALYSIS 13433 Lesson 20 PERFORM BREAKEVEN ANALYSIS 14241 Lesson 21 PERFORM LINEAR COSTVOLUME PROFIT AND BREAKEVEN ANALYSIS 14746 PERFORM LINEAR COSTVOLUME PROFIT AND BREAKEVEN ANALYSIS 15049 STATISTICAL DATA REPRESENTATION 1587 STATISTICAL REPRESENTATION 16362 STATISTICAL REPRESENTATION 17170 STATISTICAL REPRESENTATION 18079 STATISTICAL REPRESENTATION 18988 MEASURES OF DISPERSION 20099 MEASURES OF DISPERSION 207 MEASURE OF DISPERASION 217 LINE FITTING 22524 TIME SERIES AND 24039 TIME SERIES AND EXPONENTIAL SMOOTHING 25352 FACTORIALS 26059 COMBINATIONS 269 ELEMENTARY PROBABILITY 27675 Lesson 37PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS 27978 Lesson 38PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS Lesson 41 Lesson 42 Lesson 43 Lesson 44 Lesson 45 28483 Lesson 39PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS 29796 Lesson 40PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS 302 ESTIMATING FROM SAMPLES INFERENCE 314 ESTIMATING FROM SAMPLE INFERENCE 320 HYPOTHESIS TESTING CHISQUARE DISTRIBUTION 325 HYPOTHESIS TESTING CHISQUARE DISTRIBUTION 328 PLANNING PRODUCTION LEVELS LINEAR PROGRAMMING 335 2 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MTH 302 LECTURE 1 COURSE OVERVIEW COURSE TITLE The title of this course is BUSINESS MATHEMATICS AND STATISTICSquot Instructor s Resume The instructor of the course is Dr Zahir Fikri who holds a PhD in Electric Power Systems Engineering from the Royal Institute of Technology Stockholm Sweden The title of Dr Fikri s thesis was Statistical Load Forecasting for Distribution Network Planningquot Obiective The purpose of the course is to provide the student with a mathematical basis for personal and business financial decisions through eight instructional modules The course stresses business applications using arithmetic algebra and ratioproportion and graphing Applications include payroll costvolumeprofit analysis and merchandising mathematics The course also includes Statistical Representation of Data Correlation Time Series and Exponential Smoothing Elementary Probability and Probability Distributions This course stresses logical reasoning and problem solving skills Access to Microsoft Excel software is required for the course Course Outcomes Successful completion of this course will enable the student to 1 Apply arithmetic and algebraic skills to everyday business problems 2 Use ratio proportion and percent in the solution of business problems 3 Solve business problems involving commercial discount markup and markdown 4 Solve systems of linear equations graphically and algebraically and apply to cost volume profit analysis 5 Apply Statistical Representation of Data Correlation Time Series and Exponential Smoothing methods in business decision making 6 Use elementary probability theory and knowledge about probability distributions in developing profitable business strategies Unit Outcomes ResourcesTestsAssiqnments Successful completion of the following units will enable the student to apply mathematical methods to business problems solving Required Student Resources lncludinq textbook nd workbooks Text Selected books on Business Mathematics and Statistics Optional Resources Handouts supplied by the professor Instructor s Slides Online or CD based learning materials Prereguisites The students are not required to have any mathematical skills Basic knowledge of Microsoft Excel will be an advantage but not a requirement Evaluation In order to successfully complete this course the student is required to meet the following evaluation criteria Full participation is expected for this course All assignments must be completed by the closing date Overall grade will be based on VU existing Grading Rules All requirements must be met in order to pass the course Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU COURSE MODULES The following are the main modules of this course Module 1 0 Overview Lecture 1 0 Perform arithmetic operations in their proper order Lecture 2 0 Convert fractions their percent and decimal equivalents Lecture 2 o Solve for any one of percent portion or base given the other two quantities Lecture 2 0 Using Microsoft Excel Lecture 2 Calculate the gross earnings of employees paid a salary an hourly wage or commissions Lecture 3 0 Calculate the simple average or weighted average given a set of values Lecture 4 Perform basic calculations of the percentages averages commission brokerage and discount Lecture 5 0 Simple and compound interest Lecture 6 0 Average due date interest on drawings and calendar Lecture 6 Module 2 o Exponents and radicals Lecture 7 o Solve linear equations in one variable Lecture 7 o Rearrange formulas to solve for any of its contained variables Lecture 7 o Solve problems involving a series of compounding percent changes Lecture 8 0 Calculate returns from investments Lecture 8 0 Calculate a single percent change equivalent to a series of percent changes Lecture 8 o Matrices Lecture 9 o Ratios and Proportions Lecture10 0 Set up and manipulate ratios Lecture11 o Allocate an amount on a prorata basis using proportions Lecture11 0 Assignment Module 12 Module 3 0 Discounts Lectures 12 0 Mathematics of Merchandising Lectures 1316 Module 4 0 Applications of Linear Equations Lecture 1718 0 Breakeven Analysis Lecture 1922 0 Assignment Module 34 0 MidTerm Examination Module 5 0 Statistical data Lectures 23 0 Measures of central tendency Lectures 2425 0 Measures of dispersion and skewness Lectures 2627 Module 6 0 Correlation Lectures 2829 0 Line Fitting Lectures 3031 0 Time Series and Exponential Smoothing Lectures 3133 0 Assignment Module 56 Module 7 o Factorials Lecture 34 o Permutations and Combinations Lecture 34 0 Elementary Probability Lectures 3536 0 Patterns of probability Binomial Poisson and Normal Distributions Lecture 3740 Module 8 0 Estimating from Samples Inference Lectures 4142 0 Hypothesis testing ChiSquare Distribution Lectures 4344 0 Planning Production Levels Linear Programming Lecture 45 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 0 Assignment Module 78 0 EndTerm Examination Note The course modules are subject to change MARKING SCHEME As per VU Rules ESCRIPTION OF TOPICS RECOMMENDED NO MAIN TOPIC TOPICS READING 10 Module Applications of o Overviewew Lecture 1 Reference 1 1 Basic Mathematics Lectures 16 Reference 2 MOdUIe Earnineefi c gvzgtions amp Lecwre 2 1 U M p It E I Tool Microsoft 0 sung Icroso xce Excel Reference 2 Module 0 Calculate Gross Earnings Lecture 3 1 0 Using Microsoft Excel Tool Microsoft Excel Reference 2 Module 0 Calculating simple or Lecture 4 1 weighted averages Tool Microsoft 0 Using Microsoft Excel Excel Reference 6 0 Basic calculations of 5353 2 percentages averages commission nOdUIe brokerage and discount using g rvi gs c r 3 0 Microsoft Excel Excel Reference 2 0 Simple and compound Lecture 6 Module interest Reference 3 Ch 3 1 o Average due date interest on drawings and calendar Tool Microsoft Excel 0 Exponents and radicals Reference 2 20 x rquotsscrr111llfy algebraic Lecture 7 Module Applications of o p S I I t Reference 3 Ch 2 2 Basic Algebra variable W 39quotear equa mm m one Tool Microsoft Lectures 79 o Rearrange formulas to solve for any of its contained variables Excel Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 8 0 Calculate returns from investments Reference 2 0 Problems involving a Series Lecture 8 of Reference 3 Ch 3 compounding percent changes 0 I SIngle percent change Tool Microsoft equwalent Excel to a series of percent changes 9 Reference 2 Lecture 9 Matrices Reference 3 Ch 4 Tool Microsoft Excel 10 0 Set up and manipulate ratios Reference 2 30 0 Set up and solve proportIons Lecture 10 Applications o Express percent differences Reference 3 Ch 3 Module of Ratio and using proportions 2 Proportion o Allocate an amount on a gill ectures 10 prorata basls usmg Tool Microsoft proportlons Excel 11 Reference 2 Module 0 Set up and manipulate ratios Eig g gg 3 Ch 3 2 AHOFate an amount 0quot a Tool Microsoft prorata baSlS usmg proportIons Excel 12 40 0 Calculate the net price of an Reference 2 Merchandising item after single or multiple trade Lecture 12 Module and Financial discounts Reference 3 Ch 3 3 Mathematics 0 Calculate an equivalent single Lectures 12 discount rate given a series of Tool Microsoft 16 discounts Excel 13 Reference 2 o Solve merchandising pricing Lecwre 13 Module Reference 3 Ch 3 3 problems Involvmg markup and Tool Microsoft markdown Excel 14 Reference 2 Lecture 14 Module Reference 3 Ch 3 3 0 Financial Mathematics Part 1 Reference 5 Ch 16 Tool Microsoft Excel Module 15 o Financial Mathematics Part 2 Reference 2 3 Lecture 15 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Module 3 50 BreakEven Module Analysis 4 Lectures 17 22 Module 4 Module 4 Module 4 Module 4 Module 4 6 Statistical Module Representation 5 of Data Lectures 23 16 17 18 19 20 21 22 23 o Financial Mathematics Part 3 0 Graph a linear equation in two variables 0 Solve two linear equations with two unknowns 0 Perform linear costvolume profit and breakeven analysis 0 Using a breakeven chart 0 Perform linear costvolume profit and breakeven analysis 0 Using the algebraic approach of solving the cost and revenue func ons 0 Perform linear costvolume profit and breakeven analysis 0 Using the contribution margin approach 0 Perform linear costvolume profit and breakeven analysis 0 Using Microsoft Excel o Assignment Module 34 0 MidTerm Examination 0 Statistical Data Reference 3 Ch 3 Reference 5 Ch 16 Tool Microsoft Excel Reference 2 Lecture 16 Reference 3 Ch 3 Reference 5 Ch 16 Tool Microsoft Excel Reference 2 Lecture 17 Reference 3 Ch 3 Reference 5 Ch 16 amp 18 Tool Microsoft Excel Reference 2 Lecture 18 Reference 3 Ch 2 Reference 5 Ch 1 Tool Microsoft Excel Reference 2 Lecture 19 Tool Microsoft Excel Reference 2 Lecture 20 Tool Microsoft Excel Reference 2 Lecture 21 Tool Microsoft Excel Reference 2 Lecture 22 Tool Microsoft Excel Reference 2 Lecture 23 Reference 5 Ch 5 Tool Microsoft Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Module 5 Module 5 Module 5 Module 5 Module 6 27 24 0 Statistical Representation Measures of Central Tendency Part 1 25 o StatIstIcal RepresentatIon 0 Measures of Central Tendency Part 2 26 0 Measures of Dispersion and Skewness Part 1 27 0 Measures of Dispersion and Skewness Part 2 7 Correlation 28 Time Series and Exponential 0 Correlation Smoothing Part 1 Lectures 28 33 29 0 Correlation Part 2 30 0 Line Fitting Part 1 Excel Reference 2 Lecture 24 Reference 4 Ch 3 Reference 5 Ch 6 Tool Microsoft Excel Reference 2 Lecture 25 Reference 4 Ch 3 Reference 5 Ch 6 Tool Microsoft Excel Reference 2 Lecture 26 Reference 4 Ch 4 Reference 5 Ch 6 Tool Microsoft Excel Reference 2 Lecture 27 Reference 4 Ch 4 Reference 5 Ch 6 Tool Microsoft Excel Reference 2 Lecture 28 Reference 5 Ch 13 Tool Microsoft Excel Reference 2 Lecture 29 Reference 5 Ch 13 Tool Microsoft Excel Reference 2 Lecture 30 Reference 5 Ch 14 Tool Microsoft Excel Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 7 Elementary Probability Lectures 34 38 Module 7 Module 7 Module 7 Module 7 Module 7 31 32 33 34 35 36 37 38 0 Line Fitting Part 2 Time Series and Exponential Smoothing Part 1 Time Series and Exponential Smoothing Part 2 o Assignment Module 56 o Factorials o Permutations and Combinations 0 Elementary Probability Part 1 0 Elementary Probability Part 2 o Patterns of probability Binomial Poisson and Normal Distributions Part 1 0 Patterns of probability Binomial Poisson and Normal Distributions Part 2 Reference 2 Lecture 31 Tool Microsoft Excel Reference 2 Lecture 32 Reference 5 Ch 15 Tool Microsoft Excel Reference 2 Lecture 33 Reference 5 Ch 15 Tool Microsoft Excel Reference 2 Lecture 34 Reference 3 Ch 2 Tool Microsoft Excel Reference 2 Lecture 35 Reference 5 Ch 8 Tool Microsoft Excel Reference 2 Lecture 36 Reference 5 Ch 8 Tool Microsoft Excel Reference 2 Lecture 39 Reference 5 Ch 9 Tool Microsoft Excel Reference 2 Lecture 40 Reference 5 Ch 9 Tool Microsoft Excel Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Module 7 Module 7 8 Probability Distributions Lectures 39 Module 44 8 9 Linear Programming Lecture 45 Module 8 Module Module Module Methodology 39 40 41 42 43 44 45 0 Patterns of probability Binomial Poisson and Normal Distributions Part 3 0 Patterns of probability Binomial Poisson and Normal Distributions Part 4 o Estimating from Samples Inference Part 1 o Estimating from Samples Inference Part 2 o Hypothesis testing Chi Square Distribution Part 1 o Hypothesis testing Chi Square Distribution Part 2 0 Production Planning Linear Programming o Assignment Module 78 0 End Term Examination Reference 2 Lecture 41 Reference 5 Ch 9 Tool Microsoft Excel Reference 2 Lecture 41 Reference 5 Ch 9 Tool Microsoft Excel Reference 2 Lecture 42 Reference 5 Ch 10 Tool Microsoft Excel Reference 2 Lecture 43 Reference 5 Ch 10 Tool Microsoft Excel Reference 2 Lecture 44 Reference 5 Ch 11 Tool Microsoft Excel Reference 2 Lecture 45 Reference 5 Ch 1 1 Tool Microsoft Excel Reference 2 Lecture 45 Reference 5 Ch 18 Tool Microsoft Excel There will be 45 lectures each of 50 minutes duration as indicated above The lectures will be delivered in a mixture of Urdu and English The lectures will be heavily supported by slide presentations The slides for a lecture will be made available on the VU website for the course a few days before the actual lecture is televised This will allow students to carry out preparatory reading before the lecture The course will be provided its own page on the VU s web site This will be used to 10 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU provide lecture and other supporting material from the course to the students The page will have a link to a webbased discussion and bulletin board for the students Teaching assistants will be assigned by VU to provide various forms of assistance such as grading answering questions posted by students and preparation of slides Gradin There will be a term exam and one final examination There will also be 4 assignments each covering two modules The final exam will be comprehensive These will contribute the following percentages to the final grade Mid Term Exam 35 Final 50 4 Assignments 15 Text and Reference Material The course is based on material from different sources Topics for reading will be indicated on course web site and in professor s handouts also to be posted on the course web site A list of reference books will also be posted and updated on the course web site The following material will be used by the students as reference Reference 1 Course Outline Instructor s Power Point Slides Business Mathematics amp Statistics by Prof Miraj Din Mirza Elements of statistics amp Probability by Shahid Jamal Quantitative Approaches in Business studies by Clare Morris Microsoft Excel Help File Schedule of Lectures Given above is the tentative schedule of topics to be covered Minor changes may occur but these will be announced well in advance 11 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 2 Applications of Basic Mathematics Part 1 OBJECTIVES The objectives of the lecture are to learn about 0 Different course modules 0 Basic Arithmetic Operations 0 Starting Microsoft MS Excel 0 Using MS Excel to carry out arithmetic operations COURSE MODULES This course comprises 8 modules as under 0 Modules 14 Mathematics 0 Modules 58 Statistics Details of modules are given in handout for lecture 01 BA SIC ARIT HME TIC OPERA T IONS Five arithmetic operations provide the foundation for all mathematical operations These are 0 Addition 0 Subtraction o Multiplication 0 Division 0 Exponents Example Addition 12 5 17 mple Subtraction 12 5 7 mple Multiplication 12 x 5 60 Example Exponent 4quot2 16 4quot12 2 4quot12 14quot12 12 05 MICROSOFT EXCEL IN BUSINESS MATHEMATICS amp STATISTICS Microsoft Corporation s Spreadsheet software Excel is widely used in business mathematics and statistical applications The latest version of this software is EXCEL 2002 XP This course is based on wide applications of EXCEL 2002 It is recommended that you install EXCEL 2002 XP software on your computer If your computer has Windows 2000 and EXCEL 2000 even that version of EXCEL can be used as the applications we intend to learn can be done using the earlier version of EXCEL Those of you who are still working with Windows 98 and have EXCEL 97 installed are encouraged to migrate to newer version of EXCEL software 12 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Startinq EXCEL 2000 XP EXCEL 2000 XP can be started by going through the following steps Click Start on your computer Click All Programs Click Microsoft Excel The following slides show the operations i EIi39l39laiil r391III39iII li Horosol t Wire Horosol t Excel r Horosol t Frooieoe 7h wif Re note existence M E Adobe Fieecler en Horosol t PamePoin L l allquot Humor E New ISIFFiEI Eizcurl39u39lt ti opanorricaoommt Elf Set Program notes and Defaults lEia Window Cider Win nite Ltdate Geniee h Acceeezries b lr39icroiofl IIIFFico Toois Iquot F39rlrtlle Ji39itemel F39rntlng Stortup If Adzhe Fieezler EI Internet EEcplz er ll39icroiofl Fliozoii rillroeor39t Ezozel lr39itroeoft Dutlozit lr39lcroeofl Word rr39icrosoft FrotF39ege ll39icroioil PowerPoint I39I39EHExpluzirer i3 MoohEIp39cee Remote Meetmce window li39ieizlle Flafer 39iilinzlziio lihsionuer Snain I 3945 IZIIZIIII39 i The EXCEL window opens and a blank worksheet becomes available as shown below Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU M Icmsnf l Esme Harald Elle gdt Elanu mart Fgrrnat cola gate nderu Ijelp v 39 r E If g l g il r g v gt gd ma E i itsta m 1 Big E 12 Egg 335123 3931 r 1 A E E D E F G H IT Hm Wartthunk 1r n i penamthnIt Miami3 Lecture jiEEUUIUntereal Arlthmetlt paratlnre LectureJlIi More hunch19kt lilil HEW El Blank Workbook HEW Iimnt tstin Hmikhllt E Choose workbook HEW Iimnt template General Tan rplatae Tarnplatas on my Wet Sites Tan platae on I39quotIlru IftIm Add ll tmrltFlace Hmtt Emil Help quot If Ericw at startup H i r ill llieetl shaa l39ee f M i 39ii FlIIIII A 2 3 4 E E T E 9 10 11 12 13 11 15 13 1 13 1E 33 21 22 23 9 u i il39i39l39HElL39EiIEIZEI39JE hand I3939Iiurur39uufl Excel Banal1 The slide shows a Workbook by the name book1 with three sheets Sheet1 Sheet2 and Sheet3 The Excel Window has Column numbers starting from A and row numbers starting from 1 the intersection of a row and column is called a Cell The first cell is A1 which is the intersection of column A and row 1 All cells in a Sheet are referenced by a combination of Column name and row number Example 1 B15 means cell in column B and row 15 Example 2 A cell in row 12 and column C has reference C12 A Range defines all cells starting from the leftmost corner where the range starts to the rightmost corner in the last row The Range is specified by the starting cell a colon and the ending cell Example 3 A Range which starts from A1 and ends at D15 is referenced by A1 D15 and has all the cells in columns A to D up to and including row 15 A value can be entered into a cell by clicking that cell The mouse pointer which is a rectangle moves to the selected cell Simply enter the value followed by the Enter key The mouse pointer moves to the cell below If you make a mistake while entering the value select the cell again by clicking it Enter the new value The old value is replaced by the new value 14 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU If only one or more digits are to be changed then select the cell Then double click the mouse The blinking cursor appears Either move the arrow key to move to the digit to be changed or move the cursor to the desired position Enter the new value and delete the undesired value by using the Del key suggest that you learn the basic operations of entering deleting and changing data in a worksheet About calculation operators in Excel In Excel there are four different types of operators Arithmetic operators Comparison operators Text concatenation operator Reference operators The following descriptions are reproduced from Excel s Help file for your ready reference In the present lecture you are directly concerned with arithmetic operators However it is important to learn that the comparison operators are used where calculations are made on the basis of comparisons The text concatenation operator is used to combine two text strings The reference operators include and or as the case maybe We shall learn the use of these operators in different worksheets You should look through the Excel Help file to see examples of these functions Selected material from Excel Help File relating to arithmetic operations is given in in a separate file The Excel arithmetic operators are as follows Addition Symbol Example 54 Result 9 Subtraction Symbol Example 54 Result 1 Multiplication Symbol Example 54 Result 20 Division Symbol Example 124 Result 3 Percent Symbol Example 20 Result 02 Exponentiation quot Example 5quot2 Result 25 Excel Formulas for Addition All calculations in Excel are made through formulas which are written in cells where result is required Let us do addition of two numbers 5 and 10 We wish to calculate the addition of two numbers 10 and 5 Let us see how we can add these two numbers in Excel 1 Open a blank worksheet 2 Click on a cell where you would like to enter the number 10 Say cell A15 3 Enter 10 in cell A15 4 Click cell where you would like to enter the number 5 Say cell B15 5 Click cell where you would like to get the sum of 10 and 5 Say cell C15 6 Start the formula Write equal sign in cell C15 7 After write left bracket in cell C15 8 Move mouse and left click on value 10 which is in cell A15 ln cell C15 the cell reference A15 is written 9 Write quot after A15 in cell C15 10 Move mouse and left click on value 5 which is in cell B15 ln cell C15 the cell reference B15 is written 11 Write right bracket in cell C15 12 Press Enter key The answer 15 is shown in cell C15 If you click on cell C15 the formula A15B15quot is displayed the formula bar to the right of fX in the Toolbar 15 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The main steps along with the entries are shown in the slide below The worksheet H302 lec 02 contains the actual entries 7 T Micreseft Excel Arithmetic pere ene F Elle Edit Eiew insert Fgrmat Innis Data window elp Adobe PDF TWZHEEDUESGQlDDlDi iTEiD v 51 x neee weem Ev ferial vznve ettev vf 1 E El 99 ii 5 v eeleele El D E F A r e H I J 2 ADDING TWO NUMBERS 5 and 10 eleTEPe e l 1 Entereln eell A15 e l e Enterie in cell 315 e l 3 Write eln in Cell ele i 4 Piece meuee en eell A115 3 i E type ein e l 39r eete eell Bie iEI E Preee Enter key 11 Reeult Shewn in Cell C15 12 Q Cli l39f en Cell 315 13 Reeult FermuleA15BiEie el lewn in fermuleleer 15 5 1U 15 HI l h H Sheetl etE A Sheet3 f i r r I l ean e eeeeee x a a e 41 e e v iv e E e e e l ll g iii 2 l 5 g l Tg E l3 V lllrerr Ele iceit l l3939iilrIIItt F39IZI393939El39F39IIirlt IZIIZIE39E 43 The next slide shows addition of 6 numbers 5 10 15 20 30 and 40 The entries were made in row 34 The values were entered as follows Cell A34 5 Cell B34 10 Cell C34 15 Cell D34 20 Cell E34 30 Cell F34 40 The formula was written in cell G34 The formula was 51015203040 The answer was 120 You can use an Excel function SUM along with the cell range A34F34 to calculate the sum of the above numbers The formula in such a case will be SUMA34F34 You enter quot followed by SUM followed by Click on the cell with value 5reference A34 Drag the mouse to cell with value 40reference F34 and drop the mouse Enter quot and then press the Enter key 16 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel hrithmetic peratinns Elle Edit ew lnsert Fgrmat Iools Qatar indow elp Atler PDF TWEEEUESHUW FEW 0310 v 539 X a ggg rm Ev jmial vlov viv E i3 99 Tl 7 J35 v 5 4 E c o E F c H I J H L T 19 ADDING SIX N U M BERS 51015203040 2 STEPS 21 1 Enter 5 in cell 11 Type in cell C34 or formula Iquot 22 2 Enter 10in cell 334 12 Click on cell C34 23 Enter 15 in cell C34 13 Type in cell C34 or formula ll l39 24 4 Enter 20 in cell D34 14 Click on cell D34 25 5 Enter 30 in cell E34 15 Type in cell C34 or formula ll f 39 25 3913 Enter 40 in cell F34 113 Click on cell E34 2 T Enter in cell C34 1quot Type in cell C34 or formula llquot 28 13 Click on cell 113 Click on cell F34 23 9 Type in cell C34 or formula liar 13l Fll39E Enter 3 1 Click on cell 334 Ftesult In cell C34 31 32 E312 tltl 51 15203040 in a row 51 1520304012 33 34 5 10 15 20 30 40 120 y a 7quot H 1 r H heetl 6 SheetE Sheet3 I i l Draw 3 Agto hapesquot K E D Q 3 L d quot i quot E E i 1 Ready 7 Pelicans3ft EIIEl Firit 7 El l3939liIrIIIIft F39III39IerF39IIint 393939 239E In the above two examples you learnt how formulas for addition are written in Excel Excel Formula for Subtraction Excel formulas for subtraction are similar to those of addition but with the minus sign Let us go through the steps for subtracting 15 from 25 Enter values in row 50 as follows Cell A50 25 Cell B50 15 Write the formula in cell C50 as follows A50B50 To write this formula click cell 050 where you want the result Enter Click on cell with value 25 referenceA50 Enter quot minus sign Click on cell with value 15 reference B50 Press enter key If you enter 15 first and 25 later then the question will be to find result of subtraction 15 25 17 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Arithmetic pere uns Elle Edit iew lnsert Fgrmat Innis gate lndew elp Fidelge F39DF TWDEEEIUESUDWFDM HEHD v 539 X D gci39n v Evl fnrial vznvg g s 5s E Tin esn v s 4seEise1 4391 E C Formula Elarl E F G H I J K L 44 SUBTRACTING 110i NUMBERS 25 and 15 4n n STEPS 42 1 Enter 25 in eell A55 5 Type in eell C55 er f39ermula bar 43 2 Enter 15 in sell 350 Cli l t en sell 350 44 3 Write in eell C50 F Press Enter key 45 41 Clielt en eell A50 Result In eell C50 45 39 4 4s 4s a 25 15 ml 51 Yr H heetl Sheet2 4 Sheet3 f r Drawr 33 eute hapesr I 3 Ready l3939liIrIIsIIFt EIe Firit El l3939liurIIsIIFt F39III39IerF39IIint 54 k Ijlj35 Excel Formula for Multiplication Excel formula for multiplication is also similar to the formula for addition Only the sig of multiplication will be used The Excel multiplication operator is Micrusuft Excel Arithmetic pera nns Elle Edit Eiew lnsert Fgrmat Tools gate window elp Fidelge PDF WDEEEUESUDW FDD D HEHD 539 X D 39 g r Evlferial vzsvggg g f e4 ns El 99 T113 fist it sen v 8 Aenese I 43934 E C Formula Ear E F G H I J K 44 MULTIPLYING TW NUMBERS 25 and 15 52 53 54 STEPS 55 1 Enter 25 in eell A5 5 Type in eell C5D er fermula bar 55 2 Enter 15 in cell BEE 5 Click en cell 3513 5 3 1Ilulii39rite in cell GEE T Press Enter keyr 58 4 Click en eell A Result In eell GEE 59 44 25 1 5 375 E 52 ES 541 as quoti H 4 r H Sheetl SheetESheet f DLBW39 ls setu hapesv x III 4411 3 Ready l39391ir5ft EEI Arit 7 El l39391irsFt F39IZI393939Er39F39IIiFIt 55439 quotl Let us look at the multiplication of two numbers 25 and 15 The entries will be made in row 60 Enter values as under Cell A50 25 Cell 850 15 The formula for multiplication is A50BSO 18 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Click on cell C50 to write the formula in that cell Enter Click on cell with number 25 reference A50 Enter Click on cell with number 15 reference B50 Press Enter key The answer is 375 in cell C50 Excel Formula for Division The formula for division is similar to that of multiplication with the difference that the division sign I will be used Micrueuft Excel hrithmetic pera une TEX Elle Edit Eiew insert Fgrmet IDDIS gate window Help Adelge PDF lecfiiti39mei e e e v e v Dagger v Ev ferial Tzsvg e e E 99 Q c cre v e AFEIB75 A E 3 Formula Ber E F 3 H J K ee DIVIDING A NUMBER 240 BY 15 E STEPS ES 1 Enter 24D in cell A75 5 Type l in cell 75 cr fermule her ES 2 Enter 15 in cell B75 5 Cliclt cn cell BTE 7 3 Write in cell 375 7 Freee Enter keyr r1 4 Cliclt en cell A75 Reeult In cell 375 r2 r3 r4 r5 240 15 16 H 4 r n Sheetl rf SheetE rf Sheet3 r Dtaw39 it eetn hapesr e Al 3 H Reedy I Eta l39391iurIIIft Ee Firit i l39391iIrII5IIFt F39III39IerF39IIint Let us divide 240 by 15using Excel formula for division Let us enter numbers in row 75 as follows Cell A75 240 Cell B75 15 The formula for division will be written in cell C75 as under A75B75 The steps are as follows Click the cell A75 Enter 240 in cell A75 Click cell B75 Enter 15 Click cell C75 Enter Click on cell with value 240 reference A75 Enter Click cell with number 15 reference B75 Press enter key The answer 16 will be displayed in cell C75 Excel Formula for Percent The formula for converting percent to fraction uses the symbol To convert 20 to fraction the formula is as under 20 If you enter 20 in cell A99 you can write formula for conversion to fraction by doing the following Enter 20 in cell A99 ln cell B99 enter Click on cell A99 Enterquotquot Press Enter key The answer 02 is given in cell B99 19 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrsft Excel Arithmetic pera ns Elle Edit ew insert Fgrmat Tools gate window Help Typeequeetinnforhe D ig ft refn EEr ltl aWnv Arial 110v HIE EEEE T63493 E E1IIIIII v I A El 3 D l E F e H an ERTING PERCENT 91 92 93 STEPS Em 1 Enter in cell t E15 Write in cell 399 BE IEliclt en cell A99 9 El Write 93 Frees Enter key Result cell BEE 99 2D 02 Excel Formula for Exponentigtion The symbol for exponentiation is quot The formula for calculating exponents is similar to multiplication with the difference that the carat symbol quot will be used Let us calculate 16 raised to the power 2 by Excel formula for exponentiation The values will be entered in row 85 The steps are Select Cell A85 Enter 16 in this cell Select cell B85 Enter 2 in this cell Select cell C85 Enter quot Select cell with value 16 referenceA85 Enter Select number 2 reference B85 Press Enter key The result 256 is displayed in cell C85 20 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Arithmetic pera nne Eile Edit iew insert Fgrmet leels gate indew elp adage F39DF DWEQJE v Ev jerial C85 v f3 a9HEEILElEE e e r FermuleEer E F G H I J K I r CALCULATING 15 iii FE re STEPS El 1 Enter l in eellAEEn Bi 2 Enter in eell BEE 82 Write in eell C35 83 1 Cliek en eell A35 35 1E E Type A in eell C35 erferrnule leer E Cliek en eell BEE T Preee Enter key Reeult In eell C35 H 4 r gtISheet15heet2j3heet3f lil Il Drew 3 AgteShepESTK 3 I Reedy l39391iIrII5IIFt EIE Ftr39it i IE l39391iIrII5IIt39t F39III393939Er39F39IIirIl Sill KIDNE Recommended Homework Download worksheet MTH302 Iec 02xls from the course web site Change values to see change in results Set up new worksheets for each Excel operator with different values Set up worksheets with combinations of operations 21 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 3 Applications of Basic Mathematics Part 2 OBJECTIVES The objectives of the lecture are to learn about o Evaluations 0 Calculate Gross Earnings 0 Using Microsoft Excel Evaluation In order to successfully complete this course the student is required to meet the evaluation criteria 0 Evaluation Criterion 1 0 Full participation is expected for this course 0 Evaluation Criterion 2 o All assignments must be completed by the closing date o Evaluation Criterion 3 0 Overall grade will be based on VU existing Grading Rules o Evaluation Criterion 4 o All requirements must be met in order to pass the course My There will be a term exam and one final exam there will also be 4 assignments The final exam will be comprehensive These will contribute the following percentages to the final grade Mid Term Exam 35 Final 50 4Assignments 15 Collaboration The students are encouraged to develop collaboration in studying this course You are advised to carry out discussions with other students on different topics It will be in your own interest to prepare your own solutions to Assignments You are advised to make your original original submissions as copying other students assignments will have negative impact on your studies ETHICS Be advised that as good students your motto should be 0 No copying o No cheating o No short cuts 22 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Methodology There will be 45 lectures each of 50 minutes duration The lectures will be delivered in a mixture of Urdu and Englis The lectures will be heavily supported by slide presentations The slides available on the VU website before the actual lecture is televised Students are encouraged to carry out preparatory reading before the lecture This course has its own page on the VU s web site There are lecture slides as well as other supporting material available on the web site Links to a webbased discussion and bulletin board will also been provided Teaching assistants will be assigned by VU to provide various forms of assistance such as grading answering questions posted by students and preparation of slides Text and Reference Material This course is based on material from different sources Topics for reading will be indicated on course web site and in professor s handouts A list of reference books to be posted and updated on course web site You are encouraged to regularly visit the course web site for latest guidelines for text and reference material PROBLEMS If you have any problems with understanding of the course please contact mth302vuedupk Types of Employees There may be three types of employees in a company 0 Regular employees drawing a monthly salary 0 Part time employees paid on hourly basis 0 Payments on per piece basis To be able to understand how calculations of gross earnings are done it is important to understand what gross earnings include GROSS EARNLNGSSALARY Gross salary includes the following 0 Basic salary o Allowances Gross salary may include 0 Basic salary 0 House Rent 0 Conveyance allowance 0 Utilities allowance Accordance to the taxation rules if allowances are 50 of basic salary the amount is treated as tax free Any allowances that exceed this amount are considered taxable both for the employee as well as the company Example 1 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs 23 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU What is the taxable income of employee Is any add back to the income of the company Allowances 500010000 x 100 50 Hence allowances are not taxable Total taxable income 10000 Rs Add back to the income of the company 0 Example 2 The salary of an employee is as follows Basic salary 10000 Rs Allowances 7000 Rs What is the taxable income of employee Is any add back to the income of the company Allowances 700010000 x 100 70 Allowed nontaxable allowances 50 05 x 10000 5000 Rs Taxable allowances 70 50 7000 5000 2000 Rs Hence 2000 Rs of allowances are taxable Total taxable income 10000 2000 12000 Rs Add back to the income of the company 20 allowances 2000 Rs Structure of Allowances The common structure of allowances is as under 0 House Rent 45 o Conveyance allowance 25 0 Utilities allowance 25 Example 3 The salary of an employee is as follows Basic salary 10000 Rs What is the amount of allowances if House Rent 45 Conveyance allowance 25 and Utilities allowance 25 House rent allowances 045 x 10000 4500 Rs Conveyance allowance 0025 x 10000 250 Rs Utilities allowance 0025 x 10000 250 Rs Thus total allowances are 4500250250 5000Rs Provident Fund According to local laws a company can establish a Provident Trust Fund for the benefit of the employees By law 111th of Basic Salary per month is deducted by the company from the gross earnings of the employee An equal amount ie 111th of basic salary per month is contributed by the company to the Provident Fund to the account of the employee The company can invest the savings in Provident Fund in Government Approved securities such as defense saving Certificates Interest earned on investments in Provident Fund is credited to the account of the employees in proportion to their share in the Provident Fund Example 4 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs 24 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU What is the amount of deduction on account of contribution to the Provident Trust Fund What is the contribution of the company What is the total saving of the employee per month on account of Provident Trust Fund Employee contribution to Provident Fund 111 x 10000 9091 Rs Company contribution to Provident Fund 111 x 10000 909 1 Rs Total savings of employee in Provident Fund 909 1 909 1 18182 Rs Gratuity Fund According to local laws 3 company can establish a Gratuity Trust Fund for the benefit of the employees By law 111th of Basic Salary per month is contributed by the company to the Gratuity Fund to the account of the employee Thus there is a saving of 111 h of basic salary on behalf of the employee in Gratuity Fund The company can invest the savings in Gratuity Fund in Government Approved securities such as defence saving Certificates Interest earned on investments in Gratuity Fund is credited to the account of the employees in proportion to their share in the Gratuity Fund Example 5 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs What is the contribution of the company on account of gratuity to the Gratuity Trust Fund Company contribution to Gratuity Fund Total savings of employee in Gratuity Fund 111 x 10000 909 1 Rs Leaves All companies have a clear leaves policy The number of leaves allowed varies from company to company Typical leaves allowed may be as under 0 Casual Leave 18 Days per year o Earned Leave 18 Days per year 0 Sick Leave 12 Days per year Example 6 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs What is the cost on account of casual earned and sick lea ves per year if normal working days per month is 22 What is the total cost of lea ves as percent of gross salary Gross salary 10000 5000 15000 Rs Cost of casual leaves per year 18 22 x 12 x 15000 x 12 122727 Rs Cost of earned leaves per year 18 22 x 12 x 15000 x 12 122727 Rs Cost of Sick leaves per year 12 22 x 12 x 15000 x 12 81818 Rs Total cost of leaves per year 122727 122727 81818 327273 Rs Total cost of leaves as percent of gross salary 32727312 x 15000x 100 182 Social Charqes 25 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Social charges comprise leaves group insurance and medical Typical medicalgroup insurance is about 5 of gross salary Other social benefits may include contribution to employee s children s education club membership leave fare assistance etc Such benefits may be about 58 Leaves are 182 of gross salary as calculated in above example Total social charges therefore may be 182 5 58 29 of gross salary Other companies may have more social benefits The 29 social charges are quite common Example 7 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs What is the cost of the company on account of leaves 182 group insurancemedical 5 and other social benefits 58 Leaves cost 0 182 x 15000 2730 Rs Group insurancemedical 005 x 15000 750 Rs Other social benefits 0058 x 15000 870 Rs Total social charges 2730 750 870 4350 Rs W Summary of different components of salary is as follows Basic salary Allowances 50 of basic salary Gratuity 909 of basic salary Provident Fund 909 of basic salary Social Charges 29 of gross salary Gross remuneration is pay or salary typically monetary payment for services rendered as in an employment It includes Group insurance medical etc Miscellaneous social charges 1 Basic Salary 2 House rent allowance 3 Conveyance allowance 4 Utilities 5 Provident fund 6 Gratuity fund 7 Leaves 8 9 Bene ts can also include more factors and are not limited to the above list The purpose of the benefits is to increase the economic security of employees Example 8 The salary of an employee is as follows Basic salary 6000 Rs 26 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The calculations are shown in the slide below There is mistake in calculating gross remuneration in example given below Total amount of leaves are 19636 which is the amount for 1 year not 1 month So divide the amount of leaves by 12 and then calculate gross remuneration Microsoft Excel liliHIi 2lec Eile Edit Eiew insert Fgrn39iat Innis gate indow elp D n j g v 11 559611 gzv l l l mnvo Ariel no 32g E E 2341 J15 v 1 3 A E G D E F G H 2 EK rMPLE GRSS ShLARY 3 Eaeicealaly 1311111 1 House Rent Allowance 1145 22111 5 Gouveyauce allowance 2511 151 E Utlitiee tllowauce 2511 151 7 Total llowaucee 15 311111 E F rovitleut Fuutl 555 555 El wu contribution 555 55151 1 Gratuity fuutl 555 5515 11 Earuetl Leave 15 tlaye 22134 12 Gaeual leave 11 tlaye 22134 13 Sick leave 12 tlaye 45115 14 Group luequotlrletlical 511 3111 l ieceocial Gltargee 53 522 15 Total llowaucee 511111 1 Gl ealaly 5111111 15 F rovitleut Fuutl I3 5515 19 Gratuityr Fuutl 5515 2 Leavee 21 Iifiitltereocial Gltargee 13122 22 Total Social Gltargee 211455 23 Grow remuneration 211551quot Convertinq frgction to percent Calculate percent by multiplying fraction by 100 and put the percent sign Percent Fraction X 100 Example 9 Convert 0 1 to percent 01 X 100 10 Common Frgction Common fraction is a fraction having an integer as a numerator and an integer as a denominator For example 12 10100 are common fractions Converting percent into Common Fraction Example 11 20 20 100 02 Decimal fraction Any number written in the form an integer followed by a decimal point followed by a possibly in nite string of digits For example 25 39 etc Converting percent into decimal fraction 27 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 20 02 Percent 20 or 2010002 M Percentage is formed by multiplying a number called the base by a percent called the rate Thus Percentage Base x Rate Example 13 What percentage is 20 x of 120 Here rate 20 20 100 02 Base 120 Percentage 20100 x 120 0r 02 X 120 24 Example 14 What Percentage is 6 of 40 Percentage Rate X Base 006 X 40 24 Base Base PercentageRate Example 15 Find base if Rate 240 024 Percentage 96 Base 96024400 28 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 4 Applications of Basic Mathematics Part 3 OBJECTIVES The objectives of the lecture are to learn about Review Lecture 3 Calculating simple or weighted averages Using Microsoft Excel Gross Remuneration The following slide shows worksheet calculation of Gross remuneration on the basis of 6000 Rs basic salary As explained earlier house rent is 45 of basic salary Conveyance and Utilities Allowance are both 25 of basic salary Both Gratuity and Provident fund are 111th of basic salary The arithmetic formulas are as follows Excel formulas are within brackets Basic salary 6000 Rs House rent 045 x 6000 2700 Rs Excel formula B93045 Conveyance Allowance 0025 x 6000 150 Rs Excel formula B930025 Utilities allowance 0025 x 6000 150 Rs Excel formula B930025 Gross salary 6000 2700 150 150 9000 Rs Excel formula SUM893BQ6 Gratuity 111 x 6000 545 Excel formula ROUND111B930 In the Excel formulas the sign is used before the row and column reference to fix the location of the cell Since house rent CA utilities gratuity and provident fund are calculated with respect to basic salary so by using B93 we fixes the location of cell 893 This feature can be used for quick and correct calculation of all allowances and benefits 39 Microsoft Excel Gross ren39iuneration Elle Edit Eiew insert Fgrmet Tools gate window elp s e ii 3i it s as few M e v vi 2 viii ilit s 125 v Enrial 71o THBIQHEEE i a I i gj39g iiazi quotquotquot 3944quotth H Hse a e e c D E F G EGROSS REMUNERATION Rs 91 3 Amount in Rs Percentage Basic Salary 6000 House Rent 200 045 E13131 150 0025 EUtilities 150 0025 E Gratuity 5454545455 009090909 ER Fund 5454545455 009090909 E E Follow these steps 1 In cell 394 write B93DQ4 2 Press Enter 3 Click on the cell 894 E 4 Move your cursor to the bottom righthand cornor of cell 394 E till the cursor symbol become a plus sign 5 Hold the left mouse button and drag downward till 398 E 6 This lls all the cells by multiplying 393 cell yalue with the m respective percentage 109 29 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Now let us see cell by cell calculation EmitStart39Eastl7tiiiii i ii i iai mats quotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquot quot Elle Edit Eiew insert Fgrmal nulls Eata induw elp Adulge PDF ViEWFDrl39I39IUlEIE stav r E39i SUM 139 X J 52 El93 45 a E c D 91 GROSS REMUNERSTIGN Rs 92 a Basic salary l EUUUISEharges 1740 gHouse Rent B93m45l 118130 a CA 15039 as Utilities 150 a Total salary SUD as Gratuity 545 sgPFund 545 IIIIEI satiaastaa its iiiii iiii ii itjittii jiietjtiti quotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquotquot quot Elle Edit Elsiii insert Fgrmat Inuls Eata induw elp Adulge F DF ViEWFEIrlTIUIEIE SET Ei SUM 1quot X J f3 El93l25 a E C D a GROSS REMUNERATIGN Rs 92 93 Basic salary SUUU SCl1arges 1MB 94 House Rent 27011 G Remun 1181310 ea B93ra025 as Utilities 150 a Total salary 9000 as Gratuity 545 as P Fund 545 IIIlEl Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 engage ear quoti a ai g a seagemeJed E Eile Edit yew insert Fgrmet eels gate llIn ow elp ACIDQE F39DF yiewfermulee a 3 e v 2 v are 39 SUM v 4 9quot f3 SLJMlElE13zElEIEj a e e u 91 GROSS REMUNERETIDN Rs 92 as Basic salary 5000 SCharges 1MB 94 Huse Rent 2WD G Remun 11830 as CE 150 as Utilities 15l Tetal salary SUMBQ3BQE ee Gratuity 5amp5 as P Fund 545 lIIIIII In Gratuity and provident calculations the function ROUND is used to round off values to desired number of decimals In our case we used the value after the semicolon to indicate that no decimal is required If you want 1 decimal use the value 1 for 2 decimals use 2 as the second parameter to the ROUND function The first parameter is the expression for calculation 111B93 ailar aaetaa I1mmtastelessea Elle Edit ierv insert Fgrrnet eels gate window Help adage F39DF yiewforrnulee a re e v 2 v i sum r X J a eeurlurnnlreeauj a e e u a REMUNERATIDN Rs 92 as Basic salary 5000 SEharges 1740 a Heuse Rent 2700 G Remun 11830 as CA 150 as Utilities 150 a Total salary 9000 Gratuity eeunnu1r11lrea30l as P Fund 545 1IIIII Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In the calculation for social charges the formula is B9329100 Here 29100 means 29 social charges The sign was not used here to explain another feature of excel If the formula in cell D93 is copied to cell E93 say the cell reference B93 in formula changes to C93 B93 would be needed to fix the value of basic salary in cell E93 Microsoft Excel AddinLNurnherLEaernplesEel Elle Edit Eiew insert Format eels gate window Help Fidelge F39DF ViEWFUrmulaE 139 5 K ya in 21 an r x u e eeeteenee 77777777777 N e e C e E T a REMUNERATIGN as Basic salary SGhares 393201100 a House Rent G Remun 11830 as 150 as Utilities 150 a Total salary ee Gratuity 545 as P Fund 545 Microsoft Excel Adding umherijarnplesEx1 Eile Edit iew insert Format eels gate window Help Ade e F39DF yiewfermulee 1r 5 x DE Elt Ev f Ariel v1e g f g gamp f E a 1quot use v e s e e e E j 91 GROSS REMUNER iTIGN Rs as as Basic salary 5000 SCharges 1740 at House Rent 2700 G Remun 11830 as CA 150 as Utilities 150 a Total salary 9000 as Gratuity 545 as P Fund 545 me 139 32 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU AVERAGE Average Arithmetic Mean Sum N Sum Sum of all data values N number of data values EXAMPLE 1 Data 10 7 9 27 2 Sum 1079272 55 There are 5 data values Average 555 11 ADDING NUMBERS USING MICROSOFT EXCEL Add numbers in a cell Add all contiguous numbers in a row or column Add noncontiguous numbers Add numbers based on one condition Add numbers based on multiple conditions Add numbers based on criteria stored in a separate range Add numbers based on multiple conditions with the Conditional Sum Wizard NQWEWNT A I Add numbers in a cell Type 510 in a cell Result 15 See Example 2 Microsoft Excel Hool Elle Edit ew Insert Fgrmat Iools Qatar window elp Ader PDF 1 to m Eve 1 o SLIM r X oquot r 51E A a C n E F 3 1 2 NUM1NUM2 3 4 5 51I 2 Add all contiguous numbers in a row or column using AutoSum If data values are in contiguous cells of a column click a cell next to last data value in the same column If data values are in contiguous cells of a row then click a cell at right side of last data value Click AutoSum symbol 2 in tool bar 33 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Press ENTER This will add all the data values See Example sasisaiariquotssi Lhasa ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff quot Eile Edit ew insert Fgrmat Inuls Qata induw Help Adulge F39DF Iris Evil 3 Arial vIIII e as ENE 1r 5 A E I D E F 3 H r s ADDING AutSum 11 12 10 13 3915 14 E 3 Add noncontiguous numbers Use the SUM function See Example Micrnsnft Excel Baum Eile Edit ew insert Fgrmat Inuls ats induw Help Adobe PDF it PM Ev Elf SLIM v E J 5 Sumt iiz t123 A a c D E F G H s ADDING USING AutSum Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 4 Add numbers based on one condition Use the SUMIF function to create a total value for one range based on a value in another range Micruauft Excel Addini umher5ExamplesEx1 Elle Edit ew insert Fgrrnat Innls Eata indcw Help indulge F39DF E galI T E T I SLIM v x u a SUMIFia3FM2quotBuchananquot5334234I A E c n I 35 Salcancracn lnucicc 3 Buchanan 38 Buchanan 39 Suuama an Suuama 41 Buchanan 42 Dcdawcrth 22500 43 a quotBuchananquotB3 4 B42 Microst Excel iiiuninitialleheragExampllEsE11 E Ede Edit yaw naert F grmat incl esta Diet ndalgae REF an Ariai urn i a 1 a Halal 139 e Surn Elf inunices fur Buchanan 29mm I A Fnrmula Ear 35 SEIES E II SOM Invoice 3 Buchanan MENU 3 Buchanan 39 Suvama BUM a Suuama 20000 41 Buchanan lm 42 odlawcrth 43 I15 Sum aiinveicecicr Buchanan 39 quot 45 5 Add numbers based on multiple conditions Use the IF and SUM functions to do this task 6 Add numbers based on criteria stored in a separate range Copyright Virtual University of Pakistan VU 35 Business Mathematics amp Statistics MTH 302 VU Use the DSUM function to do this task DSUM Adds the numbers in a column of a list or database that match conditions you specify Syntax DSUMdatabasefieldcriteria Database is the range of cells that makes up the list or database Field indicates which column is used in the function Criteria is the range of cells that contains the conditions you specify DSUM EXAMPLE DSUMA4E10quotProfit A1F2 The total profit from apple trees with a height between 10 and 16 75 AVERAGE USING MICROSOFT EXCEL AVERAGE Returns the average arithmetic mean of the arguments Syntax AVERAGEnumber1number2 Number1 number2 are 1 to 30 numeric arguments for which you want the average Calculate the average of numbers in a contiguous row or column Elle Edit Elevll insert Fgrrnat IDEIIS Eata indnw elp FadInge F39DF DEEJE tag u Ev l 3 Arial 43 39E El 9amp1 T 1 A4 139 E A El 3 D 55 HER tGE EE 5 Data 53 1 ES F39III m 27 2 a 1 1 31 I I 5 TH Calculate the average of numbers not in contiguous row or column 36 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Hiicrnsnft Excel AddingNun1her5Exan1plesE151 Eile Edit ew Lnsert Fgrmat luuls Qata induw Help adage F39DF using autnsum cit E v I r E r 3 SLIM v H v a A ERAGE AFEz EEI BEj a a r U E F as VERAGE OF NUMBERS NOT CONTIGUOUS a Data a 3910 F9 El a 2 82 33 a AUERAGEM73MBUmBS BE Micrnsn Excel hddini umherijampleaE31 Eile Edit ew insert Fgrmat Iciclls gaita window Help Adnge F39DF using autDSUITI r Ev l 1fj iiirial 1DTBEEEEE quot El ti EI V r a E t D E F as AVERAGE OF NUMBERS NOT CONTIGUOUS a Data a 3910 Ei El a 2 82 EB an L5 EEI I HI i 37 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU WEIGHTED AVERAGE Weighted average is one type of arithmetic mean of a set of data in which some elements of the set carry more importance weight than others If X1 X2 X3 Xn is a set of n number of data and w1 wz W3 wn are corresponding weights of the data then Weighted average X1W1 X2W2 X3W3 XnXWn Be careful about one thing that the weights should be in fraction Grades are often computed using a weighted average Suppose the weightage of homework is 10 quizzes 20 and tests 70 Here weights of homework quizzes tests are already in fraction ie10 01 20 02 70 07 respectively If Ahmad has a homework grade of 92 a quiz grade of 68 and a test grade of 81then Ahmad39s overall grade 01092 02068 07081 795 Let us see another example Labor hours per Grade of Labor unit of labor Hourly wages Rs Skilled 6 300 Semiskilled 3 200 Unskilled l 100 Here weights Labor hours per unit of labor are not in fraction So rst we convert them to fraction Total labor hours 6 3 1 10 Grade of Labor hours per unit of labor labor in fraction Skilled 6 10 06 Semiskilled 3 10 03 Unskilled l 10 01 Weighted average 06300 03200 01100 38 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 250 Rs per hour LECTURE 5 Applications of Basic Mathematics Part 4 OBJECTIVES The objectives of the lecture are to learn about 0 Review of Lecture 4 0 Basic calculations of percentages salaries and investments using Microsoft Excel PERCENTAGE CHANGE Monday s Sales were Rs 1000 and grew to Rs 2500 the next day Find the percent change METHOD Change Final value initial value Percentage change Change initial value x 100 CALCULATION Initial value 1000 Final value 2500 Change 1500 Change 15001000 x 100 150 The calculations using Excel are given below First the entries of data were made as follows Cell C4 1000 Cell C5 2500 In cell C6 the formula for increase was C5 C4 The result was 1500 In cell C7 the formula for percentage change was C6C4100 39 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The result 150 is shown in the next slide Micruen Excel Perce ntageji hange Eile Edit Eiew insert Fgrmat Inels gate indew Help adage F39DF 3957quot Evif Furial 20E e as T lg ml L B 79 l 1 i163 39III39 E 139 39 A El 393 I D E F 393 2 PERCENTAGE CHANGE R5 3 4 Men ay 5 Next Day 5 Increase r Increase 150 a 1 5 j l 40 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EXAMPLE 1 How many Percent is Next Day s sale with reference to Monday s Sale Monday s sale 1000 Next day s sale 2500 Next day s sale as 25001000 x 100 250 Two and a half times Nicrnxnft Excel MTHEElec handnut E Eile Edit ew Lnserl Fgrmal Innls gate window elp El if x v r v E v 43 1 I SLIM v E J 5 D13fD121DD x a E E F e H many Percent is Next Day s sale m 1 ith refernee t Mndey s Sale 11 12 Mndey39e sale 13 Next day39s Sale 14 Next day39s sale as 15 f Mnday s sale D13jD12t1UU IE 41 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Hicrnsuft Excel tumIBElec handout Eile Edit ew insert Fgrmat Innls Qatar indnw elp nemaaaenses iiirial 3qu e Di 1 A E r E F 9 Hw many Percent is Next Day s sale mith refernce t Mnday s Sale 5 F E Inc El El 43 H mm M iii 11 12 Mnay s sale 13 Next ay39s sale H Next day39s sale as f Mnay39s sale 250 15 15 EXAMPLE 2 In the making of dried fruit 15kg of fresh fruit shrinks to 3 kg of dried fruit Find the percentage change Calculation Original fruit 15 kg Final fruit 3 kg Change 315 12 change 1215x 100 80 Size was reduced by 80 42 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Percentagejihange Eile Edit iew insert Fgrmat Innis Qatar indow Help Fido e F39DF El gt 21 up SLIP391 139 X all t D2HD19 UU A E C D E F G 19 riinal fruit 15 20 Final fruit 21 Chane in meiht 12 Chane DZ1jD191UU 23 21 Microsoft Excel Percentagejihange Eile Edit ew insert Fgrmat Iools Eata window Help Fider F DF D l i a r 3 3 E T 5 TIDTB at v r A E I13 1 lN WEIGHT IE 19 riinal fruit 15 2n Final fruit 21 Chane in xElht 12 thehange 22 23 a 25 Calculations in Excel were done as follows Data entry Cell D19 15 Cell D20 3 Formulas 43 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Formula for change in Cell D21 D20 D19 Formula for change in Cell D22 D21D19100 Resuhs Cell D21 12 kg Cell D22 80 EXAMPLE 3 After mixing with water the weight of cotton increased from 3 kg to 15 kg Find the percentage change CALCULATION Original weight 3 kg Final weight 15 kg Change 153 12 change 123 X 100 400 Weight increased by 400 Micrnenft Excel ercentageEhange E Eile Edit ew insert Fgrn39iat Innls Qatar indnw Help Filing3 F DF a asquot I iv 2 if Firial 10 w a a t in a JED 139 r A E C D I E F 23 a CHANGE IN WEIGHT 25 25 riginal weight f cttn kg 2 Final weight f cttn 15 kg 23 Change in weight 12 kg 29 all change 400 la y l 31 Calculations in Excel were done as follows Data entry Cell D26 3 Cell D27 15 Formulas Formula for change in Cell D28 D27 D26 Formula for change in Cell D29 D28D26100 Resuhs 44 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Cell D28 12 kg Cell D29 400 EXAMPLE 4 A union signed a three year collective agreement that provided for wage increases of 3 2 and l in successive years An employee is currently earning 5000 rupees per month What will be the salary per month at the end of the term of the contract Calculation 50001 31 21 1 5000x 103x 102x 101 5306 Rs Calculations using Excel are shown in the following slides Microsoft Excel Percentage hange Eile Edit ew insert Fgrmalz Iools gate window elp Fider F39DF e v 2 v 39L 39JJZ E 1 SLIM v 1 J r RDUNDEC351C3l3l lIIIIIIIIIjI a El e D E F 3392 39 33 SALARY IN YEAR 1 2 AND 31 35 Salary year 1 Rs 35 Increase year 1 3 In SaIary year 2 RUNDC351C36l ea 1 D 39 45 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Percentage l1ange Eile Edit ew insert Fgrmet leels gete indew Help Adelge F39DF El as 2 L e 7e TE it v M v x J r eeumeressrne nnnnmnj I e E F I A El 33 SALARY IN YEAR 1 2 AND 34 as Salary year 1 Re 35 Increase year 1 3 Salary year 2 5150 Re 33 Increase year 2 2 39 Salary year Re 4 Increase year 1 Salary end f year RUNDCSQ1C4UI 42 1 I RDUHDI numter numdigitsl I LI39I 1 43 Ufa micrusuft Excel ercentage hange Eile Edit Eiew insert Fgrmet eels gete indew elp Adege PDF ll E a El E 1quot f PurlEll 139 1D 1 B e 1 i L quotIquot 2 1quot 39 i hr Itquot 4 e D E F A El 33 SALARY IN YEAR 1 2 AND 34 as Salary year 1 35 Increase year 1 3 Salary year 2 33 Increase year 2 39 Salary year 4 Increase year 1 41 Salary end 1quot year Re E I I 43913 46 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Calculations in Excel were done as follows Data entry Cell C35 5000 Cell C36 3 Cell C38 2 Cell C40 1 Formulas Formula for salary in year 2 in Cell C37 ROUNDC351C361000 Formula for salary year 3 in Cell C39 ROUND C371C381000 Formula for salary at the end of year 3 in Cell C41 ROUNDC391C391000 Resu s Cell C37 5150 Rs Cell C39 5253 Rs Cell C41 5306 Rs EXAMPLE 5 An investment has been made for a period of 4 years Rates of return for each year are 4 8 10 and 9 respectively If you invested Rs 100000 at the beginning of the term how much will you have at the end of the last year Micrnsuft Excel ercentage l1ange Eile Edit Eiew insert Fgrmat Innls Qatar indnw elp Fadage F39DF E E v E 3 1 3939 sum v x J a RDUNDECdE1C4F I1IIIIIIJHIIJ A El 3 n E F 45 INVESTMENT HT THE END IF 4 YEHRS 15 Investment year 1 100000 Rs 4 Increase year 1 4 3911 VaIue in year 2 RUNIZIDCLE IEii11 Cir171i 49 Increase year 2 1000i a Value in year 112320 Rs 51 Increase year 10 In 52 Value in year 101088 Rs 53 Increase year 4 36 54 Ualyeendyear 110130 Rs 47 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel percentage hange E Eile Edit View insert Fgrmat Incls Qatar indcw Help Rulings P39DF D g v v Evf Arial 20 s a is 1L 391 a 54 v s RDUNDEC521C53I1IIIIIIJJII a E c D E F 45 INVESTMENT HT THE END NF 4 YEHRS 45 Investment year 1 100000 Rs 4 Increase year 1 4 fe a Value in year 2 104000 Rs 49 Increase year 2 Va 5 Value in year 112320 Rs 51 Increase year 3910 Va 52 Value in year 101088 Rs 53 Increase year 4 In 54lVaIue end year 4 110185le Calculations in Excel were done as follows Data entry Cell C46 100000 Cell C47 4 Cell C49 8 Cell C51 10 Cell C53 9 Formulas Formula for value in year 2 in Cell C48 ROUNDC461C471000 Formula for value in year 3 in Cell C50 ROUNDC481C491000 Formula for value in year 4 in Cell C52 ROUNDC501C511000 Formula for salary end of year 4 in Cell 054 ROUNDC521CS31000 Resu s Cell C48 104000 Rs Cell C50 112320 Rs Cell C52 101088 Rs Cell C54 110186Rs 48 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 6 Applications of Basic Mathematics Part 5 OBJECTIVES The objectives of the lecture are to learn about o Review Lecture 5 0 Discount 0 Simple and compound interest o Average due date interest on drawings and calendar REVISION LECTURE 5 A chartered bank is lowering the interest rate on its loans m 9 to 7 What will be the percent decrease in the interest rate on a given balance A chartered bank is increasing the interest rate on its loans m 7 to 9 What will be the percent increase in the interest rate on a given balance As we learnt in lecture 5 the calculation will be as follows Decrease in interest rate 79 2 decrease 29 x 100 222 Increase in interest rate 97 2 decrease 27 x 100 286 The calculations in Excel are shown in the following slides DECREASE IN RATE Data entry Cell F4 9 Cell F5 7 Formulas Formula for decrease in Cell F6 F5F4 Formula for decrease in Cell F7 F6F4100 Resu s Cell F6 2 Cell F7 222 INCREASE IN RATE Data entry Cell F14 7 Cell F15 9 Formulas Formula for increase in Cell F16 F15F14 Formula for increase in Cell F17 F16F14100 Resu s Cell F16 2 Cell F17 286 49 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU IIIIicrsft Excel IHITHIIELecE handut Eile Edit Eiew Insert Fgrmat Innls gate indnw Help nenseees rarer IEreiI as vwv aueeeaeessaees Ii Big 2 D E F G H 2 DEC IN INTEEST 4 rim inal Interest ate 5 rise Interest te a Decrease D wa s E39E Atii r nrial 10 BI E eg i iii i i BEE quotF F3 a E r n E F e H 12 IN INTEEST 13 14 rizj inal Interest Rate 15 Revise Interest Rate re Increase 1 Increase 235 re 50 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The Definition of a Stock Plain and simple stock is a share in the ownership of a company Stock represents a claim on the company39s assets and earnings As you acquire more stock your ownership stake in the company becomes greater Whether you say shares equity or stock it all means the same thing Stock yield With stocks yield can refer to the rate of income generated from a stock in the form of regular dividends This is often represented in percentage form calculated as the annual dividend payments divided by the stock39s current share price Earnings per share EPS The EPS is the total profits of a company divided by the number of shares A company with 1 billion in earnings and 200 million shares would have earnings of 5 per share Priceearnings ratio A valuation ratio of a company39s current share price compared to its pershare earnings Calculated as Market Value per Share Earnings per Share EPS For example if a company is currently trading at 43 a share and earnings over the last 12 months were 195 per share the PE ratio for the stock would be 2205 43195 Outstanding shares Stock currently held by investors including restricted shares owned by the company39s officers and insiders as well as those held by the public Shares that have been repurchased by the company are not considered outstanding stock Net current asset value per shareNCAVPS NCAVPS is calculated by taking a company39s current assets and subtracting the total liabilities and then dividing the result by the total number of shares outstanding Currentquot laaeta Tatal Lialiilitiea H iul39PE a t Eltarea uta tan ling Current Assets The value of all assets that are reasonably expected to be converted into cash within one year in the normal course of business Current assets include cash accounts receivable inventory marketable securities prepaid expenses and other liquid assets that can be readily converted to cash 51 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Liabilities A company39s legal debts or obligations that arise during the course of business operations Market value The price at which investors buy or sell a share of stock at a given time Face value Original cost of a share of stock which is shown on the certificate Also referred to as quotpar valuequot Face value is usually a very small amount that bears no relationship to its market price Dividend Usually a company distributes a part of the profit it earns as dividend For example A company may have earned a profit of Rs 1 crore in 200304 It keeps half that amount within the company This will be utilised on buying new machinery or more raw materials or even to reduce its borrowing from the bank It distributes the other half as dividend Assume that the capital of this company is divided into 10000 shares That would mean half the profit ie Rs 50 lakh Rs 5 million would be divided by 10000 shares each share would earn Rs 500 The dividend would then be Rs 500 per share If you own 100 shares of the company you will get a cheque of Rs 50000 100 shares x Rs 500 from the company Sometimes the dividend is given as a percentage i e the company says it has declared a dividend of 50 percent It39s important to remember that this dividend is a percentage of the share39s face value This means if the face value of your share is Rs 10 a 50 percent dividend will mean a dividend of Rs 5 per share BUYING SHARES If you buy 100 shares at Rs 6250 per share with a 2 commission calculate your total cost m 100 Rs 6250 Rs 6250 002 Rs 6250 125 Total Rs 6375 RETURN ON INVESTMENT Suppose you bought 100 shares at Rs 5225 and sold them after 1 year at Rs 68 With a 1 commission rate of buying selling the stock and 10 dividend per share is due on these shares Face value of each share is 10Rs What is your return on investment Bought 52 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 100 shares at Rs 5225 522500 Commission at 1 5225 Total Costs 5225 5225 527725 Sold 100 shares at Rs 68 680000 Commission at 1 6800 Total Sale 6800 68 673200 Gain Net receipts 673200 Total cost 527725 Net Gain 6732 527725 1 45475 Dividends 1001010 10000 Total Gain 145475 100 155475 Return on investment 155475527725100 2946 The calculations using Excel were made as follows BOUGHT Data entry Cell B21 100 Cell B22 5225 Formulas Formula for Cost of 100 shares at Rs 5225 in Cell B23 B21B22 Formula for Commission at 1 in Cell B24 B23001 Formula for Total Costs in Cell B25 B23B24 Resu s Cell B23 5225 Cell B24 5225 Cell B25 527725 53 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel lullllH ELec handout Elle Edit Eiew insert Format Innis gate window elp neusssos s E E39ilil al i39ic i nrial Tan T g EH 1 r EEEIEEE1DD A E i an B uiht 21 Shares 10 22 Rate 5225 23 Cost of1 shares at Rs 5225 5225 24 Commission 5225 25 Total Costs 527725 25 s a 3393 so m Data entry Cell B28 68 Formulas Formula for sale of 100 shares at Rs 68 in Cell B29 B21B28 Formula for Commission at 1 in Cell B30 B29001 Formula for Total Sale in Cell 831 829BBO Resu s Cell B29 6800 Cell B30 68 Cell B31 6732 lulicrnsnft Excel iiilTl39lE ELecll handout Eile Edit glow insert Format Innis Qatar indnw elp D o El Hist E if it is Arial 39239 ratr E E as v or Bssrsssnnn A E C quot quot quot ss so 2 ss t 2v 23 29 100 shares atRs68 6800 an Commission at1i 68 31 Total Sale 54 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU GAIN Formulas Formulator Net receipts in Cell B34 B31 Formula for Total cost in Cell B35 B25 Formula for Net Gain in Cell 836 BB1825 Formula for Gain in Cell BB7 B36B35100 Resu s Cell B34 6732 Cell B35 527725 Cell B36 145475 Cell B37 2757 DISCOUNT Discount is Rebate or reduction in price Discount is expressed as of list price Example List price 2200 Discount Rate 15 Discount 2200 x 0 15 330 Calculation using Excel along with formula is given in the following slide Microsoft Excel lil39l39IIEDELec handout Eile Edit iew insert Fgrmal Innis gate window elp iVquotVii W Dena am it w azvmiii newera v Arial m HIE EEE EEEE E v v n E51 139 i3 A El 3 D E I F G 33 QJSCUNT 39 List price 4 Discunt Rate 15 a 11 Discunt Frmula BBtB lUi1UU 42 43 NET COST PRICE Net Cost Price List price Discount Example List price 4500 Rs Discount 20 Net cost price 55 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Net cost price 4500 20 of 4500 4500 02 x4500 4500 900 3600 Rs Calculation using Excel along with formula is given in the following slide Microsoft Excel MTllii ELoo handout Eile Edit Eiew insert Formal Tools gate window elp gene eee it e m eaveee new iirial TIDTBIHEEE E133 3 39 D54 1 ii A E C D E F Iiii 45 NET CST PRICE 45 d 4 List price Rs 43 in Diseunt est 1 3934 19 Net Get Price Re El 51 Fr mula Cell 19 Bd IB4Bi1 52 SIMPLE INTEREST P Principal R Rate of interest pereent per annum T Time in years I Simple interest then I P R T 100 Thus total amount A to be paid at the end of T years P 1 Example P Rs 500 T 4 years R 11 Find simple interest I P x Tx R 100 500x4x 11100 RS 220 Calculation using Excel along with formula is given in the following slide 56 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU lulicruenft Excel MTHE ELec handuut Eile Edit ew insert Fgrmal IIZIIZIIS Eata induw Help D tttevr v Elii Furial vl vBI 39 gf g g FEE T A E I D E 55 SIMPLE INTEREST 5 5E Principal P Rs 59 Time Jeritl T 1 Year El R 1 1 51 Interest Rs 52 53 Fr mula in Cell 61 53i39559i39i t1 Ed COMPOUND INTEREST Compound Interest also attracts interest Example P 800 Interest year 1 0 1 x 800 80 New P 800 80 880 Interest on 880 0 1 X 880 88 New P 880 88 968 Calculation using Excel along with formula is given in the following slide 57 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel MTHIi ELec handuut Eile Edit Eiew insert Fgrmat eels Qatar window Help D r 5E1 iiirial 1 1 r E I H HFE 1quot 33 A 3 EE CMPUND INTEREST Ei39 res Principal P Rs EB Interest 10 quotA re Interest year1 80 Rs Frmule BEB tBBB100 r1 New P 880 Rs Frmule BEBBTU 2 Interest n 880 88 Rs Frmula B1BEBI100 3 New P 368 Frmule B1B2 r4 Co Compound Interest Formula S Money accrued after n years also called compound amount P Principal r Rate of interest n Number of periods S P1 r100 n Compound interest S P Example Calculate compound interest earned on Rs 750 invested at 12 per annum for 8 years S P1r1008 750112100A8 185 7 Rs Compound interest 1857 750 1107 Rs esvesttl memen v ee 3 use ee 39i v 3 D E F 3 H 58 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Calculation using Excel along with formula is given in the following lide lulicrueuft Excel MWB ZLec handuut Eile Edit Eiew insert Fgrmat eels gate window elp D 5ev E El r mne E 3395 BEES iii atrial v 1 1 B I H em 1 r e e D E F rr CMPUND INTEREST USING FE re Princiel P 5 Re El Interest 12 e1 Perio 8 Years 32 Meney accrue 1857 Rs 33 34 Formula RUNDB7Q1BBUI1UU BB1U 35 III 59 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 7 Applications of Basic Mathematics OBJECTIVES The objectives of the lecture are to learn about 0 Scope of Module 2 Review of lecture 6 Annuity Accumulated value Accumulation Factor Discount Factor Discounted value Algebraic operations Exponents Solving Linear equations Module 2 Module 2 covers the following lectures Linear Equations Lectures 7 Investments Lectures 8 Matrices Lecture 9 Ratios amp Proportions and Index Numbers Lecture 10 Annuity It some point in your life you may have had to make a series of fixed payments over a period of time such as rent or car payments or have received a series of payments over a period of time such as bond coupons These are called annuities Annuities are essentially series of fixed payments required from you or paid to you at a specified frequency over the course of a fixed period of time An annuity is a type of investment that can provide a steady stream of income over a long period of time For this reason annuities are typically used to build retirement income although they can also be a tool to save for a child s education create a trust fund or provide for a surviving spouse or heirs The most common payment frequencies are yearly once a year semiannually twice a year quarterly four times a year and monthly once a month Qlculatinq the Future value or accumulated value of an Annuitv If you know how much you can invest per period for a certain time period the future value of an ordinary annuity formula is useful for finding out how much you would have in the future by investing at your given interest rate If you are making payments on a loan the future value is useful for determining the total cost of the loan Let39s now run through Example 1 Consider the following annuity cash flow schedule 60 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU End f each period ifquot i n 1 1 3 4 5 l l l l l ll l l H H H ll 5mm 51 EDD 51am 51 EDD 51mm in J Payment paid er minivan at end of each patina In order to calculate the future value of the annuity we have to calculate the future value of each cash flow Let39s assume that you are receiving 1000 every year for the next five years and you invested each payment at 5 The following diagram shows how much you would have at the end of the five yearpenod El 1 E 3 4 5 ll lll lll lll lll lll II III III III III III 51993 Ei E l Ei E l H Si litEJE it e sin oilmai39 a Magnet 3 5t tl t 5l stinger 1 5i iliioeilil warm Ei it f gma i Future allu nf an ndinary Ann witty S a d Since we have to add the future value of each payment you may have noticed that if you have an annuity with many cash flows it would take a long time to calculate all the future values and then add them together Fortunately mathematics provides a formula that serves as a short cut for finding the accumulated value of all cash flows received from an annuity anquot 1 Iquotquot39IICII39Iinalr39I IEIII39Inu39rlvI I Hi CPayment per period or amount of annuity i interest rate n number of payments 1 in 1 i is called accumulation factor for n periods Accumulated value of n period payment per period x accumulation factor for n periods If we were to use the above formula for Example 1 above this is the result 1 005315 1 Fri 1000 liltlruzllnar39I F39II39II39I39J39t39r39 El 05 1ooo553 552563 61 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Note that the 001 difference between 552564 and 552563 is due to a rounding error in the first calculation Each of the values of the first calculation must be rounded to the nearest penny the more you have to round numbers in a calculation the more likely rounding errors will occur So the above formula not only provides a shortcut to finding FV of an ordinary annuity but also gives a more accurate result Calculating the Present Value or discounted value of an Annuity If you would like to determine today39s value of a series of future payments you need to use the formula that calculates the present value of an ordinary annuity For Example 2 we39ll use the same annuity cash flow schedule as we did in Example 1 To obtain the total discounted value we need to take the present value of each future payment and as we did in Example 1 add the cash flows together u 1 1 s 4 s lll m lll m lll lll ll ll ll ll ll Ill smuu smuu smuu smuu smuu iltlili z sususs nus sumu Lusii smuu E assumius nusi i39h iii sass t 3 us m mums trust Allis Prusun ltiiutallmu uf an rdiuuw nmuity 54319913 Again calculating and adding all these values will take a considerable amount of time especially if we expect many future payments As such there is a mathematical shortcut we can use for PV of ordinary annuity l 1i39quot F39I39i39lllCtlrzlinstquotI Finnuit39I I I C Cash flow per period i interest rate n number of payments 1 1 in i is called discount factor for n periods Thus Discounted value of n period payment per period x discount factor for n period The formula provides us with the PV in a few easy steps Here is the calculation of the annuity represented in the diagram for Example 2 Fquot39quot 1000 Hi ICirzllnsirlI anurtlI D IDS 1ooo433 432948 1 1 un t 5 62 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NOTATIONS The following notations are used in calculations of Annuity R Amount of annuity N Number of payments Interest rater per conversion period S Accumulated value A Discounted or present worth of an annuity ACCUMULATED VALUE The accumulated value S of an annuity is the total payments made including the interest The formula for Accumulated Value S is as follows S r 1iquotn 1i Accumulation factor for n payments 1 in 1 i It may be seen that Accumulated value Payment per period x Accumulation factor for n payments The discounted or present worth of an annuity is the value in today s rupee value As an example if we deposit 100 rupees and get 110 rupees ie 10 interest on 100 Rs which is 10010100 10 Rs so total amount is 10010 110 or simply 100 110100 100 1 01 10011 110 after one year the Present Worth or of 110 rupees will be 100 Here 110 will be future value of 100 at the end of year 1 The amount 110 if invested again can be Rs 121 after year 2 Le 10 of110 is 11010100 11 so total amount is 11011 121 The present value of Rs 121 at the end of year 2 will also be 100 DISCOUNT FACTOR ANQISCOUNTQ VA When future value is converted into present worth the rate at which the calculations are made is called Discount factor rate In the previous example 10 was used to make the calculations This rate is called Discount Rate The present worth of future payments is called Discounted Value EXAMPLE 1 ACCUMULATION FACTOR AF FOR n PAYMENTS Calculate Accumulation Factor and Accumulated value when rate of interest i 425 Number of periods n 18 Amount of Annuity R 10000 Rs Accumulation Factor AF 1 00425quot18100425 2624 Accumulated Value S 10000x 2624 260240 Rs EXAMPLE 2 DISCOUNT VALUE DV In the above example calculate the value of all payments at the beginning of term of annuity ie present value or discounted value Discount rate 425 Number of periods 18 63 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Amount of annuity 10000 Rs Value of all payments at the beginning of term of Annuity or discounted value Payment per period x Discount Factor DF Formula for Discount Factor 111iquotni 11100425quot1800425 124059 Discounted value 10000 X 124059 124059 Rs EXAMPLE 3 DISCOUNTE VALUE DV How much money deposited now will provide payments of Rs 2000 at the end of each half year for 10 years if interest is 11 compounded sixmonthly Amount of annuity 2000Rs Rate of interest i 11 2 0055 Number of periods n 10 X 2 20 DISCOUNTED VALUE 2000 x 11 10055A20 0055 2000 X1195 2390077 ALGEBRAIC OPERATIONS Algebraic Expression indicates the mathematical operations to be carried out on a combination of NUMBERS and VARIABLES The components of an algebraic expression are separated by Addition and Subtraction In the expression 2x2 3x 1 the components 2x2 3x and 1 are separated by minus sngn Algebraic 3 I H H E aperations J a I H 54 4IIquotquot39T39E quotTI 39 quot quotquot1 Mammal Binnmial TrinnmialiPa numial HUT F iEIHg I I I y y 55 mailE than H f 1Tern1 41 rizl J 39 xi 33 3 xy4 f In algebraic expressions there are four types of terms 0 Monomial ie 1 term Example 3x2 0 Binomial ie 2 terms Example 3x2xy o Trinomial ie 3 terms Example 3x2xy6y2 o Polynomial ie more than 1 term Binomial and trinomial examples are also polynomial Algebraic operations in an expression consist of one or more FACTORs separated by MULTIPLICATION or DIVISION sign Multiplication is assumed when two factors are written beside each other Example xy xy Division is assumed when one factor is written under an other Example 36x2y 50xy2 64 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Algebraic 51233 31 I peratinns39 Hiquot In 51 1 I i 39 i 535 llitlr Term each BE in an Ergmssmr f le 139 n 7 II I I7 AETEEE sqmmte hjr r mural two when on fan hrs arew thn factor I39E 39 i39h39l39l 39I 39I 39I I I ZITE eat1 ENEEIquot u mfe 39 an I rFE rr39i Factors can be further subdivided into NUMERICAL and LITERAL coefficients r Algebraic wilt1 a H I H H AIDE Illii a I 1 lgebraic EHpI EEEi I I n i iiiT in 397 39i 1 TEI ITIEI n 39i uiilliil IHLi a i w nmial Binmi l Trinumial Pa mumial IiEHigur IL 41 fitquot rt FEET HES L 393 39 Numerical Literal Eingf itm E39mgf c m39 There are two steps for Division by a monomial 1 Identify factors in the numerator and denominator 2 Cancel factors in the numerator and denominator Example 36x y 60xy2 36 can be factored as 3 x 12 60 can be factored as 5 x 12 x2y can be factored as xxy xy2 can be factored as xyy Thus the expression is converted to 3 x 12xxy 5 x 12xyy 65 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 12xxy in both numerator and denominator cancel each other The result is 3x5y a 7 E59111 3395 1 39 5E Rim it Fam FACTUES H1 m u FulfilEf i f 11 if 1 39 39 139439539 i39EI39J NEEHIE n4 swimmer r nittracer Factors f in u FEEJFFEEi39tli f 11 I 39 Ill1 i E L3939 iquotiquotE 2 F212 n4 quot Another example of division by a monomial is 48a2 32ab8a Here the steps are 1 Divide each term in the numerator by the denominator 2 Cancel factors in the numerator and denominator 48a2 8a 8x6aa 8a 6a 32ab 8a 4x8ab 8a 4b The answer is 6a 4b War he Ex wh 777i7rir 7 I 7 7 39I39quot39quot39 t J H HitI 39I I II J39 I139I IiiE1 I LI J J I I I 39I 39l m 39E rzzirrze1mtl1lrr a Jr E 39I 39 NEE Ill39E39 i ELT39i i Hi Eili i39 II39 quot mud191 Factors 7 l El 1 mill FEEJf39f39iEh li i and i 13942quot i E I393939 i i E 2 Hill 13quot How to multiply polynomials Look at the example x2x2 3x 1 Here each term in the trinomial22x2 3x 1 is multiplied by x X2X X3X X1 2x3 3x x 66 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Please note that product of two negatives is positive Mll pljting Putin mi elf Emmett5 KE3I it 1139 3116tfxIExltliffllli39u Thepm citiet efttee negeiitie quantities is peeiiiiie 3X6y3 X2232 Exponent of a term means calculating some power of that term In the example we are required to work out exponent of 3x6y3 x223 to the power of 2 The steps in this calculation are 1 Simplify inside the brackets first 2 Square each factor 3 Simplify In the first step the expression 3x6y3 x223 is first simplified to 3x4y323 In the next step we take squares The resulting expression is 32X42y32z32 9X8 y6 z6 67 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Step 1 Step 2 Step 3 mp image Sg39mzre ench z ctnr mp the brackets mt LINEAR EQUATION If there is an expression A 9 137 how do we calculate the value of A A 137 9 128 As you see the term 9 was shifted to the right of the equality To solve linear equations 1 Collect like terms 2 Divide both sides by numerical coefficient Step 1 X 34125 0025X X 0025X 34125 X10025 34125 0975X 34125 Step 2 X 341 250975 350 68 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU l 25 Divicfe bath itquot SEE 95 by H975 69 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 8 Compound Interest Calculate returns from investments Annuities ExcelFunc ons OBJECTIVES The objectives of the lecture are to learn about 0 Review of lecture 7 0 Compound Interest 0 Calculate returns from investments 0 Annuities 0 Excel Functions m Returns the cumulative interest paid on a loan between startperiod and endperiod If this function is not available and returns the NAME error install and load the Analysis ToolPak addin The syntax is as follows CUMIPMTratenperpvstartperiodendperiodtype Rate interest rate Nper total number of payment periods Pv present value Startperiod first period in the calculation Endperiod last period in the calculation Type timing of the payment Type Timing 0 zero Payment at the end of the period 1 Payment at the beginning of the period CUMIPMTEXAMPLE Following is an example of CUMIPMT function In this example in the first case the objective is to find total interest paid in the second year of payments for periods 13 to 24 Please note there are 12 periods per year The second case is for the first payment penod In the first formula the Annual interest rate 9 is cell A2 not shown here The Years of the loan are given in cell A3 The Present value is in cell A4 For the Start period the value 13 was entered For the End period the value 24 has been specified The value of Type is 0 which means that the payment will be at the end of the period Please note that the annual interest is first divided by 12 to arrive at monthly interest Then the Years of the loan are multiplied by 12 to get total number of months in the Term of the loan The answer is 1113523 In the second formula which gives Interest paid in a single payment in the first month 1 was specified as the Start period For the End period also the value 1 was enteredThis is because only 1 period is under study All other inputs were the same The answer is 93750 70 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Data Description 9 Annual interest rate 30 Years of the loan 125000 Present value CUMIPMTA212A312A413240Total interest paid in the second year of payments periods 13 through 24 1113523 CUMIPMT A212A312A4110lnterest paid in a single payment in the first month 93750 CUMPRINC The CUMPRINC function returns the cumulative principal paid on a loan between two penods The syntax is as under CUMPRINCratenperpvstartperiodendperiodtype Rate interest rate Nper total number of payment periods Pv present value Startperiod period in the calculation Payment Endperiod last period in the calculation Type timing of the payment 0 or 1 as above CUMPRINC EXAMPE Following is an example of CUMPRINC function In this example in the first case the objective is to find the total principal paid in the second year of payments periods 13 through 24 Please note there are 12 periods per year The second case is for the principal paid in a single payment in the first month In the first formula the Interest rate per annum 9 is in cell A2 not shown here The Term in years 30 is given in cell A3 The Present value is in cell A4 For the Start period the value 13 was entered For the End period the value 24 has been specified The value of Type is 0 which means that the payment will be at the end of the period Please note that the interest is first divided by 12 to arrive at monthly interest Then the years of loan are multiplied by 12 to get total number of months in the term of the loan The answer is 9341071 In the second formula which gives the principal paid in a single payment in the first month 1 was specified as the start period For the end period also the value 1 was enteredThis is because only 1 period is under study All other inputs were the same The answer is 6827827 EXAMPLE Data Description 900 Interest rate per annum 30 Term in years 125000 Present value CUMPRNCA212A312A413240The total principal paid in the second year of payments periods 13 through 24 9341071 CUMPRNCA212A312A4110The principal paid in a single payment in the first month 6827827 EFFECT Returns the effective annual interest rate As you see there are only two inputs namely the nominal interest Nominalrate and the number of compounding periods per year Npery 71 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EFFECTnominalratenpery Nominalrate nominal interest rate Npery number of compounding periods per year EFFECTEXAMPLE Here Nominalrate 525 in cell A2 Npery 4 in cell A3 The answer is 0053543 or 53543 You should round off the value to 2 decimals 535 525 Nominal interest rate 4 Number of compounding periods per year EFFECTA2A3 Effective interest rate with the terms above 0053543 or 53543 percent FV Returns the future value of an investment There are 5 inputs namely Rate the interest rate Nper number of periods Pmt payment per period Pv present value and Type FVratenperpmtpvtype Rate interest rate per period Nper total number of payment periods Pmt payment made each period Pv present value orthe lumpsum amount Type number 0 or 1 due FVEXAMPLE 1 In the formula there are 5 inputs namely Rate 6 in cell A2 as the interest rate 10 as Nper number of periods in cell A3 200 notice the minus sign as Pmt payment per period in cell A4 500 notice the minus sign as Pv present value in cell A4 and 1 as Type in cell A6 The answer is 258140 mm Dweri im EMWJJLE l it Aliuml intimatequot rate 391 Iii H un39iJH ef Ili i39l39l i5 Jill An39mnt ef the peymmt 45 P ri ent H39 lll 1 P Hyman ie tlue atthe beguiling if the uselied W quot l A3 A4 A5 e6 Future value ef an inveetm entwi 39i the ebeue term 5 2531 4131 Fiuinretenperpmtputype FVEXAMPLE 2 72 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 In the formula there are 3 inputs namely Rate 12 in cell A2 as the interest rate 12 as Nper number of periods in cell A3 1000 notice the minus sign as Pmt payment per period in cell A4 Pv present value and Type are not specified Both are not required as we are calculating the Future value of the investment The answer is 1268250 EXAMPLE 12 nnuel intereet rate 12 Number ef paymente 1 Ameunt ef the neyment FW lEl1E Ad Future HEIUE ef en ineeetrnent With the eleeve terrne thslrete per enttpmtype FVEXAMPLE 3 In the formula there are 4 inputs namely Rate 11 in cell A2 as the interest rate 35 as Nper number of periods in cell A3 2000 notice the minus sign as Pmt payment per period in cell A4 1as Type in cell A5 The value of Pv was omitted by entering a blank for the value note the double commas The answer is 8284625 EXAM PL E 11 Annual intereet r e 35 Number ef pnymente eeee meunt ef the nnylnent 1 Payment ie tlue at the heginning ef the perietl FWHEP12 A3 A4 A5 Future unlue ef an investment with the nheve terlne 132 Erie 25 FWrntem1erilntiuteen VU 73 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU FV SCHEDULE Returns the future value of an initial principal after applying a series of compound interest rates FVSCHEDULEprincipal schedule Principal present value Schedule an array of interest rates to apply FV SCHEDULEEXAMPLE In this example the Principal is 1 The compound rates 009 01101 are given within curly brackets The answer is 133089 FVSCHEDULEprincipalschedule FVSCHEDULE100901101 Future value of 1 with compound interest rates of 00901 101 133089 IPMT Returns the interest payment for an investment for a given period lPMTratepernperpvfvtype Rate interest rate per period Per period to find the interest Nper total number of payment periods Pv present value or the lumpsum amount Fv future value or a cash balance Type number 0 or 1 ISPMT Calculates the interest paid during a specific period of an investment lSPMTratepernperpv Rate interest rate Pen penod Nper total number of payment periods Pv present value For a loan pv is the loan amount NOMINAL Returns the annual nominal interest rate NOMINALeffectratenpery Effectrate effective interest rate Npery number of compounding periods per year NPER Returns the number of periods for an investment NPERrate pmt pv fv type Rate the interest rate per period Pmt payment made each period Pv present value or the lumpsum amount Fv future value or a cash balance Type number 0 or 1 due NPV Returns the net present value of an investment based on a series of periodic cash flows and a discount rate lts syntax is NPVratevalue1value2 Rate is the rate of discount over the length of one period Valu1value2 are 1 to 29 arguments representing the payments and income 74 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU M Returns the periodic payment for an annuity PMTratenperpvfvtype Rate interest rate Nper total number of payments Pv present value Fv future value Type number 0 zero or 1 PPMT Returns the payment on the principal for an investment for a given period PPMTratepernperpvfvtype Rate interest rate per period Per period and must be in the range 1 to nper Nper total number of payment periods Pv the present value Fv future value 0 Type the number 0 or 1 due PV Returns the present value of an investment PVratenperpmtfvtype Rate interest rate per period Nper total number of payment periods in an annuity Pmt payment made each period and cannot change over the life of the annuity Fv future value or a cash balance Type number 0 or 1 and indicates when payments are due RATE Returns the interest rate per period of an annuity RATEnperpmtpvfvtypeguess Nper total number of payment periods Pmt payment made each period Pv present value Fv future value or a cash balance 0 Type number 0 or 1 due Guess 10 RATEEXAMPLE Three inputs are specified 4 as years of loan in cell A5 200 as monthly payment in cell A6 and 8000 as amount of loan in cell A7 The answer is 00924176 or 924 75 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E icrnsuft Excel Lecture jiscuuntlnterest Eile Edit Eiew insert Fgrmat eels gate indnw elp Adn e F39DF FinanEi lfuntti ns 539 atav m Eva eve 39JL 391 a SLIM r X J 5 FEATEIIAE12 EMJII 4 E C D E F 1 2 RATE RHTEmperpmtpvfutypeguess 3 4 Data Descriptin 5 at Year39s f the Ian 5 200 Mnthly payment r BUUUAmunt f the Ian 3 ReTE 512A6A7 1 g 0092 Pmnual rate f the Ian 00924175 r 924 1 76 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 9 Matrix and its dimension Types of matrix OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 8 o Matrices QUESTIONS Every student wonders why he or she should study matrices There are manty important ques ons Where can we use Matrices Typical applications What is a Matrix What are Matrix operations Excel Matrix Functions There are many applications of matrices in business and industry especially where large amounts of data are processed daily TYPICAL APPLICATIONS Practical questions in modern business and economic management can be answered with the help of matrix representation in Econometrics Network Analysis Decision Networks Optimization Linear Programming Analysis of data Computer graphics WHAT IS A MATRIX A Matrix is a rectangular array of numbers The plural of matrix is matrices Matrices are usually represented with capital letters such as Matrix A B C For example 1 1 El 1 52 FE SE A El II III 3 El 52 33 EB 15 5 2 The numbers in a matrix are often arranged in a meaningful way For example the order for school clothing in September is illustrated in the table as well as in the corresponding matrix 4 3 Size Youth S M L XL Sweat Pants 0 10 34 40 12 Sweat Shirts 18 25 29 21 7 Shorts 19 13 48 36 9 Tshirts 27 7 10 24 14 The data in the above table can be entered in the shape of a matrix as follows 77 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU III ll 34 ill 12 13 25 23 21 F 13 13 EH3 3B 3 2 F ll 24 H DIMENSION Dimension or Order of a Matrix Number of Rows x Number of Columns Example Matrix T has dimensions of 2x3 or the order of matrix T is 2x3 X is just the notation it do not mean to multiply both of them T E 2 l eruwl all III F frmWE 3 l 393 call EDIE cn3 ROW COLUMN AND SQUARE MATRIX Suppose n 1234 A matrix with dimensions 1xn is referred to as a row matrix For example matrix A is a 1x4 row matrix A matrix with dimensions nx1 is referred to as a column matrix For example matrix B in the middle is a 2x1 column matrix A matrix with dimensions nxn is referred to as a square matrix For example matrix C is a 3x3 square matrix 2 2 3 A12 i ll 9 El E a i 5 u IDENTITY MATRIX An identity matrix is a square matrix with 139s on the main diagonal from the upper left to the lower right and 039s off the main diagonal An identity matrix is denoted as Some examples of identity matrices are shown below The subscript indicates the size of the identity matrix For example I represents an identity matrix with dimensions ni i n Di f Di iii iii CIDDH EDI II l fl fgziill Equotiii i101 MULTIPLICATIVE IDENTITY With real numbers the number 1 is referred to as a multiplicative identity because it has the unique property that the product a real number and 1 is that real number In other words 1 is called a multiplicative identity because for any real number n 1 n n and n 1n With matrices the identity matrix shares the same 78 Copyright Virtual University of Pakistan Business Mathematics amp unique property as the number 1 In other words a 2iquot 2 identity matrix is a multiplicative Statistics MTH 302 inverse because for any 2K 2 matrix A I 39 A A and A39 I A Examgle Given the 23quot 2 matrix A 10 2 1 32 11 3 4 2 11 11 Nib 3 4 01 Work r1c1 12 03 2 r2c1 02 13 3 r1c1 21 10 2 r2c1 31 40 3 r1c2 11 04 1 r2c2 01 14 4 r1c2 20 11 1 r2c2 30 41 4 392 1 3 4 ii 15 ii iii where r is for row and c is for column Copyright Virtual University of Pakistan VU 79 Business Mathematics amp Statistics MTH 302 VU LECTURE 10 MATRICES OBJECTIVES The objectives of the lecture are to learn about o Review Lecture 9 o Matrices EXAMPLE 1 An athletic clothing company manufactures Tshirts and sweat shirts in four differents sizes small medium large and xlarge The company supplies two major universities the U of R and the U of S The tables below show September39s clothing order for each university University of S39s September Clothing Order II S l M l L l XL Tshirts 1oo 3oo 5oo 3oo sweat shirts quot150 quot400 quot450 quot250 University of R39s September Clothing Order II 3 II M l L l XL Tshirts 60 250 4oo 250 sweat shirts quot100 quot200 quot350 quot200 Mtrix Representation The above information can be given by two matrices S and R as shown below 1m 30C 50C 30C 8 150 400 450 250 El 250 400 250 R 100 200 350 220 MATRIX OPERATIONS The matrix operations can be summarized as under Organize and interpret data using matrices Use matrices in business applications Add and subtract two matrices Multiply a matrix by a scalar Multiply two matrices Interpret the meaning of the elements within a product matrix 80 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PRODUCTION The clothing company production in preparation for the universities39 September orders is shown by the table and corresponding matrix P below II 8 II M l L l XL Tshirts 3oo 7oo 9oo 5oo sweat shirts quot300 quot700 quot900 quot500 EDD TDD EDD EDD P EDD TDD EDD EDD ADDITION AND SUBTRACTION OF MATRICES The sum or difference of two matrices is calculated by adding or subtracting the corresponding elements of the matrices To add or subtract matrices they must have the same dimensions PRODUCTION REQUIREMENT Since the U of S ordered 100 small Tshirts and the U of R ordered 60 then altogether 160 small Tshirts are required to supply both universities Thus to calculate the total number of Tshirts and sweat shirts required to supply both universities add the corresponding elements of the two order matrices as shown below 1ED 4DD 4ED EED 1DD EDD EED 22D 15D EED E39DD EED IDD EDD EDD EDD 5D EED 4DD ED 4 EED EDD EDD 4TD OVERPRODUCTION Since the company produced 300 small Tshirts and the received orders for only 160 small Tshirts then the company produced 140 small Tshirts too many Thus to determine the company39s overproduction subtract the corresponding elements of the total order matrix from the production matrix as shown below EDD TDD EDD EDD 16D EED E39DD EED 14D IED D ED EDD TDD EDD EDD EED EDD EDD 4TD ED IDD IDD ED MULTIPLY A MATRIX BY A SCALAR Given a matrix A and a number 0 the scalar multiplication cA is computed by multiplying the scalar c by every element of A For example 3 2 1n n V 26 3 4 21 a 3 2 4 2 T 2 1 4 81 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MULTIPLICATION OF MATRICES To understand the reasoning behind the definition of matrix multiplication let us consider the following example Competing Companies A and B sell juice in 591 mL 1 L and 2 L plastic bottles at prices of Rs160 Rs230 and Rs310 respectively The table below summarizes the sales for the two companies during the month of July 591mL 1 1L 1 2L 1 quotCompany A 20000 5500 quot10600 quotCompany B quot18250 7000 11000 What is total revenue of Company A What is total revenue of Company B Matrices may be used to illustrate the above information As shown at the right the sales can be written as a 2X3 matrix S the selling prices can be written as a column matrix P and the total revenue for each company can be expressed as a column matrix R 150 00000 5500 10500 230 00510 10050 1000 11000 39 00400 3 510 R P Since revenue is calculated by multiplying the number of sales by the selling price the total revenue for each company is found by taking the product of the sales matrix and the price matrix 100 20000 5500 10000 230 22510 K 10250 2000 11000 20400 Consider how the first row of matrix S and the single column P lead to the first entry of R 150 5000001150155001050110500r51015 00 000 5500 10 500 00 510 0 050 t L t 18250 000 11000 20400 310 Pr0du0t 0f Pr00LIOt 0f Pr0000t 0f First Errtri00 0000110 Entri00 Third Entri00 With the above in mind we define the product of a row and a column to be the number obtained by multiplying corresponding entries first by first second by second and so on and adding the results MULTIPLICATION RULES 82 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU If matrix A is a m K n matrix and matrix B is a n K p matrix then the product AB is the m 3quot p matrix whose entry in the ith row and the jth column is the product of the ith row of matrix A and the jth row of matrix B The product of a row and a column is the number obtained by multiplying corresponding elements first by first second by second and so on To multiply matrices the number of columns ofA must equal the number of rows of B W Given the matrices below decide if the indicated product exists And if the product exists determine the dimensions of the product matrix quot 5 ti 4 A 12 3 iii 15 1 391 4 3 9 i 5 T MULTIPLICATION CHECKS The table below gives a summary whether it is possible to multiply two matrices It may be noticed that the product of matrix A and matrix B is possible as the number of columns ofA are equal to the number of rows of B The product BA is not possible as the number of columns of b are not equal to rows of A Does a product exist Product Dimensions of Is it possible to multiply Dimensions of the Matrices the given lProduct Matrix matrices in this order A 31 3 B 31 2 Yes the product exists T T since the AB inner dimensions match 3quot 2 inner dimen MS of columns of A of rows of B No the product does not B33ii2 A3gtlt3 exist I 1 since the inner BA dimensions do na inner dimena39ons not match of columns of B ti of rows of A MULTIPLICATIVE INVERSES Real Numbers Two nonzero real numbers are multiplicative inverses of each other if their products in both orders is 1 Thus 1 1 1 1 the multiplicative inverse ofa real number x is I or 539 Since x 39 I 1 and I 39 x 1 83 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 1 The multiplicative inverse of 5 is 5 since 1 1 539 51and 53951 Matrices Two 2 2 matrices are inverses of each other if their products in both orders is a 23quot 2 identity matrix Thus the multiplicative inverse of a 2i 2 matrix A is 31 1 since A39 111 1 f2 and 111 1 quot A 32 Example 2 1 3 1 The multiplicative inverse of a matrix is 5 3 5 2 2 1 3 1 1 0 5 3 5 2 o 1 3 1 2 1 1 0 5 2 5 3 0 1 84 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 11 MATRICES OBJECTIVES The objectives of the lecture are to learn about o Review Lecture 10 o Matrix functions in Excel 0 Set up and manipulate ratios 0 Allocate an amount on a prorata basis using proportions MATRIX FUNCTIONS IN MS EXCEL The Matrix Functions in Microsoft Excel are as follows 1 MINVERSE 2 MDETERM 3 MMULT MINVERSE Returns the inverse matrix for the matrix stored in an array Syntax MINVERSEarray array is a numeric array with an equal number of rows and columns Remarks 0 Array can be given as a cell range such as A1 03 as an array constant such as 1 23456789 or as a name for either of these 0 If any cells in array are empty or contain text MINVERSE returns the VALUE error value 0 MINVERSE also returns the VALUE error value if array does not have an equal number of rows and columns 0 Formulas that return arrays must be entered as array formulas o Inverse matrices like determinants are generally used for solving systems of mathematical equations involving several variables The product of a matrix and its inverse is the identity matrix the square array in which the diagonal values equal 1 and all other values equal 0 0 As an example of how a tworow twocolumn matrix is calculated suppose that the range A1 82 contains the letters a b c and d that represent any four numbers The following table shows the inverse of the matrix A1 82 Column A Column B Row 1 dadbc bbcad Row 2 cbcad aadbc o MINVERSE is calculated with an accuracy of approximately 16 digits which may lead to a small numeric error when the cancellation is not complete 0 Some square matrices cannot be inverted and will return the NUM error value with MINVERSE The determinant for a noninvertable matrix is 0 85 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MINVERSE M INVE REE array rray i5 a nun merit arragyr with an equal IuIInIJer sf raw5 f I39IiI39I c lulnna luvBrae 0f the In ri 3E1 132 C llll39l39ll39l C lllll39lll E FE W quotI nilquotiatilIJ HE IiiquotIJ J WC iil Raw illI39l 391 a quotIi aquotIa 39I39I I quotC rray f rl39l39illl 1 F2 2 Enter fuzannula 3 Etrl Shift Enter MlNVERSEEXAMPLE Find the inversion or multiplicative inverse of following matrix 4 1 2 0 The excel formula in the example must be entered as an array formula Otherwise a single result 0 will appear Please note that formulas that return mvs must be entered as array formulas The steps of finding multiplicative inverse of above matrix is as follows 1 Enter data of array to be inverted in Cells A4B ie in cells A4 B4 A5 BS 2 Click on cell A6 3 Keeping left mouse button pressed drag it to cell B7Four cells A6 BG A7 B7 will be selected 4 Press F2 from your keyboard Key F2 is selected to enter Edit Mode in the active cell It s a keyboard shortcut Even if you don39t press F2 you can write the formula 5 Type the formula MINVERSEA4B5 This will appear in cell A6 6 Press Ctrl Shift Enter keys simultaneously from your keyboard 7 Multiplicative inverse of the matrix will appear in cells A6 A7 BB B7 Now click any cell among A6B7 in the formula bar you can see curly brackets round the formula That is MINVERSEA4BS This shows that you have gone through the right procedure of entering array formula 86 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU k lillicrnanft Excel Lecture1lllllatricea Elle Edit iew insert Fgrmat Innls Qata indnw Help D l g y E Eiiii amm39 aa mnaaeeeaaaalaltaa Java 12 v r A 2 MINVERSE a Data Data 4 4 1 5 2 0 U 05 MINVERSA435 1 2 T EEI IT Cl El MDETERM Returns the matrix determinant of an array Syntax MDETERMarray array is a numeric array with an equal number of rows and columns Remarks 0 Array can be given as a cell range for example A1 03 as an array constant such as 1 23456789 or as a name to either of these o If any cells in array are empty or contain text MDETERM returns the VALUE error value 0 MDETERM also returns VALUE if array does not have an equal number of rows and columns 0 The matrix determinant is a number derived from the values in array For a three row threecolumn array A1 C3 the determinant is defined as MDETERMA1C3 A1 BZC3B3CZ A2B3C1B1C3 A3B1CZBZC1 0 Matrix determinants are generally used for solving systems of mathematical equations that involve several variables 0 MDETERM is calculated with an accuracy of approximately 16 digits which may lead to a small numeric error when the calculation is not complete For example the determinant of a singular matrix may differ from zero by 1E16 87 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MDETERMEXAMPLE The example shows an array of dimension 4 x 4 in cell range A14D17 The formula was entered in cell A18 The result of this calculation is 88 Microsoft Excel Book l Eile Edit iew insert Formal Iools Eata indow Help Fuder PDF El 2 it v a v 2 v v s L a T SLIM v x or 151 MDETERMAHD1 A E r D E F 11 MDETERM 12 13 Data Data Data Data 11 1 5 5 1 E 1 5 1 1 1 I 1 Iquot 10 2 IE MDETERMA1 ID1 19 Determinant f the matrix abide 88 There are other ways also for using this function For example you can enter the matrix as an array constant MDETERM3611103102 Determinant of the matrix as an array constant 1 You can calculate the determinant of the matrix in the array constant MDETERM3611 Determinant of the matrix in the array constant 3 Unequal number of rows and columns results in an error MDETERM13851361 Returns an error because the array does not have an equal number of rows and columns VALUE MMULT Returns the matrix product of two arrays The result is an array with the same number of rows as array1 and the same number of columns as array2 Syntax MMULTarray1 array2 Array1 array2 are the arrays you want to multiply Remarks 88 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU o The number of columns in array1 must be the same as the number of rows in array2 and both arrays must contain only numbers 0 Array1 and array2 can be given as cell ranges array constants or references o If any cells are empty or contain text or if the number of columns in array1 is different from the number of rows in array2 MMULT returns the VALUE error value 0 The matrir product array a of two arrays b and c is at Ebece 31 where i is the row number andj is the column number o Formulas that return arrays must be entered as array formulas MMULTEXAMPLE Let 1 3 2 0 A B Find AB 7 2 0 2 To find the product AB follow these steps 7 1 Enter arra 1 in cell range A25B26 and arra 2 in cell range D25E26 I Miereseft Excel Beet1 Elle Edit Elev I lnserl Fgrmel leels gate window Help i l ldel le vze lev I a De III E i riai til I E H lg E 39 13 v i e I a I I E I F I 23 MMULT 24 array1 array2 25 1 3 2 El 25 T 2 El 2 2 2E Preduet ef array1 and array2 29 2 E MMULTA25 BEEDEE E25 3D 14 4 91 2 Find the dimension of AB matrix Here as A and B are 2X2 matrices so AB is also a 2X2 matrix Click on cell A29 Keeping left mouse button pressed drag it to cell B30Four cells A29 B29 A30 B30 will be selected Press F2 from your keyboard Key F2 is selected to enter Edit Mode in the active cell It s a keyboard shortcut Even if you don39t press F2 you can write the formula Type the formula MMULT Select array1 Put comma Select array2 590 0quot 99 51 89 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 10 Close bracket 11 Press Ctrl Shift Enter keys simultaneously from your keyboard 12 Product AB will appear in cells A29 A30 829 B30 RATIO A Ratio is a comparison between things If in a room there are 30 men and 15 women then the ratio of men to women is 2 to 1 This is written as 21 and read as two is to one is the notation for a ratio Be careful order matters A ratio of 21 is not the same as 12 In the form of fraction we can write 21 as 21 The method of calculating ratios is as under 1 Find the minimum value 2 Divide all the values by the smallest value In the above example the smallest value was 15 Division gives 30 15 2 for men and 15 15 1 for women The ratio is therefore 21 for men and women RATIOEXAMPLE Three friends Ali Fawad and Tanveer are doing business together To set up the business Ali invested Rs 7800 Fawad Rs 5200 and Tanveer Rs 6500 What is the ratio of their investments As discussed above the smallest value is 5200Rs All values are divided by 5200 The results are 15 for Ali 1 for Fawad and 125 for Tanveer The answer is 15 l 125 90 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EE Micrusuft Excel Lectu re1 Ratiu5P ru pa rtiu n5 E Elle Edit Eiew insert Fgrmat Innls Qatar indnw Help Adobe PDF SEE v Ev jf 1 LIME3 T v E J EIEBIEIEE A E I D I 52 53 54 CALCULATING RETIRE 55 55 RptTIW 5 Ni 15 SE Fawad 5200 1 59 Tanveer 353l353 an 125 El This example can be solved in Excel The formula is as under Cell D57 B57 B58 Cell D58 B58 B58 Cell D59 B59 B58 The result for cell D59 is shown in cell D60 because the cell D59 is used to display the formula PROPORTION A proportion is an equation with a ratio on each side It is a statement that two ratios are equal 34 68 OR 34 68 is an example of a proportion When one of the four numbers in a proportion is unknown cross products may be used to find the unknown number This is called solving the proportion or ESTIMATING USING RATIO EXAMPLE Ratio of sales of Product X to sales of Product Y is 43 The sales of product X is forecasted at Rs 180000 What should be the Sales of product Y to maintain the ratio of sales between the two products CALCULATION Ratio sales X Y 4 3 Insert the value for forecasted sale for X 180000 Y 4 3 It can be rewritten as 180000Y 43 91 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Cross multiply 180000x 3 4 x Y Rewrite to bring the unknown to the left of the equality 4xY180000x3 Solve Y 180000 x 34 Y 135000 Rs Qlculations usinq EXCEL In cells B70 and B71 the ratios of Product X and Y were entered The value of forecast of product X was entered in cell D70 Before writing down the formula in excel it was derived as follows 1 Ratio of X cell B70 2 Ratio of y cell B71 3 Sale of X cell D70 4 Sale on cell D71 Now Ratio X Y cell B70 cell B71 Ratio of sales cell D70 cell D71 Crossmultiply cell B70 x cell D71 cell B71 x cell D70 Cell D71 is unknown Hence cell D71 cell B71 x cell D70 cell B70 Or cell D71 cell B71 cell B70 cell D70 Thus the formula was B71B70D70 Please note that actually we are using the ratio Y to X as it is easier to think of ratio of unknown to the known Microsoft Excel Lecture39l Ratin5l3rnpnrtinn Eile Edit EiEl n39 insert Fgrmat Idols gate window elp littler F39DF El rm Ev 3 L a a i SLIM r K vquot r EI1IEIDDEI A E C D E SE a ESTIMATING RATIS EE ES RIBITI SIDILES m Prduct 14 4 I 180000 1 Prduct t I BT1ITUiiD70 2 1135000 F3 ESTIMATING USING RATIOEXAMPLE 2 In a 500 bed hospital there are 200 nurses and 150 other staff If the hospital extends by a new wing for 100 beds then what additional staff is needed 92 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Let 500 beds B1 and 100 beds 32 Staff nurses N1 is 200 and other staff O1 is 150 What is the value of N2 and O for 32 Obviously the ratio of beds will be used As pointed out above think of the ratio of unknown to known In other words ratio 3231 or 3231 Ratio of nurses would be N2N1 Ratio of other staff would be 0201 Now N2N1 3231 Or N2 3231N1 or N2 Nurses 0201 3231 Or N2 3231O1 or Oz 100500150 30 other staff Calculation Beds Nurses Other staff 500 200 150 100 X Y Nurses 500 200 100 X 500 X 200 x 100 X 200 x 100500 40 Other staff Y 150 x 100500 30 Qlculation usinq EXCEL The calculation using EXCEL was done in a similar fashion as the previous example The calculation is selfexplanatory Hicrnsuft Excel Lectur21 atiusrupurtinns Eile Edit Eiew insert Fgrmal Tools gate window Help adage F39DF a t 133 1 H a SLIM 1 X of f3 DFID5EEI A a t D E LI 3 H 1 2 R lTIS 3 4 Hspital dditin 5 Beds 500 100 5 Nurses 200 9 quot jtherstaff 150 IDE EEI a El ESTIMATING USING RATIOEXAMPLE 3 A Fruit Punch recipe requires mango juice apple juice and orange juice in the ratio of 321 To make 2 liters of punch calculate quantity of other ingredients Again we shall use the ratio of unknown to the unknown The unknowns are mango and apple juice Consider first ratio of required mango juice 3 to total quantity of punch 6 This was calculated from 321 Now the quantity of 93 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU required mango for 2 liter would simply be 362 Similarly the required quantity of apple juice is 262 m Mango juice Apple juice Orange juice 3 2 1 Total 6 X Y 2 Total 2 litre Mango juice X 3 6 2 1 litre Apple juice Y 2 6 2 067 litre Orange juice 2 1 6 2 033 liter Calculation usinq EXCEL Here also the similar ratios were used Mango BZOBZ3D23 Apple BZ1BZ3D23 Orange BZZBZ3D23 Mir rusuft Excel Lent ure1 atiu5P ru pa rtiu I15 Eile Edit iew insert Fgrmat IDEIIS Qatar induw Help adage F DF 539 i E v TL a T T SUM 139 K 39J 5 EIEEIEIEED23 A El 3 D E F 3 IE 1 USING RATIS IE 19 RATI 2n Mang juice 239 1 21 Apple juice 2 2 arrange juice I 1 I 3221323mz3 23 Ttal Litre E 2 24 94 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 12 RATIO AND PROPORTION MERCHANDISING OBJECTIVES The objectives of the lecture are to learn about 0 Module 3 Review Lecture 11 Ratio and Proportions Merchandising Assignment 1A and 1B MODULE 3 Module 3 has the following content 0 Ratio and Proportions o Merchandising Lectures 12 0 Mathematics of Merchandising Lectures 1316 ESTIMATING USING RATIOSEXAMPLE1 In the previous lecture we studied how ratios can be used to determine unknowns Here is another example with a slightly different approach Here ratios between the quantitiesand data of only one quantity is known We will estimate the total quantity that can be made It is the quantity of orange juice that will determine the total quantity that can be made Again the method is to use the ratio of the unknown to the known W In a punch the ratio of mangojuice applejuice and orangejuice is 321 If you have 15 liters of orange juice how much punch can you make m Mangojuice Applejuice Orangejuice 3 2 39 1 Total 321 6 X say Ysay Zsay 15litres Total litre Mangojuice X 31gtlt15 45 litre Applejuice Y 21gtlt15 30 litre Orangejuice Z 15 litre Punch 45 30 15 9 litres EXCEL calculation The method used is the same as in previous examples 95 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Mic rueuft Excel Lect ure1 Ratiue ru pa rtiu I15 Elle Edit iew insert Fgrmat Idols gate window Help littler F39DF E Iii iiiF ri quot E quot if L H a 39 SLIM T E uquot f EIEMEIEE 1333 A E I D E F G 2 23 RATIMS 29 an RATIW 31 Mang juice 39Pquot 32 pple juice 2 39Pquot 33 range juice 1 15 15 Ttal Litre 35 BE ESTIMATING USING RATIOSEXAMPLE2 In a punch ratio of mango juice apple juice and orange juice is 3 2 15 If you have 500 litres of orange juice find how much mango and apple juices are required to make the punch ME The ratio of mangojuice applejuice and orangejuice is 3 2 15 If you have 500 milliliters of orange juice how much mango juice and apple juice is needed Mangojuice Applejuice Orangejuice 3 2 1 5 Total 65 X say Ysay Z say 9 39 39 500 litres Total litres Mangojuice X 315500 1000 litre Apple juice Y 21 5500 667 litre Orange juice Z 500 litre Punch 1000 667 500 2167 litre EXCEL Calculation Here also ratios were used Mango juice B45B47D47 Apple juice B46B47D47 Orange juice D47 96 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU micrnenft Excel Lecture l Ratiu5Prnpurtinne E Elle Edit ew insert Fgrmat Innle Qatar indnw elp Adnge F39DF n it m 2 v L H a SLIM r X d it El lEfEl lFD l A El C D E I F 41 42 USING RnTIS 43 44 Rth 1000 Mang juice B45lB47i Df l7 e ipple juice 2 007 1 range juice 15 500 500 18 Ttal MI 05 2107 39 EXAMPLE In a certain class the ratio of passing grades to failing grades is 7 to 5 How many of the 36 students failed the course The ratio quot7 to 5quot or 7 5 or 75 tells you that of every 7 5 12 students five failed That is 512 of the class failed Then 512 36 15 students failed PROPORTION ab old the values in the quotbquot and quotcquot positions are called the quotmeansquot of the proportion while the values in the quotaquot and quotdquot positions are called the quotextremesquot of the proportion A basic defining property of a proportion is that the product of the means is equal to the product of the extremes In other words given ab old it is a fact that ad bc PROPORTIONEXAMPQ ls 24140 proportional to 30176 M 140gtlt30 4200 24X176 4224 So the answer is that given ratios are not proportional PROPORTION EXAMPLE 1 Find the unknown value in the proportion 2 x 3 9 2 x 3 9 First convert the colonnotation ratios to fractions 2x 39 Cross multiply 18 3x 6 x 97 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PROPORTION EXAMPLE 2 Find the unknown value in the proportion 2x 1 2 x 2 5 2x12x25 First convert the colonnotation ratios to fractions 2x 12 x 25 Then solve 52x 1 2x 2 10x 5 2x 4 8x 1 x 18 MERCHANDISING What does merchandising cover Understand the ordinary dating notation for the terms of payment of an invoice Solve merchandise pricing problems involving mark ups and markdowns Calculate the net price of an item after single or multiple trade discounts Calculate a single discount rate that is equivalent to a series of multiple discounts 0 Calculate the amount of the cash discount for which a payment qualifies STAKEHOLDERS IN Merchandising Who are the stakeholders in merchandising The main players are Manufacturer Middleman Retailer Consumer There are discounts at all levels in the above chain 0 O O O MIDDLEMAN A middle man is a person who buys a product directly from the manufacturer and then either sells the product at retail prices to the public or sells the product at wholesale prices to a distributor There can often be more than one middle man when the latter practice is adopted A middle man can purchase from the manufacturer and then work with another middle man who buys for the distributor The manufacturer often views the middle man as the alternative to direct distribution gst price or Retail price List price refers to the manufacturer39s suggested retail pricing It may or may not be the price asked of the consumer Much depends on 1 the product itself 2 the builtin profit margin 3 Supply and demand A product that is in high demand with low availability will sometimes sell higher than the list price though this is less common than the reverse Virtually all products have a suggested retail or list price Resellers middleman retailer buy products in bulk and get a substantial discount in order to be able to get profit from selling the product at or below list price 98 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Trade Discount Let L is the list price then amount of trade discount is some percentage of this price List price less amount of discount is the net price In mathematical terms we can write Amount ofdiscount d x L Where d Percentage of Discount L List Price Net Price L Ld L1 d Net Price List Price Amount of Discount Copyright Virtual University of Pakistan VU 99 Business Mathematics amp Statistics MTH 302 VU LECTURE 13 MATHEMATICS OF MERCHANDISING OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 12 o Solve merchandising pricing problems involving markup and markdown MARKUP Markup is an amount added to a cost price while calculating a selling price Especially an amount that takes into account of overhead and profit Markup can be expressed as 1 Percentage of cost 2 Percentage of sale 3 Rs Markup Markup as Percentage of Cost MUC Here markup is some percentage of cost price For simplicity it is also named as Markup on cost The relation between markup on cost cost price and selling price is Selling Price Cost price Cost price x Markup on cost Cost price 1 Markup on cost Markup as Percentage of Sale price MUS Here markup is some percentage of selling price For simplicity it is also named as Markup on sale The relation between markup on sale cost price and selling price is Selling Price Cost price Selling price x Markup on sale Cost price Selling price Selling price x Markup on sale Selling price 1 Markup on sale Rs Markup Markup in terms of rupees is called Rs markup The relations between Rs markup cost price and selling price are 1 Selling Price Cost price Rs Markup 2 Rs Markup Markup on cost x Cost price 3 Rs Markup Markup on sale x Selling price Any of the above formula can be used to find Rs Markup For example The cost price of certain item is 80Rs and its selling price is 100Rs Then Rs Markup Selling price Cost price 100 80 20 Rs Remember If some percentage is given as markup without mentioning that whether it is markup on cost or markup on sale it is evident that markup on cost is under consideration EAMPLE 1 A golf shop pays its wholesaler 2400Rs for a certain club and then sells it for 4500Rs What is the markup rate Calculation of Markup 100 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Cost price 2400Rs Selling price 4500Rs Selling Price Cost price Cost price x Markup on cost Markup on cost Sellinq price Cost price X100 Cost price Since Rs Markup Selling Price Cost price Markup on cost Rs markup X100 Cost price 1 First calculate Rs markup Rs markup 4500 2400 2100Rs Then markup on cost Markup on cost Rs markup X100 Cost price 21002400 X100 875 Remember to convert this fraction value to a percent The markup rate is 875 Qlculgtion usinq EXCEL Enter wholesale price 2400 in cell B5 Enter sale price 4500 in cell BG Enter formula for Rs Markup BGB5 in cell B7and press enter The answer is 2100 Enter formula for markup B7B5100 in cell B8 and press Enter The answer is 875 shown in cell BQ Microsoft Excel Bool Eile Edit Eiew insert Fgrmat Tools gate window Help Atler F39DF Q 53935 r 2 1 3939 In T SLIM v x of f ElF39IElE1IIIEI A E I C D l l 2 MARKUP 3 4 5 Whlesale price ErrI00 a Sale price 1500 Markup Rs 2100 3 Markup Brro t1oo 9 85 IEI Copyright Virtual University of Pakistan 101 Business Mathematics amp Statistics MTH 302 VU MARKUPEXAMPLE 2 A computer software retailer used a markup rate of 40 Find the selling price of a computer game that cost the retailer Rs 1500 Markup The markup is 40 on the cost so as Rs Markup Markup on cost x Cost price Rs Markup 040 1500 Rs 600 Selling Price Then the selling price being the cost plus markup that is Selling Price Cost price Rs Markup 1 500 600 Rs 2100 The item sold for Rs 2100 Qlculgtion usinq EXCEL Here we use the following formula to show an alternate method of solving the above problem Selling Price Cost price 1 Markup on cost Enter wholesale price 1500 in cell B17 Enter Markup in cell B18 Enter formula 1B18100B17 in cell B19 Here the term 1B18100 is the multiplication factor B18100 is the markup in fraction The answer 2100 is shown in cell B20 We could have calculated the multiplication factor separately But as you see it is not necessary as the entire calculation can be done in one line E Micrusu Excel Baum Elle Edit ew insert Fgrmat pols gate window Help adage PDF is at 3 e v E a 4 in 39L 3939 If T SLIM v E I f 1B1BI1EIEITEI I A E If D 39 12 13 M MARKUP 15 Example 2 15 1 Whlesale price 1500 18 sh Markup 1e1r1oora1rv 2 2100 21 MARKDOWN Markdown means a reduction from the original sale price to l 02 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 1 stimulate demand or 2 take advantage of reduced costs or 3 force competitors out of the market Markdown can be expressed as 9 v Percentage of current selling price 00 Rs markdown Markdown as Percentage of current selling price Here markdown is some percentage of current selling price For simplicity it is also named as percent markdown markdown The relation between current selling price markdown and new selling price is New selling price Current selling price Current selling price x markdown Current selling price 1 markdown Rs Markdown Markdown in terms of rupees is called Rs markdown 1 New selling price Current selling price Rs Markdown 2 Rs markdown Current selling price x markdown Let us look at an example to understand how markdown is calculated MARKDOWNEXAMPLE1 An item originally priced at 3300 Rs is marked 25 off What is the sale price Markdown First find the Rs markdown The markdown is 25 of the original price or current selling price as Rs markdown original price x markdown 0253300 825Rs Selling Price Then calculate the sale price by subtracting the markdown from the original price New Selling price 3300 825 2475Rs The sale price is 2475 Rs Qlculgtion usinq EXCEL Enter original price 3300 in cell B28 Enter Markdown 25 in cell B29 Enter formula for Rs Markdown B29100B28 in cell B30 Here the term B29100 is the markdown in fraction The result of this part of the calculation is 825 Enter formula for new sale price B28 B30 in cell B31 This formula is not shown in the slide We could have calculated the new sale price directly also by writing just one formula 1B29100B28 by using following formula New selling price Current selling price 1 markdown In other words the multiplication factor is calculated as 1025 075 and multiplied with the original price 3300 The answer would be the same By breaking the calculation in parts you can check the intermediate result and avoid errors But if you become very conversant with formulas then you may wish to reduce the number of unnecessary steps in the calculations 103 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ejj icrusuft Excel Boom Elle Edit ElevJ insert Fgrmal Ippls gate window Help littler PDF 1 are m 2 ve 1quot Fe SLIM v x 4quot f ElEEiflIIIIIIElEE A 25 MARKDWN 25 Example 1 2 EB riginal price 29 We BEBI1DWBZB 31 Sale price 2MB 32 DISCOUNT Discount is a reduction in price which the seller offers to the buyer There are different types of discount 1 Trade discount 2 Cash discount 3 Seasonal discount etc TRADE DISCOUNT When a manufacturer or wholesaler offers goods for sale a list price or retail price is set for each item This is the suggestion price to be charged from the ultimate consumer A discount on the list price granted by a manufacturer or wholesaler to buyers in the same trade is called trade discount Trade discount represents a reduction in list price in return for quantity purchases Thus Rs Trade discount list price x discount rate Net price list price Rs trade discount There are two main types of trade discounts 1 Single trade discount 2 Multiple or series trade discount 104 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU SINGE TRADE DISCOUNTEXAMPL 1 The price of office equipment is 3000 Rs The manufacturer offers a 30 trade discount Find the net price and the trade discount amount Diseeunt Net Price L1 d 30001 03 300007 2100 Rs Amount of discount dL 03 x 3000 900 Rs Qlculgtion usinq EXCEL Enter price of equipment 3000 in cell B39 Enter trade Discount 30 in cell B40 Enter formula for Rs Discount B40100B3 in cell B41 Here the term B40100 is the discount in fraction The result of this part of the calculation is 900 Enter formula for net price B39B41 in cell B42 This formula is not shown in the slide The result is 2100 as shown in cell B42 Ejjdicrusuft Excel Baum Elle Edit Elevi insert Fgrmal Duels Qatar induw Help Filippa F39DF 259 3 iv e v 2 v v e 1 3939 FE T SLIM r K J F EldelIIIIIIEl39 A E C D 35 35 DISCUNT 3 Example 1 33 39 Price f equipment 4n 111 Trade discunt e4ur1ocreae 42 Net price 2100 43 1 05 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 14 MATHEMATICS OF MERCHANDISING PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 13 Financial Mathematics Part 1 SERIES TRADE DISCOUNT This refers to the giving of further discounts as incentives for more sales Usually such discount is offered for selling product in bulk lf series discount of 15 10 5 are offered on list price say L of an item then net price is calculated as follows Subtract 15 of L from L Let the new price is L1 L1L L X15 Then subtract 10 of L1 from L1 Let the new price is L2 L2 L1L1x10 Then subtract 5 of L2 from L2 The new price is net price of an item N L2 sz 5 Or alternatively N L 1 15 1 10 1 5 Let d1 15 d2 10 d3 5 then above formula becomes N L 1 di1 d21 d3 Remember total discount is not 15 10 5 30 SERIES TRADE DISCOUNTEXAMPLE The price of of ce furniture is Rs 20000 The series discounts are 2010 5 What is the net price For series trade Discount Net price 1d1 ldz 1d3 Here d1 20 d2 10 d3 5 So Net price 2000010210101005 200000809095 2000006840 13680 Rs 106 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU H39i39emmefl lEamEI bloom 333 Fr L 39 iiiaw lrncri ngrma Ind gate 135mm Align FDF r Em En an 2 v El Te 39239 quot39 7 lil39i l J j39i EFF39Ei39EFEHDZIE 39EW1U jmm uui 5 E 1 5395 IF39E TRl rlEl E 39DlllElGLl39MT h 3 Grass price R5 2mm 3 First EE unlt 3 9 Next discount Eta ln first 1 a Herrt discount quot3 train ti r51 7 5 all earrtnaarar 3339 a tutti lETEHDEI 3 Net prime E5 1350 LEBH 35 LIST PRICE An order for power tools has a Rs 2100 net price after a 30 trade discount What is the list price Net Price Net Price L1 d L N 1 d 21001 03 2100 07 3000 Rs EXCEL Calculation EXCEL formula for list price was based on the calculation 21001 03 The net price was entered in cell 867 Trade discount was entered in B69 The formula for list price was entered in cell B71 as BG7 1869100 The answer is shown in cell C72 as 3000 1 07 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Elle Edit ew ll39lSE rt Fgrmal IDDIS Qatar window elp Adnge PDF 33 it a v w v E v T e 1 3939 Fe SLIM r K d 15quot EIEF It1ElEBI1EIEI A E C D 53 54 LIST PRICE 55 55 a Net price 2100 EB 59 Trade discunt 39an m eerrr1 59 oo 2 73 TRADE DISCOUNTEXAMPLE 2 Find the single discount rate that is equivalent to the series 15 10 and 5 Trade Discount Apply the multiple discount to a list price of Rs 100 Net price 1d11d21d3 1001151101 5 100085 09 095 10007268 7268 Discount 100 7268 2762 EXCEL calculation EXCEL formula for net price was based on the calculation 10010151011005 The formula for net price was entered in cell F8 The formula is shown in cell F8 The answer is shown in cell F12 108 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnsnft Excel Bunk1 Elle Edit ElemI Lnsert Fermet leels gate indew elp Fidelge F39DF it a v m w E v Q 1 TL 39JLJZ Fe SLIM 1quot X J 5 llFfli ll1F5leDjI1FEI1EIDIII1IJD A E C D E F I G H J K 1 2 SERIES DISCUNTS 3 4 First Discth 15 ill 5 Next Diecunt 10 a 5 Next Discunt 5 le F ne price r1F4r1normFsiworhFEI in manned 11 12 727 13 In the following slide the net price was calculated in cell F8 Then the discount was calculated assuming the list price was 100 This is a common method to assume 100 as the list price when no price is given but you are required to calculate the net discount Micrusuft Excel Bunl Elle Edit iew insert Fgrmet eels gate indnw Help delge F39DF cit IE v an v E v E 3 TL 391 FerE EUM 1quot H 39439 1UElFE A e e e E 4 3 H 2 SERIES DISCUNTS 3 4 Firet Dieeunt 15 quote 5 Next Diseunt 10 In 5 Next Diseunt 5 quota F e Net price 72T Dieeunt 1 TRADE DISCOUNTEXAMPLE3 The price of car parts is Rs 20000 The series discounts are 20 8 2 What is the single equivalent discount rate Also find Rs discount 1 09 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Let Rs 100 is the list price then Net price 1001 021 0081 002 10008092098 10007213 7213 Single equivalent discount rate 100 7213 2787 Rs Discount 0278720000 5574 Rs EXCEL calculation EXCEL formula for net price was based on the calculation 10010210081002 The formula for net price was entered in cell F21 The formula is not shown Price of car parts was entered in cell F23 Formula for discount was based on 0278720000 and is shown in cell F24 The answer is shown in cell F26 as 5574 Microsoft Excel Baum Elle Edit ew insert Fgrmat Innis Data window Help Adobe PDF clquot LE v E v E v 3 1 3939 quotEa SLIM v E u 75 F23F2271IIIIZI A El I3 D E F G H 15 SINGLE EUIVALENT DISCUNT RATE IE 1 First Discount 20 3937 IE Next Diecunt 8 7a 19 Next Diecunt 2 3917 ED 21 Net price 721 3937 22 Discunt 279 7 23 Price of car parts Rs Diecunt F23F22l 100i 25 25 5574 27 CASH DISCOUNT A seller always desires to be paid by the buyer as soon as possible A discount given for the prompt payment of the dues is called Cash Discount Such a discount is an advantage to both the seller and the buyer The buyer has a saving of money while the seller has funds at his disposal Cash Discount is allowed on Invoices Returned Goods Freight Sales Tax and A common business phrase for a cash discount is quot310 net30quot meaning that a 3 discount is offered if the amount due is paid within 10 days otherwise 100 of the amount due is payable in 30 days For example if the amount due is 100Rs the buyer may pay 97Rs within 10 days or 100 Rs within 30 days 110 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU DISCOUNT PERIODS Discount Periods are periods for the buyer to take advantage of Discount Terms CREDIT PERIODS Credit Periods are periods for the buyers to pay invoices within specified times CASH plSCOUNT EXAMPE Invoice was dated May 1st The terms 210 mean that 2 discount is offered if invoice is paid up to 10thMay What is the net payment for invoice value of Rs 50000 if paid up to 10th May Cash Discount N L1 d 500001002 50000098 49000 Rs EXCEL Calculation EXCEL formula for net price was based on the calculation 500001002 However here an IF condition was applied that means that if the payment date in cell D31 sign is put in front of row and column to fix its location is less than or equal to 10 May then the discount will be as given in cell D30 Here also sign was used to fix the location of the cell In cell D38 the date was changed to 11 May and the same formula was applied again The result as shown in cell D39 and D40 are 0 for discount and 0 Rs for Rs iscount 39 Miereenft Excel Bunk Eile Edit yew insert Fgrrnat IDDIS gate indnaI Help in lgeniliaaliii itlasel vailev Avlazvt Emil rm vII a IE 2 E aal 1 a at at me v a a e e a E F e 29 CASH DISCDUNT an Dieeeunt 2 quotii 31 Last date 10 W39layF 32 aa lnveiee value 50000 Re 34 35 Payment date 0 allayF a3 Dieeeunt IFD3539D31D300 a 1000 R5 3a Payment date 11 allayF ae Dieeeunt 0 quotit an 0 R5 1 1 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 15 MATHEMATICS OF MERCHANDISING PART 3 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 14 0 Financial Mathematics Part 2 PARTIAL PAYMENTS When you buy on credit and have cash discount terms part of the invoice may be paid within the specified time These part payments are called Partial Payments Let us look at an example You owe Rs 40000 Your terms were 310 3 discount by 10th day Within 10 days you sent in a payment of Rs 10000 Rs 10000 was a part payment How much is your new balance First we will find the amount that if 3 discount is given on it the net amount is 10000Rs Let that amount is tThen 10000 t 1 003 This implies t 10000 1 003 Thus t 10309Rs This means that although you pay 10000Rs due to 3 cash discount 10309Rs among 40000Rs is paid Hence the new balance 40000 10309 29691 Rs MARKETING TERMS There are a number of marketing terms First of these is the Manufacturer Cost This is the cost of manufacturing Next is the price charged to middlemen in The Distribution Chain The DistributorgtWholesalergtRetailer is a chain The next term is the Selling Price This is the price charged to Consumers by Retailers It may or may not be the same as list price MARKETING OPERATING EXPENSES AND SELLING PRICE Gross Sales less Cost of Goods sold gives the Gross Profit The gross Profit less the Operating Expenses gives the Net Profit Operating Expenses Expenses the company incurs in operating the business eg rent wages and utilities is called operating Expenses Selling Price Selling Price is composed of Cost and Rs Markup Selling Price S Cost C Rs Markup M MARGIN While determining Sale Price a company includes the operating expenses and profit to their own cost This amount is called the margin of the company It is usually calculated 112 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU as percentage but can also be expressed as rupees It is also named as markup on sale Margin or markup on sale Selling price Cost price X100 Selling Price Selling price Cost price Rs Margin Margin and markup confuse many By margin company evaluates that for every rupee generated in sales how much is left over to cover basic operating costs and profit Markup represents the amount added to a cost to arrive at a selling price Markup on cost Selling price Cost price X100 Cost price For example an item costs 50Rs and is sold for 100Rs Markup 100 50 x100 100 50 Margin 100 50 X100 50 100 Note Remember unless it is mentioned that markup is on sale simple markup means markup on cost Example A computer s cost is 9000Rs An amount of Rs 3000 was added to this cost by the retailer to determine the sale price for the consumer Thus the selling price 9000 3000 Rs 12000 Rs Rs 3000 is Margin available to meet Expenses and make a Pro t MARKUP If the Markup on cost 33 then Selling Price S Cost C Cost C X Markup on cost MUC S C C X MUC MARKUPEXAMPLE You buy candles for Rs 10 You plan to sell them for Rs15 What is your Rs Markup What is your percent Markup on cost Rs Markup Selling price cost price Selling price Cost 15 10 Rs Markup Rs 5 Markup on cost Selling price Cost price X100 Cost price Markup 510100 50 SELLING PRICE ll3 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Fawad s Appliances bought a sewing machine for Rs 1500 To make the desired profit he needs a 60 Markup on Cost What is Fawad s Rs Markup What is his Selling price Selling Price Rs Markup Cost price x Markup on cost Rs Markup 1500 x 06 900 Rs Selling Price S Cost C Rs Markup M Selling Price 1500 900 2400 Rs 9r Alternatively since Selling Price S Cost C Cost C X Markup on cost MUC SCCgtltMUC C1MUC Selling price 15oo x 106 1500 X 16 2400 Rs EXCEL Calculation Here 1500 is the Sewing machine cost in cell F4 and 60 is the Percent Markup on cost in cell F5 EXCEL formula in cell F6 for Rs Markup was based on the calculation 601001500 The Selling price was calculated in cell F7 by using the formula F4F6 The answer 2400 is shown in cell F7 Micrnenft Excel Ennk1 Elle Edit iew insert Fgrmel Icicle gate indnw elp ureter eerieleela Jam were initial 310 r l E I H I E E 3 We 539 TEE13933 I 39 FiE v e A e r D E t i l 2 MARKUP SELLING PRICE 3 4 Sewing machine ceet 1500 Re 5 A Markup an east 60 5 Re Markup F5I100F4 Re r Selling price 2400 Re 8 114 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU RS MARKUP AND PERCENT ON COST Tanveer s flower business sells floral arrangements for Rs 35 To make his desired profit Tanveer needs a 40 Markup on cost What do the flower arrangements cost Tanveer What is the Rs Markup Rs Markup and Percent Markup on Cost Sale price S Cost C C XMarkup on cost MUC S C 040C 35 1 40C C 3514 25 Rs Rs Markup 25 x 04 10 Rs EXCEL Calculation Here 35 is the Selling pricefloral arrangement in cell H15 Markup on cost is in cell H16 EXCEL formula in cell H18 for Cost was based on the calculation 3514 The Rs Markup was calculated in cell H19 by using the formula H18H16100 The answer as shown in cell H19 was 10 El Hii f39i ILIEIEIEII Llllitlrljflil39l LiliaI ELEElilli lil iJ ilrliil rli i39 5 Elli FigJP rillmg lam Eng all Fl 35quot Fl F Ea E Ell Z 7111 1a rm 4 tr 3 iingllariur at E t n E F 3 H l I l l 11 IE 13 R5 MAEHUF AIME FEEEEHT on EDET H 15 Sellng priceaflnral arrangement I EEIH E 1E llllrltup en EDE E 4 til 1quot ttest Ltll15fltlll1 l1 ll m 5 Martin 1 E5 33 21 Formula 5 E 4quot IM 2 E rr s l 23 1 Jill I 15 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MARKUP AgAIN You buy candles for 2 Rs You plan to sell them for 250 Rs What is your Rs Markup What is your Percent Markup on Selling Price Rs Markup Rs Markup 25 2 05 Rs Percent Markup on Sellinq Price As explained in lecture 13 Cost price Selling price 1 Markup on sale Markup on selling price Selling price Cost price X100 Selling price Markup on Selling Price 0525 X100 20 EXCEL calculation Here 2 is the Purchase price in cell E30 Sale price is entered in cell E31 Rs Markup on Purchase Price was calculated by using the formula E31E30 in cell E32 The Markup on sale price was calculated in cell E33 by using the formula E32E31100 The answer as shown in cell E35 was 49 20 116 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Mir rusuft Excel Lectu re14Marl up i5cuu ntMarl duwn Elle Edit Eiew insert Fgrn39lal Innls Qatar window Help Atler PDF El a it T m T 2 T 1T n a SLIP391 r K J f E32fE311EIEI A E C D E I F G H 2 23 MARKUP AGAIN 29 an Purchase price 2 RS 31 Sale price 25 R5 32 R5 Markup 05 R5 3 Markup n sale price E32fE31f100 3T1 39 35 20 SE SELLING PRICE Fawad s Appliances bought a sewing machine for Rs 1500 To make the desired profit he needs a 60 Markup on Selling price What is Fawad s Rs Markup What is his Selling Price Selling Price As explained in lecture 13 Selling Price Cost price Selling price x Markup on sale Selling Price S 1500 068 S 068 1500 Rs Or 048 1500 3750 Rs Rs Markup Rs Markup 3750 x 06 2250 Rs EXCEL calculation Here 1500 is the Purchase price in cell E39 Markup on Sale Price is entered as 60 in cell E40 Sale Price was calculated by using the formula E391E40100 The result 3750 is shown in cell D41 EXCEL formula in cell E42 for Rs Markup was E41E39 The result 2250 is shown in cell E42 Basic formula SC06S is shown in cell A44 In cell A45 it was simplified to 04C In cell A46 it is rewritten as SC04SC1mucC04 Here muc is the Markup on cost 117 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Lecture14liilar up i5cnunt larkdnwn Eile Edit ew insert Fgrmat Innls Qatar indnw Help ndnlge PDF El quot3 it a v 2 v 1 n n SLIM T K I f E3Em lE4IZII1IZIEIJ A E c n E F G 3 33 SELLING PRICE AGAIN 39 Purchase price 1500 Rs 4 Markup n sale price 50 A Sale price 3750 E39l1E40l100 42 R5 Markup 2250 Rs 43 14 45 45 SCi04SCI1mu Cl 04 4 RS MARKUP AND PERCENT MARKUP ON COST Tanveer s flower business sells floral arrangements for Rs 35 To make his desired profit Tanveer needs a 40 Markup on Selling Price What do the flower arrangements Cost Tanveer What is the Rs Markup Selling Price Selling Price Cost price Selling price x Markup on sale Selling Price 35 C 04 x 35 35 C 14 C 35 14 21 Rs Or alternatively C S 04 S 06 S 06 x 35 21 Rs Rs Markup Rs Markup 35 x 04 14 Rs EXCEL Calculation Here 35 is the Sale price in cell E50 Markup on Sale Price is entered as 40 in cell E51 Cost was calculated by using the formula E501E51100 The result 21 is shown in cell D52 EXCEL formula in cell E53 for Rs Markup was E50E52 The result 14 is shown in cell E53 Basic formula SC04S is shown in cell A55 In cell A56 it was simplified to 06SCS1mus 1 18 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Lecture14Mar up i5cuuntiuiar duwn Eile Edit new insert Fgrmat Iouls gate window Help Fldtl e PDF El 2 it e v 2 v 1L 39 w SUM v in uquot is E5III1E51I1IIIIII A El 3 u E F G 48 49 Rs MARKUP AND PERCENT N CST an Sale price 135 Rs 51 le Markup n sale price 40 We 52 Est 21 E501E51i100 53 Rs Markup M Rs 54 55 UESCS1mus 5 CONVERTING MARKUPS Convert 50 Markup MU on Cost to MU on Sale Formula for converting Markup on Sale mus to Markup on Cost Price muc is Markup on Selling Price mus Markup on Cost 1 Markup on Cost mus muc1muc Solution Markup on Sale mus 05 105 0515 mus 03333 3333 Convertinq Markups Converting 3333 MU on Sale to MU on C M Convert Markup on Cost muc to Markup on selling price mus Markup on cost Markup on S l Markup on S muc mus lmus Markup on cost 033331 0333 0333306666 05 50 EXCEL Calculation 119 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Here 333 is the Markup on sale in cell E61 EXCEL formula in cell E62 for Markup on cost was E611001E61100100 The result 50 is shown in cell E64 Basic formula mucmus 1mus is shown in cell A65 Micrnanft Excel Lecture l4lllarltupi5cnuntlllarlnlwrn E Elle Edit ew insert Fgrmat nulls Qata window Help H Eyeiir 51 x 39039 LE 3 i sum v x J r EElllDDllEElllDDlIJD A E C D E F G H J K l 59 CNVERTING MARKUP N SALE T MARKUP N CST El E1 Markup n sale gMarkup n est EE1I100l1E61l100100 EH El 50 ES mucmus11 mus100 55 EE ES 1 20 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 16 MATHEMATICS OF MERCHANDISING PART 4 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 15 o Markup and Markdown Financial Mathematics Part 3 MARKDOWN Reduction from original selling Price is called Markdown M Markdown Rs Markdown Selling Price original x100 MARKDOWNEXAMPLE 1 Store A marked down a Rs 500 shirt to Rs 360 What is the Rs Markdown What is the markdown Rs Markdown Let S Sale price Rs Markdown Old S New S Rs 500 Rs 360 Rs 140 Markdown Markdown Markdown Markdown x100 Old S Markdown mx100 500 028x100 28 EXCEL calculation Here 500 is the Original price in cell E73 Price after Markdown is entered as 360 in cell E74 Rs Markdown was calculated in cell E75 by using the formula E73E74 The result 140 is shown in cell D75 121 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU XCEL formula in cell E76 for Markdown was E75E73 100 The result 28 is shown Hicrnsuft Excel Lecture14Marltup i5cnunt tarkdnwn Elle Edit Eiew insert Fgrmat Innis Qatar indnw elp Adobe PDF SLIM r X a 3 3 E3 Ei r39Ji El I i E I F I3 H F EI r1 MHRKDWN 72 r3 riginal price I SHORE r4 Price after Marde 350 Rs r5 Rs Markdwn 140 E7 3E74l n3 n Wu F m 2 MARKDOWNEXAMPLE2 A variety of plastic jugs that was bought for Rs 5775 was marked up 45 of the SellingPrice When the jugs went out of production they were marked down 40 What was the Sale Price after the 40 markdown Here there are two parts to this problem First we must find the original sale price so that markdown can be calculated on that price Original Sale Price Let Selling price 100 Markup on selling price 45 Cost 100 45 55 Thus Original Sale price 10055 x 5775 105Rs Rs Markdown Markdown 40 04 Rs Markdown 105 x 04 42 Sale price after markdown Sale price after markdown 105 42 63 Rs EXCEL calculation Here 5775 is the purchase price in cell F83 Selling price is entered as 100in cell F84 Rs Markup was calculated in cell F85 using the formula F84F83 The result is shown as 45 in cell F85 Original Sale Price was calculated in cell F87 by using the formula F84F86F83 The result 105 is shown in cell E87 Markdown was entered as 40 in cell F88 The Rs Markdown was calculated using the formula F87F88100 in cell F89 The result 42 is shown in cell F89 122 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The reduced price was calculated by using the formula F87F89 in cell F90 The result is shown as 63 in cell F90 Micrusu Excel Lecture14Mar upi5uuntMarI duwn Elle Edit Elevi Insert Fgrmal Incl Qatar indnw Help Ado e F39DF site m 2 e 1quot Fe SLIM r K a fair FEWFEEFI33 A E c n E F l 3 H a MARKDWN 33 Purchase price 34 Let Selling price 35 Markup BE ESEquot at riginel Sale price as Merkdwn 39 RS Markdwn en Reduced Price Ell PROJECT FINANCIAL ANALYSIS Financial analysis is the analysis of the accounts and the economic prospects of a firm which can be used to monitor and evaluate the firm39s financial position to plan future financing and to designate the size of the firm and its rate of growth When you carry out Project Financial analysis a number of Financial Calculations are required The important ones are summarized below Cost estimates Revenue estimates Forecasts of costs Forecasts of revenues Net cash flows Benefit cost analysis Internal Rate of Return BreakEven Analysis 123 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU COST ESTIMATES In every project you will be required to prepare a cost estimate Generally such cost estimates cover calculations based on quantities and unit rates Such calculations are done in the form of tabular worksheets In large projects there may be a number of separate calculations for part projects Such component costs are then combined to calculate total cost These are simple worksheet calculations unless conditional processing is required Such conditional processing is useful if unit prices are to be found for a specific model from a large database REVENUE ESTIMATES Along with costs even revenues are calculated These calculations are similar to component costs FORECASTS OF COSTS Forecasting requires a technique for projections One of such technique Time Series Analysis will be covered later in this course Forecasting techniques vary from case to case The applicable method should be determined first Calculation of future forecasts can then be done through worksheets FORECASTS OF REVENUES These will be done similar to the forecast of costs Here also the method must be determined first Once the methodology is clear the worksheets can be prepared easily ET CASH FIOWS The difference between Revenue and Cost is called the Net Cash flow This is an important calculation as the entire Project Operation and Performance is based on its cash flows BENEFIT COST ANAIYSIS This is the end result of the Project Analysis The ratio between Present Worth of Benefits and Costs is called the Benefit Cost BC ratio For a project to be viable without profit or loss the BC Ratio must be 1 or more Generally a BC Ratio of 12 is considered acceptable For Public projects even lesser BC ratio may be accepted for social reasons MRNAI RATE OF RETURN Internal Rate of Return or lRR is that Discount Rate at which the Present Worth of Costs is equal to the Present Worth of Benefits lRR is the most important parameter in Financial and Economic Analysis There are a number of functions in EXCEL for calculation of lRR BREAKEVEN ANALYSIS In every project where investment is made it is important to know how long it takes to recover the investment It is also important to find the breakeven point where the Cash lnflow becomes equal to Cash Outflow After that point the company has a positive cash flow ie there is surplus cash after meeting expenses 124 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 17 MATHEMATICS FINANCIAL MATHEMATICS INTRODUCTION TO SIMULTANEOUS EQUATIONS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 16 0 Financial Mathematics 0 Introduction to Linear Equations MARKDOWN Module 4 Module 4 covers the following 0 Financial Mathematics Lecture 17 0 Applications of Linear Equations 0 Lecture 1718 0 Breakeven Analysis 0 Lectures 1922 0 MidTerm Examination PROJECT FINANCIAL ANALYSIS Project Financial Analysis covers the following 0 Cost estimates Revenue estimates Forecasts of costs Forecasts of revenues Net cash flows Benefit cost analysis Internal Rate of Return BreakEven Analysis EXCEL FUNCTIONS FOR FINANCIAL ANALYSIS List of Excel Financial functions is as under The name and utility of each function is given below AMORDEGRC Returns the depreciation for each accounting period If an asset is purchased in the middle of the accounting period the prorated depreciation is taken into account The function is similar to AMORLINC except that a depreciation coefficient is applied in the calculation depending on the life of the assets Syntax AMORDEGRCcostdatepurchasedfirstperiodsalvageperiodratebasis Important Dates should be entered by using the DATE function or as results of other formulas or functions For example use DATE2008523 for the 23rd day of May 2008 Problems can occur if dates are entered as text Cost cost of the asset Datepurchased date of the purchase of the asset Firstperiod date of the end of the first period Salvage salvage value at the end of the life of the asset Period period 125 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Rate rate of depreciation Basis year basis to be used Basis Date system 0 or omitted 360 days NASD method 1 Actual 3 365 days in a year 4 360 days in a year European method Remarks Microsoft Excel stores dates as sequential serial numbers so they can be used in calculations By default January 1 1900 is serial number 1 and January 1 2008 is serial number 39448 because it is 39448 days after January 1 1900 This function will return the depreciation until the last period of the life of the assets or until the cumulated value of depreciation is greater than the cost of the assets minus the salvage value The life of the asset is calculated by 1 quotratequot The depreciation coefficient depends on the life of the asset If the life of the asset is between 3 and 4 years the coefficient is 15 If the life of the asset is between 5 and 6 years then the coefficient is 2 If the life is the asset is greater than 6 years then the coefficient is 25 The depreciation rate will grow to 50 percent for the period preceding the last period and will grow to 100 percent for the last period If the life of assets is between 0 zero and 1 1 and 2 2 and 3 or 4 and 5 the NUM error value is returned See its example in next lecture AMORLINC Returns the depreciation for each accounting period If an asset is purchased in the middle of the accounting period the prorated depreciation is taken into account Syntax AMORLINCcostdatepurchasedfirstperiodsalvageperiodratebasis Cost cost of the asset Datepurchased date of the purchase of the asset Firstperiod date of the end of the first period Salvage salvage value at the end of the life of the asset Period period Rate rate of depreciation Basis year basis to be used Basis Date system 0 or omitted 360 days NASD method 1 3 4 Actual 365 days in a year 360 days in a year European method AMORQNCEXAMPE A2 A3 A4 Data Description 2400 Cost 8192008 Date purchased 12312008 End of the first period 126 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU A5 300 Salvage value A6 1 Period A7 15 Depreciation rate A8 1 Actual basis see above Formula Result Description AMOFiLNCA2A3A4A5A6A 7A8 First period depreciation 360 m Returns the cumulative interest paid between two periods For description see lecture 8 CUMPRINC Returns the cumulative principal paid on a loan between two periods For description see lecture 8 DB Returns the depreciation of an asset for a specified period using the fixeddeclining balance method Syntax DBcostsalvagelifeperiodmonth Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset Life is the number of periods over which the asset is being depreciated sometimes called the useful life of the asset Period is the period for which you want to calculate the depreciation Period must use the same units as life Month is the number of months in the first year If month is omitted it is assumed to be 12 Remarks 0 The fixeddeclining balance method computes depreciation at a fixed rate DB uses the following formulas to calculate depreciation for a period cost total depreciation from prior periods rate where rate 1 salvage cost quot 1 ife rounded to three decimal places 0 Depreciation for the first and last periods is a special case For the first period DB uses this formula cost rate month 12 o For the last period DB uses this formula cost total depreciation from prior periods rate 12 month 12 DDB Returns the depreciation of an asset for a specified period using the doubledeclining balance method or some other method you specify Syntax DDBcostsalvagelifeperiodfactor Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset This value can be 0 Life is the number of periods over which the asset is being depreciated sometimes called the useful life of the asset Period is the period for which you want to calculate the depreciation Period must use the same units as life Factor is the rate at which the balance declines lf factor is omitted it is assumed to be 2 the doubledeclining balance method 1 27 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Important All five arguments must be positive numbers Remarks 0 The doubledeclining balance method computes depreciation at an accelerated rate Depreciation is highest in the first period and decreases in successive periods DDB uses the following formula to calculate depreciation for a period Min cost total depreciation from prior periods factorlife cost salvage total depreciation from prior periods 0 Change factor if you do not want to use the doubledeclining balance method MIRR Returns the modified internal rate of return for a series of periodic cash flows MIRR considers both the cost of the investment and the interest received on reinvestment of cash Syntax MIRRvaluesfinanceratereinvestrate Values is an array or a reference to cells that contain numbers These numbers represent a series of payments negative values and income positive values occurring at regular periods Values must contain at least one positive value and one negative value to calculate the modified internal rate of return Otherwise MIRR returns the DlVO error value If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included Financerate is the interest rate you pay on the money used in the cash flows Reinvestrate is the interest rate you receive on the cash flows as you reinvest them INTERNAI RATE OF RETURN IRR Returns the internal rate of return for a series of cash flows Syntax lRRvaluesguess Values is an array or a reference to cells that contain numbers for which you want to calculate the internal rate of return 0 Values must contain at least one positive value and one negative value to calculate the internal rate of return 0 IRR uses the order of values to interpret the order of cash flows Be sure to enter your payment and income values in the sequence you want o If an array or reference argument contains text logical values or empty cells those values are ignored Guess is a number that you guess is close to the result of IRR 0 Microsoft Excel uses an iterative technique for calculating IRR Starting with guess IRR cycles through the calculation until the result is accurate within 000001 percent lf IRR can39t find a result that works after 20 tries the NUM error value is returned 0 In most cases you do not need to provide guess for the IRR calculation lf guess is omitted it is assumed to be 01 10 percent o If IRR gives the NUM error value or if the result is not close to what you expected try again with a different value for guess IRREXAMPLE In the slide the Excel worksheet is shown 128 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In cell A97 the investment of 70000 is entered with minus sign to denote negative cash flow In cell A98 to A102 revenue per year 449 5 is entered IRRA97A101 formula in cell A103 only years 1 to 4 were selected from the revenue stream The IRR is 2 in this case In the next formula in cell A105 the entire revenue stream was considered The IRR improved to 9 Next only first 2 years of revenue stream were considered with an initial guess of 10 The result was 44 r J Microsoft Excel Eoolt1 Eile Edit Elevr insert Fgrmet Idols gate window elp seller v sflev as E slil Esrial vmvlsso Es3s rfsEs39iEIEEEE 390 END v H a i s I E l I l D as lRRyaluesguess as Description sr 7oooo Initial cost of a business as 12 Net income for the rst year as 1sooo Net income for the second year 1o 1sooo Net income for the third year an 21ooo Net income for the fourth year 1o2 zsooo Net income for the fth year its ms lRRfAQA1 1 IRR after 4 years 2 ice IRRA9A1DE IRR after 5 years 9 me lRRfAQAQQ1 IRR after 2 years in include a guess 44 391I39lll 129 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 18 MATHEMATICS FINANCIAL MATHEMATICS SOLVE TWO LINEAR EQUATIONS WITH TWO UNKNOWNS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 17 o Solve two linear equations with two unknowns AMORDEGRCEXAMPLE itirm EamlLmture1 IFinzamialFum nna E A I L Eli Ejlt fallow lnaort Egmat Elma Eater lrdaw Ijolpi ningol iE39F W Shaitiiu l im l f etr m E H ELM 1 a a a aaaaIaaacaamaararamaazuaaii a a 13 AMDHDEGRCcoatdatapurchaaadfiratpariadaavagapariodratabaais 14 EData a DE EE39Irilpti ll I 13 tg Data piu rchaaad 1104231 End of the first perind 1 300 Salvage valua 1a a uii39l Depreciation rate 21 1 mammal baaia AMQRDEGRGIM 5IA15I First ariad de 39reeiation WE MEWS quot AMORLINCEXAMPLE 1 3 0 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft IEmal Lecturej EF39i39lil i lFl Il1 l rl E IEIIE Edit ew insert Format Ionls Esta lndnw liens 41der FDF mrkshee tfunttionsbgrtat w i sea 39 a a vevar EUN 1 X r KRMDRLIIHICEA1EAIEA1Fgma i g em A I E 13 Data Description 15 2400 Cost IE 030319 Date purchased 1 031231 End of the first period 13 300 Salvage value 19 1 Period 33 015 lDepreciation rate 21 391 Actual basis AMORLINC A15II 6 22 A17A131 AZUmZ ill 23 First period depreciation 350i DBEXAMPLE Microsoft Excel Lecture1EFinancialFunctiune Elle Edit iew insert Format eels Data window elp Ado eF DF worksheet functions hr tat v E gt 39 f l 11s it 303 3 a o g 3 i SLIM r X o a DEII EF QE EEIIFI a a 25 DBcostsalvagelifeperiodmonth 25 Data Description 2 1000000 Initial cost 23 100000 Salvage value 29 0 quotLifetime in years so DBA27023A2017 Depreciation in first year 31 39Dims 39 I39i39iii with with only 7 months calculated a 13303333 33 ADDITIONAL DB EXAMPLES Look at the following examples to see how the DB function can be used in different ways DBA27A28A2917 Depreciation in first year with only 7 months calculated 18608333 13 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU DBA27A28A2927 Depreciation in second year 25963942 DBA27A28A2937 Depreciation in third year 17681444 DBA27A28A2947 Depreciation in fourth year 12041064 DBA27A28A2957 Depreciation in fifth year 8199964 DBA27A28A2967 Depreciation in sixth year 5584176 DBA27A28A2975 Depreciation in seventh year with only 5 months calculated 1584510 Returns the present value of an investment Syntax PVratenperpmtfvtype Rate interest rate per period Nper total number of payment periods in an annuity Pmt payment made each period and cannot change over the life of the annuity Fv future value or a cash balance you want to attain after the last payment is made Type number 0 or 1 and indicates when payments are due Microsoft Excel Lecture1EFinancialFunctinns Elle Edit ElevJ insert Format Idols Data window elp adobe F39DF worksheet functions by tat v 539 X s a 2 v a a v a v a v SLIM v X J 8 F39leEf12 12M9MF Ilj a a E 45 PWrate nper pmtfvtype 15 Data Description Money paid out of an insurance annuity at 500 1 the end of every month 008 Interest rate earned on the money paid out 48 19 20 Years the money will be paid out 5 s pvm412 Present value of an annuity with the terms i12m49m47 above 5977715 52 2 0 m Returns the net present value of an investment based on a series of periodic cash flows and a discount rate Syntax NPVratevalue1value2 Rate rate of discount over the length of one period Value1 value2 1 to 29 arguments representing the payments and income 1 32 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnsnft Excel Lecture1EFinancialFunctinns Elle Edit Eiew insert Fgrrnat eels gate indew elp adage F39DF tt v m 239 a SLIM v X J f3 NPViAEAEEA5Ei EDAEij a E aNPWratejualueWalueZ aEiMMPLE 1 55 Data Description 5 annual disceunt rate Initial cast of investment cine year from 55 teday 55 Return from first year 55 4200 Return tram secend year 55 Return from third year a NPVM57A53A59 60ai rt 55 118844 XNPV Returns the net present value for a schedule of cash flows that is not necessarily periodic Syntax XNPVratevaluesdates Rate discount rate to apply to the cash flows Values series of cash flows that corresponds to a schedule of payments in dates ates schedule of payment dates that corresponds to the cash flow pamnts Microsoft Excel Eeelt1 5 Elle Edit EielaI insert Fgrmat leels gate indew elp are e are areaia aavaia v via E at are gala am all e u l E E l s as 939 tat 433 is E 7 Eli 139 a XNPVrate5ualues5dates Example 1 rate 9 Values Dates 1 i ii ii i January 15 2mm 25 Marsh 1 EMS 425D Deteher 3D 2W3 325i February 155 2009 25l April 15 zone XHPVBSA5A9B BQ Net present value 2 85 5 1 3 3 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 19 PERFORM BREAKEVEN ANALYSIS EXCEL FUNCTIONS FINANCIAL ANALYSIS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 18 0 Perform breakeven analysis 0 MS EXCEL Financial Functions SLN Returns the straightline depreciation of an asset for one period Syntax SLNcostsalvagelife Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset Life is the number of periods over which the asset is depreciated sometimes called the useful life of the asset Miereeeft Eaeel Leeture1EFineneielFunetiene Elle Edit iew insert Fgrmat eels gate indew Help adage F39DF WDFkShEEtFUI39IEtiDI39IS E 9 lt v a v E v E E a v a v a SUM v X a a eulraaaaaaarm a a a3 SLN cestsalvaelife 5 Data Descriptin Ea Cat 59 7500 Salvage value a 10 Years f useful life SLNA68A69MD a The depreciatin allwance fr each year 2250 F3 i39 39 SYD Returns the sumofyears39 digits depreciation of an asset for a specified period Syntax SYDcostsalvagelifeper Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset Life is the number of periods over which the asset is depreciated sometimes called the useful life of the asset 1 34 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Per is the period and must use the same units as life Remark 0 SYD is calculated as follows STE 2 east aetvage 139 Ef e per39 lth ii jtil 1 Micrusnft Excel Lecture1EFinancialFunctiun5 Elle Edit ew insert Fgrmat eels Data indnw elp adage F39DF WDrkShEEHUI39IEtiEII39IS h39r Eat 139 E quot9 at E r E at E 1r 6 Ir a 139 T SUM r X J a S rD ED 31 EE1 a a as SYDcst salvage life per rr SYD CsatSalvage ifeper12llifelife1 FE ra Data Descriptin a 300 Initial est a 7500 Salvage value a 10 quotLifespan in years aa SYD lUA31 A21 a S TP F tquot at 1W3 iation allowance for the first year 409091 as I VDB Returns the depreciation of an asset for any period you specify including partial periods using the doubledeclining balance method or some other method you specify VDB stands for variable declining balance Syntax VDBcostsalvagelifestartperiodendperiodfactornoswitch Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset Life is the number of periods over which the asset is depreciated sometimes called the useful life of the asset Startperiod is the starting period for which you want to calculate the depreciation Startperiod must use the same units as life Endperiod is the ending period for which you want to calculate the depreciation Endperiod must use the same units as life Factor is the rate at which the balance declines lf factor is omitted it is assumed to be 2 the doubledeclining balance method Change factor if you do not want to use the doubledeclining balance method For a description of the doubledeclining balance method see DDB Noswitch is a logical value specifying whether to switch to straightline depreciation when depreciation is greater than the declining balance calculation o If noswitch is TRUE Microsoft Excel does not switch to straightline depreciation even when the depreciation is greater than the declining balance calculation 135 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU o If noswitch is FALSE or omitted Excel switches to straightline depreciation when depreciation is greater than the declining balance calculation All arguments except noswitch must be positive numbers All arguments except noswitch must be positive numbers Hicrusuft Excel Lecture1EFinancialFunctiuns Elle Edit ew insert Format sols Qatar window elp adage F39DF W rksheetfuntti nst El 3 a v o v E v E if E quotl39 quotl39 a 139 sum 139 x or a vuaraaaaauaairsaaaaau1j a a as VDBCcost salvage life startperid endperiod a factor noswitch 33 Data Description 39 2400 Initial cost 9 i300 Salvage value 9 10 Lifetime in years a VDBM39MUA91365 A9501 as First day39s depreciation a Excel automatically assumes that f39actr is 2 132 as E Returns the internal rate of return for a series of cash flows represented by the numbers in values These cash flows do not have to be even as they would be for an annuity However the cash flows must occur at regular intervals such as monthly or annually The internal rate of return is the interest rate received for an investment consisting of payments negative values and income positive values that occur at regular periods Syntax lRRvaluesguess Values is an array or a reference to cells that contain numbers for which you want to calculate the internal rate of return 0 Values must contain at least one positive value and one negative value to calculate the internal rate of return 0 IRR uses the order of values to interpret the order of cash flows Be sure to enter your payment and income values in the sequence you want 0 If an array or reference argument contains text logical values or empty cells those values are ignored Guess is a number that you guess is close to the result of IRR 0 Microsoft Excel uses an iterative technique for calculating IRR Starting with guess IRR cycles through the calculation until the result is accurate within 000001 percent If IRR can39t find a result that works after 20 tries the NUM error value is returned 0 In most cases you do not need to provide guess for the IRR calculation If guess is omitted it is assumed to be 01 10 percent 0 If IRR gives the NUM error value or if the result is not close to what you expected try again with a different value for guess 136 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Remarks IRR is closely related to NPV the net present value function The rate of return calculated by IRR is the interest rate corresponding to a 0 zero net present value The following formula demonstrates how NPV and IRR are related NPVRRB1B6B1B6 equals 360E08 Within the accuracy of the IRR calculation the value 360E08 is effectively 0 zero IRREXAMPLE In the slide the Excel worksheet is shown In cell A97 the investment of 70000 is entered with minus sign to denote negative cash flow In cell A98 to A102 revenue per year 1 to 5 is entered In the first formula in cell A103 IRRA97A101 only years 1 to 4 were selected for the revenue stream The HR is 2 in this case In the next formula in cell A105 the entire revenue stream was considered The IRR improved to 9 Next only first 2 years of revenue stream were considered with an initial guess of 10 not shown in slide The result was 44 Microsoft Excel Eoolt1 Elle Edit ew insert Format Iools gate windowI elp t sold cv Elev E vsltl isrial m vls s ol Elia a s assists L Eno v a s B C D E l as lRRyaluesguess as Description a JDEIIIDD Initial cost of a business as 12 Net income for the rst year as 15 Net income for the second year 1o 13 Net income for the third year 1o1 21 Net income for the fourth year 1oz E Net income for the fth year 103 ms IRRAQA1D1 IRR after 4 years 2 ms IRRA9A102 IRR after 5 years 9 ins IRRAQA991 IRR after 2 years 1o include a guess 44 XIRR Returns the internal rate of return for a schedule of cash flows that is not necessarily periodic To calculate the internal rate of return for a series of periodic cash flows use the IRR function 137 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU If this function is not available and returns the NAME error install and load the Analysis ToolPak addin To do that 1 On the Tools menu click AddIns 2 In the AddIns available list select the Analysis ToolPak box and then click OK 3 If necessary follow the instructions in the setup program Syntax XIRRvaluesdatesguess Values is a series of cash flows that corresponds to a schedule of payments in dates The first payment is optional and corresponds to a cost or payment that occurs at the beginning of the investment If the first value is a cost or payment it must be a negative value All succeeding payments are discounted based on a 365day year The series of values must contain at least one positive and one negative value Dates is a schedule of payment dates that corresponds to the cash flow payments The first payment date indicates the beginning of the schedule of payments All other dates must be later than this date but they may occur in any order Dates should be entered by using the DATE function or as results of other formulas or functions For example use DATE2008523 for the 23rd day of May 2008 Problems can occur if dates are entered as text Guess is a number that you guess is close to the result oleRR Remarks 0 Microsoft Excel stores dates as sequential serial numbers so they can be used in calculations By default January 1 1900 is serial number 1 and January 1 2008 is serial number 39448 because it is 39448 days after January 1 1900 Microsoft Excel for the Macintosh uses a different date system as its default 0 Numbers in dates are truncated to integers o XIRR expects at least one positive cash flow and one negative cash flow otherwise XIRR returns the NUM error value o If any number in dates is not a valid date XIRR returns the VALUE error value o If any number in dates precedes the starting date XIRR returns the NUM error value o If values and dates contain a different number of values XIRR returns the NUM error value 0 In most cases you do not need to provide guess for the XIRR calculation lf omitted guess is assumed to be 01 10 percent 0 XIRR is closely related to XNPV the net present value function The rate of return calculated by XIRR is the interest rate corresponding to XNPV 0 0 Excel uses an iterative technique for calculating XIRR Using a changing rate starting with guess XIRR cycles through the calculation until the result is accurate within 0000001 percent lf XIRR can39t find a result that works after 100 tries the NUM error value is returned The rate is changed until where P r I39 92 tar ail quot3 1 rate 355 di the ith or last payment date d1 the 0th payment date Pi the ith or last payment XIRR EXAMPLE Here the investment is in cell A111 The revenue stream is in cells A112 to a115 The dates for each investment or revenue are given in cells B111 to B115 Please note that the dates are in European format yearmonthday On your computer you may not have this format 138 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ei iernsnft Excel Lecture1 EFinaneialFunctiens E Eile Edit Eiew Lnsert Fgrmet Inels gate lndew elp Ade e PDF werksheetfu 9 3 CE v 2 110 e 55 sum x r r KIRHIIA I I IA115EllllEli 5El1j 2 El 109 MERWalues atesguess 10 Values Dates 111 10000 20080101 112 250 20080301 113 r4250 20081 0130 1 31250 20090215 2750 20000401 E XIRRM111 KIRR 033352535 r 3731 11 1153111 123311201 1 After entering these days in Excel you can right click on the cell You see a short cut menu as shown below E iereseft Excel Leeture1EFinaneialFunetinns Elle Edit EleniI Lnsert Fgrrnet eels gate indew Help Dg g y l rr Ev l jmws nrial 20IQEE 3EE iii E quot3quot Ellll fr DATEEUUE11II E mazegueee 11 Dates 11I20080101 3 Cut 112 20080301 113 20031030 Earl E15 Paste eeciel H 20000215 1 20000401 Clear Ientents V r a H men 0373325 34 r llr HE EermatCells 119 PiclgFrem List 1239 Field match imid tES re rperlink When you will select Format Cells the Format Cells Dialog Box appears as shown below You can then choose the desired format for the date 1 3 9 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Fu rrnat Eells Filignment 1 Fent 1 Etercler 1 Patterns 1 F39retectien 1 Qategcnry Sample General J 20080101 Number Currency Type Ficccnuntini clen 14 mars EIIIIIII T39me secures14 Percentage scarce14 133u Fractlen 0143314 Scientific 010314 1330 Text 5 Etial 143 2001 51311 J Lecale lecaticunIH 15weclish Date Fermats clisplay39 clate ancl time serial numbers as clate values Except FcIr items that have an asterislt II applied Fermats 2ch net switch clate erclers Iwith the eperating system 1 Di 1 Cancel 1 In cell A116 the formula XRRA111A115B111B11501 the range A111A115 is the cost and revenue stream The range B111B115 is the stream for dates The third term 01 is the initial guess for XIRR The answer in fraction or is given in cell B1163734 MAR QUATIONS Linear equations have following applications in Merchandising Mathematics 0 Solve two linear equations with two variables Solve problems that require setting up linear equations with two variables Perform linear CostVolumeProfit and breakeven analysis employing The contribution margin approach The algebraic approach of solving the cost and revenue functions SOL VING LINEAR EQUATIONS Here is an example of solving simultaneous linear equations 2x 3y 6 x y 2 Solve for y 2x 3y 6 2x 2y 4 5y 10 y 105 y 2 140 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Let us look at the same equations again 2x 3y 6 x y 2 We solved for x Now let us substitute y by 2 2x 32 6 2x 6 6 2x 0 x 0 Check your answer By substituting the values into each of the equations Eguation 1 2x 3y 6 x 0 y 2 LHS 2x 3y 2032 6 RHS x y 2 LHSX y022RHS The right side is equal to left hand side Hence the answer is correct 141 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 20 PERFORM BREAKEVEN ANALYSIS EXCEL FUNCTIONS FOR FINANCIAL ANALYSIS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 18 0 MS EXCEL Financial Functions 0 Perform BreakEven Analysis LINEAR QUATIONS Zain purchases the same amount of commodity 1 and 2 each week After price increases from Rs 110 to Rs 115 per item of commodity 1 and from Rs 098 to Rs 114 per item of commodity 2 the weekly bill rose from Rs 8440 to Rs 9170 How many items of commodity 1 and 2 are purchased each week Let x of commodity 1 Let y of commodity 2 Settinq up Linitr Equations Eguation 1 110x 098y 8440 1 Eliminate x in 1 by Dividing both sides by 110 1 10x 098y110 8440110 x 08909y 7673 M 115x114y917 2 Eliminate x in 2 by Dividing both sides by 115 115x 114y115 9170115 x 09913y 7974 Result 1 x 08909y 7673 3 x 09913y 7974 4 Next Subtract 4 from 3 w 01004y 301 y 30101004 Or y2998 approximantely 30 30 items of commodity 2 are purchased each week 110X 098y 8440 Substitution Substitute value of y in 1 Result 110x 0982998 8440 Solve 110x 2938 8440 110x 8440 2938 110x 5502 Resu x 5002 approximately 50 50 items of commodity 1 are purchased each week 1 42 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Check your answer New weeklv cost Commodity 1 50 x 115 5750Rs Commodity 2 30 x 114 3420 Total cost 91 70Rs TERMINOLOGY There are either Business Costs or Expenses Mk Even Analysis Break Even Analysis refers to the calculation to determine how much product a company must sell in order to get break even point The point at which no profit is made and no losses are incurred on that product Breakeven analysis provides insight into whether or not revenue from a product or service has the ability to cover the relevant costs of production of that product or service Managers can use this information in making a wide range of business decisions including setting prices preparing competitive bids and applying for loans CostVolume Profit Analysis Costvolumeprofit CVP analysis expands the use of information provided by breakeven analysis It deals with how profits and costs change with a change in volume More specifically it looks at the effects on profits by changes in such factors as variable costs fixed costs selling prices volume By studying the relationships of costs sales and net income management is better able to cope with many planning decisions For example CVP analysis attempts to answer the following questions 1 What sales volume is required to break even 2 What sales volume is necessary in order to earn a desired target profit 3 What profit can be expected on a given sales volume 4 How would changes in selling price variable costs fixed costs and output affect profits Fixed Costs FC Fixed Costs are such costs that do not change if sales increase or decrease eg rent property taxes some forms of depreciation Variable Costs VC Variable costs do change in direct proportion to sales volume eg material costs and direct labor costs Production Ca citv PC It is the number of units a firm can make in a given period Break Even Point Break Even point is a point at which neither a profit nor loss is made Revenue is exactly equal to costs Break even point can be expressed as 1 units 2 Sales or Rupees Rs 3 Percent of capacity BEP in units calculates how many units should be sold to break even lfthe product is sold in a quantity greater than this the firm will makes a profit below this point a loss BEP in units Fixed Costs Contribution Margin per unit 1 43 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU BEP in Rs calculates the revenue that must be obtained to reach break even point BEP in Rs Fixed Costs x Net Sales Contribution Margin BEP in Rs Fixed costs x Selling Price per unit Contribution Margin per unit BEP as percent of capacity calculates what percent of production capacity will be utilized to produce the number of units required to reach break even point BEP as of capacity BEP in units x 100 Production capacity Contribution Margin Contribution Margin is the Rs amount that is found by deducting Variable Costs from Sales or revenues and contributes to meeting Fixed Costs and making a Net Profit It can be calculated on a total or per unit basis Contribution Margin Net Sales Variable Cost S VC Contribution margin per unit CM Sale price per unit Variable cost per unit Contribution Rgte CR Contribution rate Contribution Margin X 100 CM X 100 Net sales S Contribution rate Contribution Margin per unit X 100 CM X 100 Sale price per unit S A CONTRIBUTION MARGIN STATEMENT Rs Net Sales Price x Units Sold x 100 Less Variable Costs x x Contribution Margin x x Less Fixed Costs x x Net Income x x The net sales are calculated by multiplying price per unit with number of units sold Net Sales Sale price per unit x number of units sold This figure is treated as 100 Next variable costs are specified and deducted from the Net sales to obtain the Contribution Margin Next Fixed costs are deducted from the contribution Margin The result is Net Income Net Income Contribution Margin Fixed Costs Under the column percentage of each item is calculated with respect to the Net Sales 1 44 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Here39s an example Net Sales 462452 Rs 100 Less Variable Costs Cost of goods sold 230934 Rs 50 Sales Commissions 58852 Rs 127 Delivery Charges 13984 Rs 3 Total Variable Costs 230934 58852 13984 303770 Rs 657 Contribution Margin 462452 303770 158682 Rs 343 Less Fixed Costs Advertising 1850 Rs 04 Depreciation 13250 Rs 28 Insurance 5400 Rs 12 Payroll Taxes 8200 Rs 18 Rent 9600 Rs 21 Utilities 17801 Rs 38 Wages 40000 Rs 86 Total Fixed Costs 18501325054008200 96001780140000 96101 Rs 208 Net Income 158682 96101 62581 Rs 135 m A firm is planning to add a new item in its product line Market research indicates that the new product can be sold at Rs 50 per unit Cost analysis provides the following information Fixed Costs FC per period Rs 8640 Variable Costs VC Rs 30 per unit Production Capacity per period 900 units How much does the sale of an additional unit of a firm s new product contribute towards increasing its net income M Contribution Margin per unit CM S VC Contribution Rate CR CMS x 100 Break Even Point BEP in Units x Rs x FC CM in Sales Rs Rs x FC CM S in of Capacity BEP in UnitsPC x 100 At Break Even Net Profit or Loss 0 Scenario 1 Summary Selling price per unit S 50 Rs Fixed Costs per period FC Rs 8640 Variable Costs VC Rs 30 per unit 145 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Production Capacity per period PC 900 units Solution of this problem is in the next lecture 146 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 21 PERFORM LINEAR COSTVOLUME PROFIT AND BREAKEVEN ANALYSIS USING THE CONTRIBUTION MARGIN APPROACH OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 18 0 Perform BreakEven Analysis 0 MS EXCEL Financial Functions 0 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 20 0 Perform linear costvolume profit and breakeven analysis 0 Using the contribution margin approach SCENARIO 1 Contribution Margin per unit CM S VC 50 30 20 Rs Contribution rate CR CMS x 100 Rs 205OX100 40 Break Even Point In Units x FC CM 864020 432 Units In Rs x FCCM S Rs 8640Rs20 Rs50 Rs21600 of Capacity BEP in units PC X100 432 900gtlt100 48 Thus by selling more than 432 units of its new product a firm can make profit SCENARIO 2 The Lighting Division of A Lighting Fitting Manufacturer plans to introduce a new street light based on the following accounting information FC per period Rs 3136 VC Rs157 per unit 8 Rs 185 per unit Production Capacity per period 320 units Calculate the break even point BEP in units in rupees as a percent of capacity Break Even Point in units FC CM CM S VC Rs185 157 Rs28 147 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU BEP in units 313628 112 Units Break Even Point in Rupees FC CM S 313628 185 20720 Rs Break Even Point as a percent of cayacitv BEP in unitsPCX100 112320gtlt100 35 Capacity SCENARIO 21 FC Rs3136 VC Rs157per unit 8 Rs185 per unit Production Capacity 320 units Determine the BEP as a of capacity if FC are reduced to Rs2688 Formula BEP as a of capacity BEP in unitsPCX100 Step1 Find CM Step 2 Find BEP in units Step 3 Find of Capacity Step 1 Find CM per unit S 185Rs per unit VC 157Rs per unit CM S VC 185 157 Rs 28 Step 2 Find EP in units BEP in units FCCM Rs 2688 Rs28 96 Units Step 3 Find of Capacity BEP as a of capacity BEP in units PCX100 96320gtlt100 30 of Capacity SCENARIO 22 FC Rs3136 VC Rs157 per unit S Rs185 per unit Production Capacity 320 units per period Determine the BEP as a of capacity if FC are increased to Rs4588 and VC reduced to 80 of S BEP as a of capacity BEP in unitsPCX100 NewVC S x 80 185 x 08 Rs148 New FC 4588 Rs 148 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Step 1 Find CM per unit S 185 Rs per unit VC 148Rs per unit CM S VC Rs 37 Step 2 Find EP in units BEP in units FCCM Rs 4588 Rs 37 124 Units Step 3 Find of Capacity BEP as a of capacity BEP in units PCgtlt100 124320gtlt100 39 of Capacity SCENARIO 2 3 FC Rs 3136 VG Rs157 per unit 8 Rs185 per unit Production Capacity 320 units per period Determine the BEP as a of capacity if S is reduced to Rs171 BEP as a of capacity BEP in unitsPCX100 Step 1 Find CM per unit S 171Rs per unit VC 157Rs per unit CM S VC Rs 14 Step 2 Find EP in units BEP in units FCCM Rs 3136 Rs 14 224 Units Step 3 Find BEP as a of Capacity BEP as a of capacity BEP in units PCgtlt100 224320gtlt100 70 of Capacity 149 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 22 PERFORM LINEAR COSTVOLUME PROFIT AND BREAKEVEN ANALYSIS OBJECTIVES The objectives of the lecture are to learn about I Review Lecture 21 I Perform Linear CostVolume Profit and BreakEven analysis Using Microsoft Excel SCENARIO 1 Let us look at different scenarios for calculation of contribution margin and net profit The explanations are given in the slides The Break Even Point in Rs is 21600 The break Even point as a of capacity is 48 iiz i 39s ll EIIEEII LEEIUTIEJ I5IquotE quotl ri iir1 link the rm Ila15 tutz Lta39li e hem Le Ensignatria a Eriftl l 39w rrmvnry 1 L EEF39tI mammal 39 i El I n E F 3 H I t i Scenar 1 3 Frtluctnh Galactle E i Eale 5 a t 5 Variable eets tilt 31 a Eentrihutien Illlariri i Elli Eluln Ell HliIHS r FiltEEI oat 1 Ft E 364m 3 in Units l FE l Ellll ll32 lllTlFlE 3 EP in HE E FE l EMF 5 Et EHlEtHi l in Ean units lPEf39l lll 4 4E EllEfllitt39lll39l SCENARIOZ The Break Even Point in Rsis 20720 The break Even Point as a of capacity is 35 150 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E H39Ermirl39 EEG LeliaLi i TiErnrrrs1 EJEWWLWFWWWeWmnn mm 5i i3EEiJEE iii Iingir EIEM I f gEIi if aquot r12 33 F FE H IIJ R if a I III E Scenario 2 i Preductien Sagacity 32 15 Sale E il iEi llTiarialzle eets VIE iii a Ennirihutinn Margin em S iii EH HI15Hfi 13 Fine East Fl area 5 El in Unita 2r FE i EM 112 Hi l i Hi EF in Rs 3 FE i EMF 5 Elii39 ii HHTHT IE N 31 EEFiin unite iFEiii 35 iiI IEi39Ii39ii iii 23 SCENARIO 21 The Break Even Point in Rs is 17760 The break Even Point as a of capacity is 30 Mitremm Ham Leelmejljtmarieg jr H 53mm L1H Ei DIESELEEIT39E39 E i i w viii ri uiy i itr r 13 j amalgam Scenarln 21 25 F Tn lmiinin Erapanzity 3 20 23 Sallie E E E 39il 2 variable GizaSite iiE iiiT na Enntrilutian Margin E Eliili Siiii 2E EHEEmiliili39ii 29 Fixed Enst Fri EF in Units in FE i EM BE HHEEJHEE 31 EP in IRS iIFC i Eiiili E 39iii39iiiliv HEilii iHEE JEF in mite ilFCiilii Elli EHE iHE iil 13 F G H I J L SCENARIO 22 The Break Even Point in Rs is 22940 The break Even Point as of capacity is 39 15 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU I39I39Iifn 39iJIE Jil Einf l L z lijfLlljtn iill i39lilfi51 39il EJEIH at am 1w Em ism ram mim rah gamer 4331 D i f evw i Emu wh fye ri r 39 I j i5 E 392 El E i Senarm mFrrndiuctiicrn apaeity F a 32G ilam 5 quot135 a Haraicile Errata iii that a 14 13 Eenirihuti ni Iiilrgin DIM F E Iuii 3 H4iilHi42 dai Fixe ail IFE 1533 5 E in Units at 1 FE i EM quot124 HaidiHai EIEF in R5 iFCi i EMF 223i EH iH l 4r iEFini units iFEi i 39 H i i39H i iii 13 F I H i I lIi SCENARIO 23 The Break Even Point in Rsis 38304 The break Even Point as of capacity is 70 E i ielrmi EEEI u Lwiurgjii juMiaiHJJ Em eii i aiw Efg Erasg rrg m gfugi 122 ii j HI I ID E iF 1 eii i J a IF39IrIirueiinm capacity EM 3 Eaie a E iii a iuiarriahile casts hit 5 L15 Gienirillzmiieri Margin 5 cm a id EHIEEeH d Fittea East 3135 EF in Limits 5 FE i EM H55giH55 EIF in 1R5 F6 i came 3331M i I iilli53 EEFEIHJ iLlll it39S iFEi i iii mg iii H5WH52WEIE ED 152 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Net Income NI or Profit Net incomeNlNumber of units sale above BEP in unitsgtltContribution Margin per unit SCENARIO 24 FC Rs 3136 VC Rs 157 8 Rs 185 Capacity 320units Determine the Net income NI if 134 units are sold Formula for Net Income NI Nl Number of units sold above BEP x CM Step 1 Find CM per unit S 185Rs per unit VC 157Rs per unit CM S VC 185 157 Rs 28 CM of Rs28 per unit Step 2 Find EP in units BEP in units FCCM Rs 3136 Rs 28 112 Units Step 3 Find units sold over BEP Units Sold 134 units BEP in units 112 units Number of units sold above BEP in units 134 112 22 Hence ompan had a net income NI of 22 Rs 28 Rs 616 I Microsoft Excel Book1 i Eile Edit ew insert Format Iools gate indow Help tale acne alaal a ale via 2 tltlll ENEl le IE I H IEEEEI 5 a 9 TaEe39iEIEE iil 39 F15 v a i a e c o E E e 39 1 Scenario 24 2 Production capacityr 320 units 3 SaleS 135 Re 4 Variable CoeteVC 15 Re 5 Contribution Margin CM 3 VC 23 F3F4 5 Fixed Coats FE 3135 Re F BEP in units FE i CM 112 FEIF5 3 Units eold 134 9 Units over BEFin uniteiUnite eoldBEFiin units 22 FEFT in HI unite over BEFin units a CM 515 FEF5 SCENARIO 25 FC Rs 3136 VC Rs157 S Rs185 Capacity 320 units How many units must be sold to generate Nl of Rs 2000 Formula for Net Income Number of Units sold above BEP in units Nl CM 1 5 3 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Step 1 Find CM per unit S 185Rs per unit VC 1 Rs per unit CM Rs 28 CM of Rs28 per unit Step 2 Find BEP in units BEP in units FCCM Rs 3136 Rs 28 112 Units Step 3 Find units over m Number of units above BEP Nl ICM Rs 2000Rs 28 71 Units Total units sold 71 Units above BEP 112 BEP Units 183 units A Hioroioll ml t loro 5ioooiooii EllI sologm BEBE Elli l Tj oa n u gg g h 39 1 In 3quot TE or 3 zrtzl Hivl A E E II E F E H J h in Production Eooooity 33D H Solo o 2 too liloriolhlo illooto E E Iii o Contribution lllllari rl Ellll ENE Eii HEEJH 3 o Fixed Eot E Fl 31 o EF in Units or 2 lF l l39 Elli 112 HEEllilE o EFl in Ho FE lC llllF E E i HE39Eil Il o EFlirl onito iFlCi1 2 till 345 HEEll ll 1li l1 so llllEUnltoooorEEiF iEMl Elilil Lini llo WET BEEP 1 llilllll il39ll Ti 9 Total oniioE EF is oooo EFFquot onito 133 EHEHHEE Net loss Net lossNLNumber of units sale below BEP in units x Contribution Margin per unit Alternatively Negative sign with Net Income means Net loss That is Net loss Net Income Number of units sale below BEP in units Number of units sale above BEP in units Thus Net loss Net income Number of units sale above BEP in unitsgtltContribution Margin per unit SCENARIO 26 1 5 4 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU FC Rs 3136 VC Rs157 S Rs185 Capacity 320 units Find the number of units sold if there is a Net Loss NL of Rs 336 M Net loss Number of units sale below BEP in units x Contribution Margin per unit Number of Units below BEP in units NLCM Step 1 Find CM per unit S 185 Rs per unit VC 157 Rs per unit CM 28 Rs CM of Rs28 per unit Step 2 Find EP in units BEP in units FCCM Rs 3136 Rs 28 112 Units Step 3 Find units below BEPNLCM Number of unit sales below BEP Rs 336Rs 28 per Unit 12 Units Hence Total Sold Units BEP in units Number of units sale below BEP 112 12 100 Alternate Method Net loss Net income Number of units sale above BEP in unitsgtltContribution Margin per unit Number of units sale above BEP in units Net loss Contribution Margin per unit 33628 12 Total sold units BEP in units Number of units sale above BEP in units 112 12 100 155 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Hittersift HEEL LII itiIQJQHEIE Eitii glaii are 512 we Meat Tami we Mil3w Dita Mia r lial 3 ea 1131 Haiti 1quot I H 3 mm EVVLBVF EIWLE F lg Eoenarlo 2 Pram31in Eapeeity EEIE e Variant Eats WE 7 Ennirihuti n Mryirn fax3211M EmitE Fired eet Fl 1 me E in Units in Fl 1quot EM E ER in Re FE i39 EZIFiin units iPC39i iiill it m me Hi Unis ever iElFiiiilii WE Limits irter iEF Hlii i i TelWEI units EEIP as above Eli units SCENARIO 27 FC Rs 3136 VC Rs157per unit 8 Rs185 per unit Production Capacity 320 units The company operates at 85 of its capacity Find the Profit or Loss M Number of units above BEP in units x CM NI Step 1 Find CM per unit S 185 Rs per unit VC 1 Rs per unit CM Rs 28 CM of Rs28 per unit Step 2 Find BEP in units BEP in units FCCM Rs 3136 Rs 28 112 Units Step 3 Find units over BEP Units produced 320085 272 Units BEP in units 112 units Number of units over BEP in units 272 112 160 Hence Net income N 160 Units 28 4480 Rs ii i H 11 quot E 333 1155 HE H EITHEH 31 11112 ETH1JDDEH ETQ E H1i 1H E 3 55 THW1JHEEWHG eiEH1iii rH 9 H39Ii EslaH il39i Copyright Virtual University of Pakistan 156 Business Mathematics amp Statistics MTH 302 VU EttaElf ml miniature iii immin fi if Ella 55 ram L39 a Briam 51 3 E E if tl Iiii 15 1 E I E E m it Iquot 53 in El I ll Scenario 2 m Preduetien Espaaei tye PI eEEtt Capaeit ETE m Elle E E E 15 m varrlahle Costs tit 1 115 an Eantrilh tttiion Mriirt EM EB Hl39i iEth 13 we Fitted East an a 31135 E in Unlts it FE mitt 1112 HHEIHHM t inE l9 at EFF 1 Es FIE i Chi E E i l111til till1 il mg EEZPirl units liF39E39i39tl ii MATE liill ltIEIH ltl1ltti mg m Units eater HEP FEEEP39 Units t li ll1tl39ileH11E Profit i Units tarer EF EM itED lill1E il ll1tl4 Company A s year end operating results were as follows Total Sales of Rs 375000 Operated at 75 of capacity Total Variable Costs were Rs 150000 Total Fixed Costs were Rs 180000 What was Company A s BEP expressed in rupees of sales 1 5 7 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 23 STATISTICAL DATA REPRESENTATION OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 22 0 Statistical Data Representation MODULE 5 Statistical data representation Lecture 23 Measures of central tendency Lectures 2425 Measures of dispersion and skewness Lectures 2627 MODULE 6 Correlation Lecture 2829 Line Fitting Lectures 3031 Time Series and Exponential Smoothing Lectures 3233 MODULE 7 Factorials Permutations and Combinations Lecture 34 Elementary Probability Lectures 3536 ChiSquare Lectures 37 Binomial Distribution Lectures 38 MODULE 8 Patterns of probability Binomial Poisson and Normal Distributions Lecture 3941 Estimating from Samples Inference Lectures 4243 Hypothesis testing ChiSquare Distribution Lectures 4445 EndTerm Examination STATISTICAL DATA Information is collected by government departments market researchers opinion pollsters and others Information then has to be organized and presented in a way that is easy to understand EASIS FOR CLASSIFICATION 1 Qualitative Attributes sex religion 2 Quantitative Characteristics Heights weights incomes etc 3 Geographical Regions Provinces divisions etc 4 Chronological or Temporal 5 By time of occurrence Time series 1 5 8 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU TYPES OF CLASSIFICATION There are different types of classifications Oneway One characteristic Population Twoway Two characteristics at a time Threeway Three characteristics at a time METHODS OF PRESENTATION Different methods of representation are Text quotThe majority of population of Punjab is located in rural areas Semi tabular Data in rows Tabular Tables with rows and columns Graphic Charts and graphs TYPES OF GRAPHS O O O O O O O O PICTU Picture graph Column Graphs Line Graphs Circle Graphs Sector Graphs Conversion Graphs Travel Graphs Histograms Frequency Polygon Cumulative Polygon or Ogive RE GRAPH or PICTOGRAPH ln picture graph or pictograph each value is represented by a proportional number of pictures In the example below one car represents 10 ca l39S PICTURE GRAPHS Ears passin school it during the day 3 112m 3 3 511 lE r L PEEL w 1amp2 11 am i E11511 Ii ELEM goron GRAPHS 159 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Sector graphs use the division of a circle into different sectors The full circle is 360 degrees For each percentage degees are calculated and sectors plotted SECTOR GRAPHS Distributien ef employment injuries by eccupatien him lithemine and al quot 39 flan 23955 i39tPJ Ether Iii1 1 a5 39t I Trudi ntju Ii Ii herj LI1rquotI FailEEEJL an m m Example of Sector Graph Leeel e Liih eri 2 W F39IeeeLi re 2 39ii Residential Tit Diner 1 Cemmerciel 31 Agriculture 37 20 COLUMN AND BAR GRAPHS The following slide gives the Proportion of households by size in the form of a Column and Bar graph 160 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Fi f HEEHIE EiEE E E quotquot quot39 rr j I 39i 3 EEJgt r a Him inquot v r i I11 Mam Hm 2131mm Famous pietl i7 LINE GRAPHS Line graphs are the most commonly used graphs A line graph plots data as points and then joins them with a line LINE GRAFHS A student39s height mm a 1m mam EISEI m tlf T a it 1quot ith 1 1 al 9 Ht 493 11 lll lll llll ll 311231SE E EE EQMLEHHLE ig fm r Hr ig i aw Example 161 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Time Needed tn heatE luitahlle Lihra r Hateriallg El Number Inf 539 Emmi13 4D AIJIIE En Lamina 3 Eu ahl Materials 3393 39 IIII I I I I I I I I 5 ID 15 El 25 3 39II39II39IIITIE iim I39I39ll nuum Line chart with multiple series 4 35 an 4 3E 25 2D 15 1D 5 III New Dec Jan Feb Mar Apr 2005 EDGE EDI EDI EDI EDI itItaly EElanain 162 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 24 STATISTICAL REPRESENTATION MEASURES OF CENTRAL TENDENCY PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 18 0 Statistical Representation 0 Measures of Central Tendency LINE GRAPHS Line graphs are the most commonly used graphs In the following graph you can see the occurrence of causes of death due to cancer in males and females You can see that after the age of 40 the occurrence of cancer is much greater in the case of males The line graph of heart diseases also shows that the disease is more prominent in the case of males As you see line graphs help us to understand the trends in data very clearly LINE GRAPHS anges of death 539 nice Hurt the e I m 1H m mn 3 Hale1 iraia E EH39jifzl T k 5 a 39 r m u at 22 3391 4393 ti E39ti fl tiquot i ii 3quot at Ilil 5quot ts fit at quotIf 3951 i 39 WM Another line graph of temperature in 4 cities A B C and D shows that although the general pattern is similar the temperature in city A is lowest followed by D B and C In city C the highest temperature is close to 30 whereas in city A and B it is about 25 The highest temperature in city D is about 28 degrees 1 63 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LINE GRAPHS Temperature efcities ill E t and i939II39I 3812143 rquot v 39 n H 3 JE l 3 1 quot Ir If 1 an W El r39 fl Jib 15311 JAE JWESI JFZHAEdTJAEElWEI t 5r 39 n 41 p 5 I 5 15 l 35 39 E 1 739 a 1n s g 33 FIEAEIHJ JAEPERI ILiquot F51ALd J JAE UWE Centrgl Tendencv The term central tendency refers to the middle value sometime a typical value of the data Measures of central tendency are measures of the location of the middle or the center of a distribution The Mean is the most commonly used measure of central tendency MEAN Also known as the arithmetic mean the mean is typically What is meant by the word average The mean is perhaps the most common measure of central tendency The mean of a variable is given by the sum of all its valuesthe number of values For example the mean of 4 8 and 9 is 4 8 93 7 Example 58 69 73 67 76 88 91 and 74 8 marks Sum 596 Mean 5968 745 Please note that the mean is affected by extreme values MEDIAN Another typical value is the median To find the median of a number of values first arrange the data in ascending or descending order then locate the middle value If there are odd number of data points then median is the middle values lfthere are even number of data points then median is mean of the two middle values Median is easier to find than the mean and unlike the mean it is not affected by values that are unusually high or low 1 64 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 3 611 141919 21 24 31 9 values In the above data there are 9 values So median is The middle value ie 19 MODE The most common score in a set of scores is called the mode There may be more than one mode or no mode at all 2212032114111220321 The mode or most common value is 1 DescriptionExplanation Advantages Disadvantages ihe sum of all the results Quick and easy to May not be representative of the Mean 1nc1uded 1n the sample d1v1ded calculate whole sample by the number of observatlons More tedious to calculate than Median the middle value of all the Takes all numbers into the other two numbers in the sample account equally Can be affected by a few very large or very small numbers the most frequently observed value of the measurements in the sample There can be more than one mode or no mode 0 for an even number of Fairly easy to calculate Mode values the median is Half of the sample Tedious to nd for a large the average of the middle two values 0 for an odd number of values the median is the middle of the all of the values normally lies above the median sample which is not in order 165 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 ORGANISING DATA There are many different ways of organizing data Orqanizinq Numerical Data Grganizing Numariaal Data I llumariml Data drama2252121 at 3821 Fratpamy li rilmliara lnlllttiaa i rilniiana I 2121 21 2321 2130223341 2 Milt 3 I133 4 1 lTalIla I Numerical data can be organized in any of the following forms o The Ordered Array and Stemleaf Display 0 Tabulating and Graphing Numerical Data 0 Frequency Distributions Tables Histograms Polygons 0 Cumulative Distributions Tables the Ogive ta Stern and Leaf Display A stem and leaf display also called a stem and leaf plot is particularly useful when the data is not too numerous NUMERICAL Data in Raw farm aa aallaatatl at as 224 a 21 2 an an 32 3 Mia rtlararl f ra IquotIquotI Sm laat ta Largaat 2124 24 5 2 a a 32 an 41 Stern LE af g V N V 14451 Stem an Laaf 3 quot3 diaplay 4 1 Copyright Virtual University of Pakistan VU 166 Business Mathematics amp Statistics MTH 302 VU Since2120110gtlt21 This is represented in the plot as a stem of 2 and a leaf of 1 The digit at the tenth place is taken as stem and the digit at units place is taken as leaf Similarly 26 is represented in the plot as a stem of 2 and a leaf of 6 Remember a stem is displayed once and the leaf can take on the values from 0 to 9 Example Consider Figure 1 It shows the number of touchdown TD passes thrown by each of the 31 teams in the National Football League in the 2000 season 37quot 33 33 32 29 EB 28 23 22 22 22 21 2121ED2C1919181E1E1E1Ei15 1414141212EEi Figure 1 Number of touchdown passes A stem and leaf display of the data is shown in the Table 1 below The left portion of the table contains the stems They are the numbers 3 2 l and 0 arranged as a column to the left of the bars As in 34 3 is stem and 4 is leaf In 16 l is stem and 6 is leaf Stem and leaf display showing the number of passing touchdowns 32337 2001112223889 l2244456888899 069 Table 1 To make this clear let us examine this Table l more closely In the top row the four leaves to the right of stem 3 are 2 3 3 and 7 Combined with the stem these leaves represent the numbers 32 33 33 and 37 which are the numbers of TD passes for the first four teams in the table The next row has a stem of 2 and 12 leaves Together they represent 12 data points We leave it to you to figure out what the third row represents The fourth row has a stem of 0 and two leaves One purpose of a stem and leaf display is to clarify the shape of the distribution You can see many facts about TD passes more easily in Figure 1 than in the Table 1 For example by looking at the stems and the shape of the plot you can tell that most of the teams had between 10 and 29 passing TDs with a few haVing more and a few haVing less The precise numbers of TD passes can be determined by examining the leaves Tabulatinq gnd Grgphinq UnivaLiate Cgteqorical Data There are different ways of organizing univariate categorical data 0 The Summary Table 0 Bar and Pie Charts the Pareto Diagram 167 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Tabulatinq and Graphing Bivariate categorical Data Bivariate categorical data can be organized as o Contingency Tables 0 Side by Side Bar charts III GRAPHICAL EXCELLENCE AND COMMON ERRORS IN PRESENTING DATA It is important that data is organized in a professional manner and graphical excellence is achieved in its presentation High quality and attractive graphs can be used to explain and highlight facts which otherwise may go unnoticed in descriptive presentations That is why all companies in their annual reports use different types of graphs to present data TABULATING NUMERICAL DATA Group data into classes In some cases it is necessary to group the values of the data to summarize the data properly The process is described below Step 1 Sort Raw Data in Ascendinq Order Data 12 13 17 21 24 24 26 27 27 30 32 35 37 38 41 43 44 46 53 58 Step 2 Find Range Range Maximum value Minimum Value Thus Range 58 12 46 Step 3 Select Number of Classes Select the number of classes The classes are usually selected between 5 and 15 In our example let us make 5 classes Step 4 Compute Class width Find the class width by dividing the range by the number of classes and rounding up Be careful of two things a You must round up not off Normally 32 would round to be 3 but in rounding up it becomes 4 b lfthe range divided by the number of classes gives an integer value no remainder then you can either add one to the number of classes or add one to the class width In our example Class width Range 92 Number of classes 5 Round up 92 to 10 Step 5 Determine Class Boundaries limits Pick a suitable starting point less than or equal to the minimum value You will be able to cover quotthe class width times the number of classesquot values Your starting point is the lower limit of the first class Continue to add the class width to this lower limit to get the lower limit of other classes 1 68 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In this example if we start with 10 we will cover 10 x 5 50 values which is close to our range So let 10 be the lower limit of the first class Continue to add 10 to this lower limit to get the lower limit of other classes 10 201010 302010 403010 504010 To find the upper limit of the first class subtract one from the lower limit of the second class Then continue to add the class width to this upper limit to find the rest of the upper limits Upper limit of first class is 20 1 19 Rest upper limits are 29 1910 39 2910 49 3910 Step 6 Compute Class Midpoints Class Midpoint Lower limit Upper limit 2 First midpoint is 10192 145 Other midpoints are 20292 245 30392 345 40492 445 50592 545 Depending on what you39re trying to accomplish it may not be necessary to find the midpoint Step 7 Compute Class Intervals First class Lower limit is 10 Higher limit is 19 We can write first class interval as 10 to 19 or 10 19 or 10 but under 20quot In 10 but under 20quot a value greater than 195 will be treated as above 20 Similarly other 4 class intervals are 20 29 30 39 40 49 50 59 Important points to remember 1 There should be between 5 and 15 classes 2 Choose an odd number as a class width if you want to have classes midpoints as an integer instead of decimals 3 The classes must be mutually exclusive This means that no data value can fall into two different classes 4 The classes must be all inclusive or exhaustive This means that all data values must be included 5 The classes must be continuous There should be no gaps in a frequency distribution Classes that have no values in them must be included unless it39s the first or last class es that could be dropped 6 The classes must be equal in width The exception here is the first or last class It is possible to have a quotbelow as a first class or and abovequot as a last class Frequency Distribution Count Observations amp Assiqn to Class lnterv Looking through the data shows that there are three values between 10 and 19 Hence frequency is 3 Similarly frequency of other class intervals can be found as follows 10 19 3 169 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 202916 30 39 5 40 49 4 50 59 2 Total frequency 3 6 5 4 2 20 Relative frequency Relative Frequency of a class Frequency of the class interval Total Frequency There are 3 observations in first class interval 10 19 The relative frequency is 320 015 Similarly relative frequency for other class intervals are calculated Percent Relative Frequency If we multiply 015 by 100 then the Relative Frequency 15 is obtained iFFIE HENevuI39smaunuus 1313112124 e26EtEtt tttteemme 5353 Class Frequerwy we 3993 we PM FratHt 113mm EH 3 IE 15 hut 3 ii in iiquot Eillhrtt mide1 till 5 35 5 i hut made 5E 1 3923 If hmtm39uier Eff I l H Tnhl II 1 llll Cumulative Frequency If we add frequency of the second interval to the frequency of the first interval then the cumulative frequency for the second interval is obtained Cumulative frequency of each class interval is calculated below 10 19 3 20 29 3 6 9 30 3936514 40 49365418 50 593654220 Percent cumulative relative frequency This can be calculated same as cumulative frequency except now percent relative frequency for each class interval is considered The percent cumulative relative frequency of the last class interval is 100 as all observations have been added Percent cumulative relative frequency of each class interval is calculated below 10 19 15 20 29153045 30 40 153025 70 40 501530252090 50 60 1530252010100 1 70 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 25 STATISTICAL REPRESENTATION MEASURES OF CENTRAL TENDENCY PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 24 0 Statistical Representation 0 Measures of Central Tendency Part 2 GRAPHING NUMERICAL DATA THE HISTOGRAM Histogram is a bar graph of a frequency distribution in which the widths of the bars are proportional to the classes into which the variable has been divided and the heights of the bars are proportional to the class frequencies Histogram of the example of frequency distribution discussed at the end of lecture 24 is given below GRAPHING NUMERIEAL DATA THE HISTDERAM 313 quot1 I dEI39E 3912 5 a 3quot 34a 2 4 Eli 3 23 3 321 35 3t 33 41 43 443 rm a i I39Iazr m i E39 I I I I I I I I Initlhzriltg MEASURES OF CENTRAL TENDENCY Measures of central tendency can be summarized as under 1 Arithmetic Mean a Arithmetic mean for discrete data i Sample Mean ii Population Mean b Arithmetic Mean for grouped data 2 Geometric Mean 3 Harmonic Mean 4 Weighted Mean 5 Truncated Mean or Trimmed Mean 6 Winsorized Mean 7 Median l 7 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU a Median for grouped data b Median for discrete data 8 Mode a Mode for grouped data b Mode for discrete data 9 Midrange 10 Midhinge As you see it is a long list However if you look closely you will find that the main measures are Arithmetic Mean Median Mode All the above measures are used in different situations to understand the behavior of data for decision making It may be interesting to know the average median or mode salary in an organization before the company decides to increase the salary level Comparisons with other companies are also important The above measures provide a useful summary measure to consolidate large volumes of data Without such summaries it is not possible to compare large selections of data EXCEL has a number of useful functions for calculating different measures of central tendency Some of these are explained below You are encouraged to go through EXCEL Help file for detailed descriptions of different functions For selected functions the help file has been included in the handouts The examples are also from the help files AVERAGE Returns the average arithmetic mean of the arguments Syntax AVERAGEnumber1number2 Number1 number2 are 1 to 30 numeric arguments for which you want the average Remarks 0 The arguments must either be numbers or be names arrays or references that contain numbers o If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included Example An example of AVERAGE is shown below Data was entered in cells A4 to A8 The formula was AVERAGEA4A8 The 11 is shown in cell A10 1 72 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Buul Elle Edit ElevJ insert Fgrrnat Innls Eata indnw Help Fading3 F39DF H at n v E I 3 v lt3 E E El SLIP391 j x J AVEHAGEMampEI W E L D E F 3 2 3 AVERAGEmumbeH numberE 4 1 I 5 F a Q r 2 2 AUERAG EA lA3 1 1 AVERAGEA Calculates the average arithmetic mean of the values in the list of arguments In addition to numbers text and logical values such as TRUE and FALSE are included in the calculation Syntax AVERAGEAvaIue1value2 Value1 value2 are 1 to 30 cells ranges of cells or values for which you want the average Remarks 0 The arguments must be numbers names arrays or references 0 Array or reference arguments that contain text evaluate as 0 zero Empty text 39quot39 evaluates as 0 zero If the calculation must not include text values in the average use the AVERAGE function 0 Arguments that contain TRUE evaluate as 1 arguments that contain FALSE evaluate as 0 zero Example A 1 Data 2 10 3 7 4 9 5 2 1 73 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 6 Not available 7 Formula Description Result Average of the numbers above and the text AVERAGEAA2A6 quotNot Availablequot The cell with the text quotNot availablequot is used in the calculation 56 AVERAGEAA2A5A7 Average of the numbers above and the empty cell 7 MEDIAN Returns the median of the given numbers The median is the number in the middle of a set of numbers that is half the numbers have values that are greater than the median and half have values that are less Syntax MEDIANnumber1number2 Number1 number2 are 1 to 30 numbers for which you want the median Remarks 0 The arguments should be either numbers or names arrays or references that contain numbers Microsoft Excel examines all the numbers in each reference or array argument o If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included o If there is an even number of numbers in the set then MEDIAN calculates the average of the two numbers in the middle See the second formula in the example Example The numbers are entered in cells A14 to A19 In the first formula MEDAN12345 the actual values are specified The median as you see is 3 in the middle In the next formula MEDANA14A19 the entire series was specified There is no middle value in the middle Therefore the average of the two values 3 and 4 in the middle was used as the median 35 Copyright Virtual University of Pakistan 174 Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Buuk1 Elle Edit ElevJ insert Fgrrnat IDDIS Eata irtjaw elp ACIDQE F39DF Eggn nvz ml fIDTB l g EEE E15 3 A E L D E F G 12 MEDIANmumber numberz n 13 Ir1 1 15 2 TE 3 I I IE IEI MEDIANH 2 1 5 5 MEDIANA14A1S El 21 22 MODE Returns the most frequently occurring or repetitive value in an array or range of data Like MEDIAN MODE is a location measure Syntax MODEnumber1number2 Number1 number2 are 1 to 30 arguments for which you want to calculate the mode You can also use a single array or a reference to an array instead of arguments separated by commas Remarks 0 The arguments should be numbers names arrays or references that contain numbers o If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included o If the data set contains no duplicate data points MODE returns the NA error value In a set of values the mode is the most frequently occurring value the median o is the middle value and the mean is the average value No single measure of central tendency provides a complete picture of the data Suppose data is clustered in three areas half around a single low value and half around two large values Both AVERAGE and MEDIAN may return a value in the relatively empty middle and MODE may return the dominant low value Example The data was entered in cells A27 to A32 The formula was MODEA27A32 The answer 4 is the most frequently occurring value 1 75 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU crusuft Excel Buuk1 Elle Edit ew insert Fgrrnal Innls Eata irtjaw Help adage F39DF DEE nvE lf1oTBEamp EEE I 535 j 24 A C D E F G 25 MDEfnumber39ir iLm berz 25 2 55 28 29 3 31 32 33 1 MDEA2A32 34 COUNT FUNCTION Counts the number of cells that contain numbers and also numbers within the list of arguments Use COUNT to get the number of entries in a number field that39s in a range or array of numbers Syntax COUNTvalue1value2 Value1 value2 are 1 to 30 arguments that can contain or refer to a variety of different types of data but only numbers are counted Remarks 0 Arguments that are numbers dates or text representations of numbers are counted arguments that are error values or text that cannot be translated into numbers are ignored o If an argument is an array or reference only numbers in that array or reference are counted Empty cells logical values text or error values in the array or reference are ignored If you need to count logical values text or error values use the COUNTA function Example A 1 Data 2 Sales 3 1282008 4 5 19 6 2224 7 TRUE 176 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 8 DIVO Formula COUNTA2A8 COUNTA5A8 COUNTA2A82 39 FREQUENCY Description Result Counts the number of cells that contain numbers in the list above 3 Counts the number of cells that contain numbers in the last 4 rows of the list 2 Counts the number of cells that contain numbers in the list and the value 2 4 Calculates how often values occur within a range of values and then returns a vertical array of numbers For example use FREQUENCY to count the number of test scores that fall within ranges of scores Remember FREQUENCY returns an array it must be entered as an array formula Syntax FR EQUENCYdataarray binsarray Dataarray is an array of or reference to a set of values for which you want to count frequencies f dataarray contains no values FREQUENCY returns an array of zeros Binsarray is an array of or reference to intervals into which you want to group the values in dataarray f binsarray contains no values FREQUENCY returns the number of elements in dataarray Remarks 0 FREQUENCY is entered as an array formula after you select a range of adjacent cells into which you want the returned distribution to appear 0 The number of elements in the returned array is one more than the number of elements in binsarray The extra element in the returned array returns the count of any values above the highest interval For example when counting three ranges of values intervals that are entered into three cells be sure to enter FREQUENCY into four cells for the results The extra cell returns the number of values in dataarray that are greater than the third interval value 0 FREQUENCY ignores blank cells and text 0 Formulas that return arrays must be entered as array formulas A B Scores Bins 79 70 85 79 78 89 85 50 81 95 88 ACOW IQUIhWNl 97 177 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Formula Description Result FREQUENCYA2A1 Number of scores 0 3235 ess than or equal to 39 7o 1 Number of scores in the bin 7179 2 Number of scores in the bin 8089 4 Number of scores greater than or equal to 90 2 Example Note The formula in the example must be entered as an array formula After copying the example to a blank worksheet select the range A13A16 starting with the formula cell Press F2 and then press CTRLSHFTENTER If the formula is not entered as an array formula the single result is 1 ARITHMETIC MEAN GROUPED DATA Below is an example of calculating arithmetic mean of grouped data Here the marks Classes and frequency are given The class marks are the class mid points calculated as average of lower and higher limits For example the average of 20 and 24 is 22 The frequency f is multiplied by the class mark to obtain the total number In first row the value of fx is 1 x 22 22 The sum of all fx is 1950 The total number of observations is 50 Hence the arithmetic mean is 195050 39 Marks Frequency Class Marks fX 2024 1 22 22 2529 4 27 108 3034 8 32 256 3539 1 1 37 407 4044 15 42 630 4549 9 47 423 5054 2 52 104 TOTAL 50 1950 n 50 SumfX 1950 Mean 195050 39 Marks EXCEL Calculation The above calculation would be common in business life Let us see how we can do it using EXCEL The basic data of lower limits is entered in cell range A54A60 The data of higher limit is entered in cells B54B60 Frequency is given in cell range D54D60 Class midpoints or class marks were calculated in cells F54F60 ln cell F54 the formula A54B542 was used to calculate the class mark This formula was copied in other cells F55 to F60 The value of fx was calculated in cell H54 using the formula D54F54 This formula was copied to other cells H55 to H60 Total frequency was calculated in cell D61 using the formula SUMD54D60 Sum of fx was calculated in cell H61 using the formula SUMH54H60 Mean was calculated in cell H62 using the formula ROUNDH61D610 178 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnsn Excel Lecturej ile Edit Eiew insert Fgrmat Innls gate indnw elp ndings F39DF Dusss 45 23 33444 TEUvBIHEEEE h v s s s H53 3 4 E r U E F G L I J K 53 Marks Frequencym Class Markm fit 54 20 24 1 22 22 D545F54 55 25 2B 4 27 108 55 30 8 132 256 5 35 39 1 1 37 40 as 40 44 15 42 5130 59 45 4B 9 47 4213 an 50 54 2 52 104 E1 TTAL 50 1950 SUMH54HEU j Mean 33 HE1IDE1 53 179 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 26 STATISTICAL REPRESENTATION MEASURES OF DISPERSION AND SKEWNESS PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 25 0 Statistical Representation 0 Measures of Dispersion and Skewness FREQUENCYEXAMPLE FREQUENCY Function calculates how often values occur within a range of values and then returns a vertical array of numbers For details see handout for lecture 25 The syntax is FREQUENCYdataarraybinsarray Microsoft Excel Frequency Elle Edit ew insert Fgrmat Innis gate window Help all5 1SST1317 Ti 139 539 X D i 3139 ci lb r SE39ESiEi i mmtr ArieI mm are E EEJEEEE Javen DH r A E C D I E F G H J H LT i FREUENCYidetearraybineerrey 2 Scere Bins 3 70 Step 1 Select the range A13 16 starting 4 85 with the f rmula cell 5 89 Step 2 Press F2 e 85 Step Fees CTRLSHIFTENTER i 50 1 FeEUENCYn3n11B3BE a 31 2 g 95 4 m 33 2 H 9 The data was entered in cells A3 to A11 The Bins array which gives the limits 70 79 and 89 were entered in cells B3 to B5 Select cells BC to 3010 one more than the limits Type the formula FREQUENCYA3A1B3B5 Then CTRLShiftEnter were pressed to indicate that we are entering an array formula The result is given in cells B7 to 810 We can interpret the result as the frequency oflt Less than or equal to 70 is 1 71 to 79 is 2 80 to 88 is 4 89 and above is 2 Application of FREQUENCY function in Frequency Distribution Consider the example at the end of lecture 24 1 80 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 12 mi 31 act Class Franyrenrr we 3996 we Elm FratHt RelatesHirer El 3 15 15 ifth If ii EH Elihu IdleI ill 5 15 25 43111311 5H 4 Elam nder Eff I ll 1 Tab El 1 llll Letus find Frequency distribution using Excel function FREQUENCY Microsoft Excel Bauld Elle g agile l ile 39 10 1 E I u I E24 v a a e i D E Data Class 12 Lower limit Upper Limit Edit Eiew insert Fgrmal Inels gate indew elp IIIIII 5 in Frequency M J J I Jquot LCI LEI DMbU Im 2 Total M C Write data in column A In column C and D write the class s lower and upper limits Here we will take class s upper limit as bins Select the cells E3 to E8 under Frequency heading one cell more than the number of classes Type FREQUENCYA2A21D3D7 Press CtrlShiftEnter Your Frequency column will get filled with above mentioned values FREQUENCY POLYGONS Frequency polygon is a line graph obtained from a frequency distribution by joining with straight lines points whose abscissa are the midpoints of successive class intervals and whose ordinates are the corresponding class frequencies 1 8 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU GRAPHING NUMERIEAL DATA THE FREQUENle PDLVGDN Data in arderad arrayI 1313133133 33 33 3quot 3quot 33 33 35 37quot 33 31 3333335353 Elam luitlminta CUMULATIVE FREQUENCY Relative frequency can be converted into cumulative frequency by adding the current frequency to the previous total In the slide below the first interval has the relative as well as cumulative frequency as 3 In the next interval the relative frequency was 6 it was added to the previous value to arrive at 9 as cumulative frequency for interval 20 to 30 What it really means is that 9 values are equal to or less than 30 Similarly the other cumulative frequencies were calculated The total cumulative frequency 20 is the total number of observations Percent Cumulative frequency is calculated by dividing the cumulative frequency by the total number of observations and multiplying by 100 For the first interval the cumulative frequency is 320100 15 Similarly other values were calculated TAULATING NUMERIC L AT l EUMLIL tTNE FREQUENCY ata l l El E array 1213112123232321213332353133313333355333 W EEillumulatiue ch35 ime iFlplrI39tiun Frpqllpnqr l hutm 39m 3 15 mhutunl 39iil 9 5 Mllutm 39 l 14 TI Whutunla39m 13 H mhl l 39 II llll CUMULATIVE POLYGONOGIVE From the cumulative relative frequency polygon that starts from the first limit not mid point as in the case of relative frequency polygons can be drawn Such a polygon is called Ogive The maximum value in an Ogive is always 100 Ogives are determining cumulative frequencies at different values not limits 1 82 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU GRAPHING NUMERICAL ATA THE EUMULATWE a PnLveeHl Data in ordered arrayr 121317 21ELLE 128212303235333 1J3M5353 iiglw Class Elcunclaries NEE I39i39i39iu39pnints TABULATING AND GRAPHING UNIVARIATQATA Univariate data one variable can be tabulated in Summary form or in graphical form Three types of charts namely Bar Charts Pie Charts or Pareto Diagrams can be prepared TALlLATl AND ERAF HIN G EATEEDRIEAL DATA UNIVARIATE ATA Emegoricnl IIItutu I Talmliiing Data Gml li Data TI39IEI Stl39lrrrtll39jlr Tallies Pie Elmrte Diagram SUMMARY TABLE A summary table is built specifically from detailed data It contains summaries of the data and is used to speed up analysis A typical Summary Table for an investor s portfolio is given in the slide The variables such as stocks etc are the categories The table shows the amount and percentage Investment Amountin Percentage Category thousand Rs Stocks 465 4227 Bonds 32 2909 Cash 155 1409 Deposit Savings 16 1455 TOTAL 110 100 1 83 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU BAR CHART The data of Investor s portfolio can be shown in the form of Bar Chart as shown below This chart was prepared using EXCEL Chart Wizard The Wizard makes it very simple to prepare such graphs You must practice with the Chart Wizard to prepare different types of graphs CHART F 119 Ml II39III quE RPS RTF Ll I39l quotilitIF39i F39IZIFTIZIIIIZI Gamngr II I B on E III Iii III 213 Elli 41339 SD mpzuntln PIE CHARTS Pie Charts are very useful charts to show percentage distribution These charts are made with the help of Chart Wizard You may notice how Stocks and bonds stand out PIE CHART fIF DR MI IH IuI E E35 RTF Ll An39hz rlllt Iwe l in H5 Savings 15 Steel5 III 1241 M39H P 39 l t fl39 r ll l l SE5 to lienaan 1 H t PARETO DIAGRAMS A Pareto diagram is a cumulative distribution with the first value as first relative frequency in this case 42 The point is drawn in the middle of bar for the first category stocks Next the category Bonds was added The total is 71 Next the savings 15 were added to 71 to obtain cumulative frequency 86 Adding the 14 for CD gives 100 Thus the Pareto diagram gives both relative and cumulative frequency 1 84 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU HAGRAM PA RETC Axis for bar chart shows EH invested in each category Axis for line Euth shears eLrnIJ atiue EH invested CONTINGENCY TABLES Another form of presentation of data is the contingency table An example is shown in the slide below The table shows a comparison of three investors along with their combined total investment TALILATING CATEGDRIEAL DATA WARIATE DATA en ngeney Table Ill39u39i l39l39 39lt in mlmmls If mum Investment huestzr P I39mester El twat3r I ctal Category Stacks 35 55 25 123 Bands 32 am 13 35 ED 155 EIII 135 13 Savings 1E E Tquot 51 113 11D la 1quot E 321 SIDE BY SIDE CHARTS The same investor data can be shown in the form of side by side charts where different colours were used to differentiate the investors This graph is a complete representation of the contingency table GRAPHING CATEGDRJEAL DATA EIVARIATE DATA SIIIEIy ERIE El rt Dem paring I1 H latrs 5 GEOMETRIC MEAN 1 85 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Geometric mean is defined as the root of product of individual values Typical syntax is as under Gx1x2x3xnquot1n Example Find GM of 130 140 160 GM 130140160 A 13 1428 HARMONIC MEAN Harmonic mean is defines as under HM n n 1x11x21xn Sum1xi Example Find HM of 10 8 6 HM 3 766 1101816 QUARTILES Quartiles divide data into 4 equal parts Syntax 1st Quartile Q1 n14 2nd Quartile Q2 2n14 3rd Quartile Q3 3n14 Grouped data Qi ith Quartile l hfSum f4i of l lower boundary h width of CI of cumulative frequency DECILES Deciles divide data into 10 equal parts Syntax 1st Decile D1n110 2nd Decile D2 2n110 9th Deciled D9 9n110 Grouped data Qi ith Decile i129 l hfSum f10i of l lower boundary h width of CI of cumulative frequency PERCENTILES Percentiles divide data into 100 equal parts Syntax 1st Percentile P1n1100 2nd Decile D2 2n1100 99th Deciled D9 99n1100 Grouped data Qi ith Decilei129 l hfSum f100i of l lower boundary h width of CI of cumulative frequency 186 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Symmetrical Distribution mean median mode Positively Skewed Distribution Tilted to left mean gt median gt mode Negatively Skewed Distribution mode lt median lt mean Tilted to right EMPIRICAL RELATIONSHIPS Moderately Skewed and Unimodal Distribution Mean Mode 3Mean Median Example mode 15 mean 18 median Median 13mode 2 mean 1315 218 15363 513 17 A trimmed or truncated mean is a measure of central tendency and is one type of modified mean In this first we sort the data Then according to the problem discard an equal number of data at both ends Most often 25 percent of the ends are discarded That is values below first quartile and above third quartile are removed Mean of the remaining data is called trimmed mean or truncated mean WINSORIZED MEAN It involves the calculation of the mean after replacing given parts of a data at the high and low end with the most extreme remaining values Most often 25 percent of the ends are replaced That is values below first quartile and above third quartile are replaced Example Find trimmed and winsorized mean 91 92 93 92 92 99 Arrange the data in ascending order 91 92 92 92 93 99 Position of Q1 614 175 Q1 2nd value approximately 92 Position of Q3 3614 525 Q3 5th value approximately 93 Trimmed Mean 92 92 92 93 4 9225 Winsorized Mean 92 92 92 92 93 93 6 9233 DISPERSION OF DATA Definition The degree to which numerical data tend to spread about an average is called the dispersion of data TYPES OF MEASURES OF DISPERSION Absolute measures Relative measures coefficients DISPERSION OF DATA Types Of Absolute Measures Range l 87 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Quartile Deviation Mean Deviation Standard Deviation or Variance Types Of Relative Measures Coefficient of Range Coefficient of Quartile Deviation Coefficient of Mean Deviation Coefficient of Variation Copyright Virtual University of Pakistan VU 188 Business Mathematics amp Statistics MTH 302 VU LECTURE 27 STATISTICAL REPRESENTATION MEASURES OF DISPERSION AND SKEWNESS PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 26 0 Measures of Dispersion and Skewness MEASURES OF CENTRAL TENDENCY VARIATION AND SHAPE FOR A SAMPLE There are many different measures of central tendency as discussed in the last lecture handout These include Mean Median Mode Midrange Quartiles Midhinge Range lnterquartile Range Variance Standard Deviation Coefficient of Variation Rightskewed Leftskewed Symmetrical Distributions Measures of Central Tendency Variation and ShapeExploratory Data Analysis FiveNumber Summary BoxandWhisker Plot Proper Descriptive Summarisation Exploring Ethical Issues Coefficient of Correlation MEANS The most common measure of central tendency is the mean The slide below shows the Mean Arithmetic Median Mode and Geometric mean Another mean not shown is the Harmonic mean Each of these has its own significance and application The mean is the arithmetic mean and represents the overall average The median divides data in two equal parts Mode is the most common value Geometric mean is used in compounding such as investments that are accumulated over a period of time Harmonic mean is the mean of inverse values Each has its own utility The slide shows the formulas for mean and geometric mean Measures nf Central Tendeney It entral TEI IdEnW I 2 33114135 Mean iquot 24339 f f39 KIIII J quotT lHarmnnit Mean l 189 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU THE MEAN The formula for Arithmetic Mean is given in the slide It is the sum of all values divided by the number In the case of mean of a sample the number n is the total sample size When the sample data is to be used for estimating the value of mean then the number is reduced by 1 to improve the estimate In reality this will be a slight overestimation of the population mean This is done to avoid errors in estimation based on sample data that may not be truly represented of the population The Mean Arithmetie Average The Arithmetic Average ef data trainee ETfn m E1I2 HI fu 71 H Sample Hean SEmPIE Size F 1 3 IN E1f2 II f 1 N H Pcleaticn Mean P ma n Size EXTREME VALUES An important point to remember is that arithmetic mean is affected by extreme values In the following slide mean of 5 values 1 3 5 7 and 9 is 5 In the second case where the data values are 1 3 6 7 and 14 the value 14 is an outlier as it is considerably different from the other values In this case the mean is 6 in other words the mean increased by 1 or about 20 due to the outlier While preparing data for mean it is important to spot and eliminate outlier values The Me an The Meat Centnten Meaeure ef Central Tentleneyr Affeeteel by Eatren39ie valuee utliere 24 3 h 0E4aaam I214 39 mm 5 I Mm 190 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU THE MEDIAN The Median is derived after ordering the array in ascending order Ifthe number of The Median lmpertant Meaeure ef Central Tendeney In an erdered array the median ie the middle number If n ie add the median ie the middle num er If Fl ie even the median ie the average at the 2 middle numbera observations is odd it is the middle value otherwise it is the the average of the two middle values It is not affected by extreme values The Median Net Affected by Extreme Values 392 If Fee llillild lie l6EFeellil l IIquot 392 I4e39e 5 tamer THE MODE The mode is the value that occurs most frequently In the example shown on the slide 8 is the most frequently occurring value Hence the mode is 8 Mode is also not affected by extreme values 191 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The Mede A Meeeure ef Centre Ter39rrJer ieyr Value that eeure Meet then Net Affected by Extreme Values E gi gee li dEEFEEEIIIII I213 IWIE B An important point about Mode is that there may not be a Mode at all no value is occurring frequently There may be more than one mode The mode can be used for numerical or categorical data The slide shows two examples where there is no mode or there are two modes The Mede There May Net lee e Mede There lil39leyr lee Several Medee Lleed fer Either Numeri eel er Eeteerieel eta eeeeeee g g 3931 3 4139 393 l39lelutmle Tmhlerl RANGE Another measure of dispersion of data is the Range It is the difference between the largest and smallest value The slides show examples where the value of range was calculated 1 92 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU ZEIU N C F ATA Range H Largest Smallest Iul alue Example Fintl range 31 13 43 13 Elquot 32quot 1E 33 33 3 24 El Largest value Elnalleet 13 Range 13 12 31 The Range Meaaure efjtifariatien Etifferenee etween Largeat Smalleat Deeeraatiena Elan 93 Largeat Elnalleat Value Igneree Haw Data Are Dietriuteel Range12T5 RargelET ecuJean 139101111 III 5 III 0 1391111112 MIDRANGE Midrange is the average of smallest and largest value In other words it is half of a range Midrange is affected by extreme values as it is based on smallest and largest values Mid range Meaaure ef Central Teniler39ieisr Aeerae ef 3m alleat and Largeat Ubaereatian I z f I Midrange w 193 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Midrange ffeeted by Extreme value 3 ESSJ J I2Ia4ua rl2l l39l Mtlmlm 5 thngaa 3 QUARTILES Quartiles are not exclusively measures of central tendency However they are useful for dividing the data in 4 equal parts each containing 25 of the data So there are three quartiles 25 data falls below first quartile 50 below second and 75 below third quartile Quartiles t a ITIEHEUI39E Elf E tl l tendency Split l dEl E data i t 4 HUEWEFE 25 25 25 25 I 2 313 Position of ith quartile 39 n 1 4 Where i 1 2 3 and n is the number of data points To understand the procedure of finding the value of each quartile let us consider an example Example Find first second and third quartile of the following data 112217161221161318 Arrange the data in ascending order 11 12 13 16 16 17 1821 22 Here n number of data points 9 Position of first quartile Q1 M 25 4 Since we get position of first quartile as decimal fraction so we proceed as follows 01 2nd value 05 x 3rd value 2nd value 12 05 x 13 12 12 05 x 1 125 Position of second quartile Q2 2 x 9 1 5 4 1 94 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU So 02 5th value 16 Position of third quartile Q3 3 x 9 1 75 4 So 03 7th value 05 x 8th value 7th value 18 05 x 21 18 1805x3 195 MIDHINGE Midhinge is the average of the first and third quartiles Midhinge Q1 Q3 2 QUARTILE DEVIATION Quartile Deviation is the average of 1st and 3rd Quartile QD Q3 Q1l 2 Example Find QD of the following data 14 10 17 5 9 20 8 24 22 13 Here number of data points n 10 Position of Q1 n14 1014 275 So Q1 2nd value 075 x 3rd value 2nd value 8075x98 8 075 x1 875 Position of Q3 3 x 10 1 4 3 x 275 825 So Q3 8th value 0259th value 8th value 20 025 x 22 20 2050 QD 2050 875 5875 2 SHAPE OF DISTRIBUTION 1 Symmetrical Distribution 2 Asymmetrical Distribution a Rightskewed or positivelyskewed distribution b Leftskewed or negativelyskewed distribution We can find the shape of distribution using 5number summary 5NUMBER SUMMARY 5 number summary is o Smallest value 1St Quartile Q1 MedianQ2 3rd Quartile 03 Largest value 195 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 90X AND WHISKER PLOT Box and whisker plot shows 5number summary Eaplaratnry ata Analysis anatantlwhialtar Graphi al iSFtlay at data uaing numlaar auntrnaryr Madian i HEWE39EH 1 EEG i 4 E 3 ll 12 The plot gives a good idea about the shape of the distribution as detailed below Box and whisker plot for symmetrical leftskewed and rightskewed distribution are shown below iatrilautian Shape Er Baaeand whiakar Plata LeftEltawad Eymmetric ightEltawad q Matlim Malian Malian 4 l lIl I 1 Symmetrical Distribution Data is perfectly symmetrical if 0 Distance from Q1 to Median Distance from Median to Q3 0 Distance from Xsmauest to Q1 Distance from Q3 to Xlargest That is Median Midhinge Midrange 2 AsymmetricalDistribution a Rightskewed distribution Distance from Xlargest to Q3 greatly exceeds distance from Q1 to Xsmauest That is Median lt Midhinge lt Midrange b Leftskewed distribution Distance from Q1 to Xsmauest greatly exceeds distance from Xlargest to Q3 1 96 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU That is Median gt Midhinge gt Midrange Example Suppose there are nine homes valued at Rs 1500000 Rs 1400000 Rsl600000 Rs1500000 Rsl600000 Rsl700000 Rsl600000 Rs1500000 and Rsl600000 in a new area There is one small empty lot in the area and someone builds small home with a valuation of Rs 200000 Find either the frequency polygon of these values is negatively skewed or positively skewed Following sample represents annual costs in 000 Rs for attending 10 conferences 130 145 149 152 152 154 156 162 17 231 Find 5 number summary and shape of the distribution Position of Q1 10 1 4 275 Q1 145 075 x 149 145 148 Position on3 3 1o 14 825 03 162 025 x 17 62 164 Median 152 154 2 153 Thus 5 number summary is o Smallest value 130 1St Quartile Q1 148 MedianQ2 153 3rd Quartile Q3 164 Largest value 231 Midrange Largest value Smallest value 231 130 1805 2 2 Midhinge Q1 Q3 148 164 156 2 2 Find the relationship between median midhinge and midrange Median lt Midhinge lt Midrange Thus the shape of distribution is right skewed SUMMARY MEASURES The slide shows summary of measures of central tendency and variation In variation there are range Interquartile range standard deviation variance and coefficient of variation The measures of central tendency have been discussed already 197 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Summary Maaurs I lll39l39ll39l39l l39jf Measures I ICentIril Tattlency I Varia un I tmtlnrtl El I Wi ti ill Interquartile Marian Im39 ml l I II Range C ef ciant c L mi 0539 I IV ri n I if 139Ii39rifil39l tllliliIFI MEASURES OF VARIATION In measures of variation there are the sample and population standards deviation and variance the most important measures The coefficient of variation is the ratio of standard deviation to the mean in Masu res Variation iiaria nn lirarime lemma Deviat39nn Coef ciemnf E l ifaria nn ME 395quot W cr 1nn Var39nme 531111 5131113111 IIItEIqIEII39 h Rang I Darth INTERQUARTIE RANGE Interquartile range is the difference between the 1st and 3rd quartile 1 98 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU lnte rquertile Range Meeeure ef erietien Alee Knewn ee Mideereed pureed in the Middle ifferer39iee etereen Thir E Firet IZZELIer39tilee Interquertile Renge Datainfh39 ugll ng39 11 1 13 15 15 139 139 11 1 E 393 1re1e55 Net A eeteel by Extreme Veluee 199 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 28 MEASURES OF DISPERSION CORRELATION PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 27 0 Measures of Dispersion 0 Correlation MODULE 6 Module 6 covers the following Correlation Lecture 2829 Line Fitting Lectures 3031 Time Series and Exponential Smoothing Lectures 3233 VARIANCE Variance is the one of the most important measures of dispersion Variance gives the average square of deviations from the mean In the case of the population the Verienee II39I I PEI NIGHT MEESU I39E Elf V311 ti SHOWS V ti b l l t tI39IE MEEI I Fer the Population a 23933 1 N Fer the Sample 5 Elia 5fl 11 1 Fer the Pepuletien uee III lF er the Sahele uee n if in the tlenemintier fl inithe tIenerniintgte31 Sum of square of deviations is divided by N the number of values in the population In the case of variance for the sample the number of observations less 1 is used STANDARD DEVIATION Standard deviation is the most important and widely used measure of dispersion The square root of square of deviations divided by the number of values for the population and number of observations less 1 gives the standard deviation 200 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example Standard emetlen Meet lmpertaht Meeeure ef Varietieh Ehewe 1I39l39r erietieh eheut the Mean Sam e Ll i39l ef n1 eeeurem em 35 the 5 hSEW ti E F X we rr Fur the Semi1e FIJIquot Fepuletieh F ch F llulatien nee H Fertiliei iaihhleiiiee hif in the lenemin m 7 in the hequotmni m rliy Let us do this for a simple dataset shown below The Number of Fatalities in Motorway Accidents in one Week Number of fatalities Day X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28 The arithmetic mean number of fatalities per day is izgz n 7 4 Taking the deviations of the Xvalues from their mean and then squaring these deviations we obtain X 4 6 2 0 3 5 8 20 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Hence 2X 02 42 is now positive and this positive value has been achieved without bending the rules of mathematics Averaging these squared deviations the variance is given by Variance The variance is frequently employed in statistical work but it should be noted that the gure achieved is in squared units of measurement In the example that we have just considered the variance has come out to be 6 squared fatalities which does not seem to make much sense In order to obtain an answer which is in the original unit of measurement we take the positive square root of the variance The result is known as the standard deviation Standard Deviation Lye W n l 265 fatalities The formulae that we have just discussed are valid in case of raw data In case of grouped data ie a frequency distribution each squared deviation round the mean must be multiplied by the appropriate frequency figure ie 202 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU COMPARLNG STANDARD DEVIATIONS In many situations it becomes necessary to calculate population standard deviation SD on the basis of SD of the sample where n1 is used for Camparing Standard amatlana Data ID 139 11 15 1397 18 18 1 143 Meanl 1 TIE dig 3 33535 N tuftind far the atantlartl HWl l ll Ia lard ar far data canairlarad aa a Sample division In the slide the same data is first treated as the sample and the value of SD is 42426 When we treat it as the population the SD is 39686 which is slightly less than the SD for the sample You can see how the sample SD will be overestimated if used for the population COMPARLNG STANDARD DEVIATIONS The slide shows three sets of data A B and C All the three datasets have the same mean 155 but different standard deviations A s3338 B s09258 and C s457 It is clear that SD is an important measure to understand how different sets of data differ from each other Mean and SD together form a complete description of the central tendency of data Camparing Standard H amatlana LEW i a a a E a a a MPH5 II I 13 III If IE IF I I3 2 2 WEE g Men 155 quot I I1 I l3 E l a a 2 2 53953 Mamii g a4 l l iii a 14 rs IE a COEFFICIENT OF VARIATION 203 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Cempering Ceeffieie rit ef Verietien Steelt ill average price laet year Re 5131 Standard deuiatien Re 5 my 1DWD age price Iaet year Re tee tandem tlevietien RS 5 lEeel fieierit ef vana en Steel t tZ lf 113 Steelt E int 5 Coefficient of variation CV shows the dispersion of the standard deviation about the mean In the slide you see two stocks A and B with CV10 and 5 respectively This comparison shows that in the case of stock A there was a much greater variation in price with reference to the mean EXAMPLE Suppose that in a particular year the mean weekly earnings of skilled factory workers in one particular country were 1950 with a standard deviation of 4 while for its neighboring country the figures were Rs 75 and Rs 28 respectively From these figures it is not immediately apparent which country has the GREATER VARIABILITY in earnings The coefficient of variation quickly provides the answer COEFFICIENT OF VARIATION For country No 1 4 5 x 100 205 per cent And for country No 2 ExlOO 373 per cent 75 From these calculations it is immediately obvious that the spread of earnings in country No 2 is greater than that in country No l and the reasons for this could then be sought MEAN DEVIATION Other useful measures are Mean Deviation about the Mean and median The mean deviation of a set of data is defined as the arithmetic mean of the deviations measured either from the mean or from the median The symbolic definition of the mean deviation about the mean is 1 MD Zx x for sample n i1 204 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 1 N MD Z x u For apopulatlon N i1 Note that first take the absolute value of difference of data point and the mean and then add those absolute values The absolute value of a number is the number without its sign Example Calculate the mean deviation from mean and median of the following set of examination marks 45 32 37 46 39 36 41 48 and 36 Solution Median 39 Q Mean 2 Mean 3609 40 marks x xi xi xi Median 32 8 8 7 36 4 4 3 36 4 4 3 37 3 3 2 39 1 1 0 41 1 1 2 45 5 5 6 46 6 6 7 48 8 8 9 Sum 330 o 40 39 lel 4 Mean Deviation from Mean 409 44 marks n lel Median Mean DeV1at10n from Med1an 399 43 marks n 205 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In excel ABS function returns the absolute value of a number Syntax ABSnumber Number is the real number of which you want the absolute value For the data organized into a grouped frequency distribution having k classes with midpoints x1 x2 x3 xk and the corresponding frequencies f1 f2 f3 fi 2 fi N the mean deviation about mean for grouped data is given by ixii MD i1 2f Similarly we can define mean deviation about median just by replacing the mean value in formula with the median REGRESSION ANALYSIS The primary objective of regression analysis is the development of a regression model to explain the association between two or more variables in the given population A regression model is the mathematical equation that provides prediction of value of dependent variable based on the known values of one or more independent variables In regression Analysis we shall encounter different types of regression models One of the main functions of regression analysis is determining the simple linear regression equation What are the different Measures of variation in regression and correlation What are the Assumptions of regression and correlation What is Residual analysis How do we make lnferences about the slope How can you estimate predicted values What are the Pitfalls in regression What are the ethical issues An important point in regression analysis is the purpose of the analysis SCATTER DIAGRAM The first step in regression analysis is to plot the values of the dependent and independent variable in the form of a scatter diagram as shown below The form of the scatter of the points indicates whether there is any degree of association between them In the scatter diagram below you can see that there seems to be a fairly distinct correlation between the two variables It appears as if the points were located around a straight line 206 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 The Scatter isg rem Flet efell is s pairs El 5391 i i it I Types of Reqression Models There are two types of linear models as shown in the slide below These are positive and negative linear relationships In the positive relationship the value of the dependent variable increases as the value of the independent variable increases In the case of negative linear relationship the value of the dependent variable decreases with increase in the value of independent variable Types ef Regression Medels Pesilise linear R iel iu l39l emtise Linear Heatien in rrI h E El 1 C IIIII CI I I I quot Copyright Virtual University of Pakistan VU 207 Business Mathematics amp Statistics MTH 302 VU LECTURE 29 MEASURES OF DISPERSION CORRELATION PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 28 0 Correlation CORRELATION Correlation is a measure of the strength or the degree of relationship between two RANDOM variables When do we use correlation It will be used when we wish to establish whether there is a degree of association between two variables If this association is established then it makes sense to proceed further with regression analysis Regression analysis determines the constants of the regression You can not make any predictions with results of correlation analysis Predictions are based on regression equations CORRELATION ANALYSIS To analyze the strength of the relationship or covariation between two variables we use correlation analysis Correlation analysis contributes to the understanding of economics behavior aids in locating the critically important variables on which others depends may reveal to the economist the connections by which disturbances spread and suggest to him the path through which stabilizing forces may become effective CORRE LATION WHEI I d USE correlation DIE NEE it t lli I I I Iil TE THE strength f HSS Gi Wl i bl D312 H fUSE it ifyou WEI t predict tI39IE WEIUE f I QWEI I if lquot WEE F EI SE II Il t li I ii II 1 r pl 1 i ful i 1 395 Enrrel39amn kl Rigniacin II 1 SIMPLE LINEAR CORRELATION VERSUS SIMPLE LINEAR REGRESSION The calculations for linear correlation analysis and regression analysis are the same In correlation analysis one must sample randomly both X and Y Correlation deals with the association importance between variables whereas Regression deals with prediction intensity The slide shows three types of correlation for both positive and negative linear relationships In the first figure r 09 the data points are practically in a straight 208 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU line This kind of association or correlation is near perfect This applies to negative correlation also The graphs where r 05 the points are more scattered there is a clear association but this association is not very pronounced ln graphs where r 0 there is no association between variables TVF IE L RREL TI N CORRQATION COEFFICIM For calculation of correlation coefficient 1 A standardised transform of the covariance sxy is calculated by dividing it by the product of the standard deviations of X sx amp Y sy 2 It is called the population correlation coefficient is defined as r SXysty 01 r C0vX Y VarX VarY Where covariance of X and Y is de ned as C0vXY Y Y Y n This formula is a bit cumbersome to apply Therefore we may use the following short cut formula Short Cut Formula pry 220mm lZXZZXlzw ZYZ Zlera 209 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU It should be noted that r is a pure number that lies between 1 and l ie l lt r lt 1 Actually the mathematical expression that you have just seen is a combination of three different mathematical expressions Case 1 Positive correlation 0 lt r lt 1 In case of a positive linear relationship r lies between 0 and 1 Y n X In this case the closer the points are to the UPWARDgoing line the STRONGER is the positive linear relationship and the closer r is to 1 Case 2 No correlation r 0 2 l 0 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The extreme of dissociation zero correlation r 0 V n V In such a situation X and Y are said to be uncorrelated Case 3 Negative correlation l lt r lt 0 Warning Existence of a high correlation does not mean there is causation which means that there may be a correlation but it does not make things happen because of that There can exist spurious correlations And correlations can arise because of the action of a third unmeasured or unknown variable In many situations correlation can be high without any solid foundation 21 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EXAMPLE Suppose that the principal of a college wants to know if there exists any correlation between grades in Mathematics and grades in Statistics Suppose that he selects a random sample of 9 students out of all those who take this combination of subjects The following information is obtained 30 Marks in Statistics In order to compute the correlation coef cient we carry out the following computations 25 20 15 10 Marks in Marks in Mathematics Statistics Student Total Marks 25 Total Marks 25 X Y A 5 11 B 12 16 C 14 15 D 16 20 E 18 17 F 21 19 G 22 25 H 23 24 I 25 21 SCATTER DIAGRAM X 5 10 15 20 25 30 Marks in Mathematics 212 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU SK 6 Ya Cr 5 11 25 121 55 12 16 144 256 192 14 15 196 225 210 16 20 256 400 320 18 17 324 289 306 21 19 441 361 399 22 25 484 625 550 23 24 529 576 552 25 21 625 441 525 156 168 3024 3294 3109 2 XY 2 XXX Yn lizxz z X2ni 2 Y2 z Y2ni 3109 1561689 3024 1562 9 3294 1682 9 3109 2912 3024 2704 3294 3136 197 197 088 320x158 22486 There exists a strong positive linear correlation between marks in Mathematics and marks in Statistics for these 9 students who have been taken into consideration EXCEL Tools For summary of sample statistics use Tools Data Analysis Descriptive Statistics For individual sample statistics use Insert Function Statistical and select the function you need EXCEL Functions In EXCEL use the CORREL function to calculate correlations The correlation coefficient is also given on the output from TOOLS DATA ANALYSIS CORRELATION or REGRESSION 2 l 3 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU tter Diagram Two Variables Yo Ehart Wizard Step 1 of 4 Ehart Type Standard Types Custam Types Qhart type lChart subtype M Calumn a E Ear Daughnut r F39adar Surface 539 Bubble Em Stpclt gt0 W Scatter Campares pairs at yalues iii Press and led tp ielaI Sample I Cancel HESII I Einish I The slide shows a scatter diagram of Advertisement and Sales over the years The graph was made using EXCEL chart Wizard As you can see one cannot draw any conclusions about the degree of association between advertising from this graph Microsoft Excel Lecture Eile Edit ew insert Fgrmat pols Qhart window Help 39 39 v 39 peeeeevs m perm vwvnru Chari3 V r A B C D E F G H J 8 I g SCATTER DIAGRAM TWO VARIABLES i 3 E 15000 11 E U 15 lg 10000 ADV I l l I 13 n 1 5000 39 SALES 20 NJ 06 21 gt 0 0 22 D U o O 6 23 lt3 24 1988 1988 1999 1992 25 g YEARS 28 I 29 SALES VERSUS ADVERTISEMENT The scatter diagram for sale versus advertisement shows a fairly high degree of association The relationship appears to be positive and linear 214 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micreseft Excel Lecture Eile Edit Eiew Lnsert Fgrmat eels Qhart window elp El 1 De i tttt v material ZDBIH t Chadd 139 f3 A E C 0 E F e H j 3839 I j SALES VS ADVERTISEMENT 12 33 8 15000 45 a 40 j 3 10000 4 40 2 E 5000 52 53 g I I I 2 U 200 400 600 OD ADVERTISEMENT RS 000 01 r CORRQATION COEFFICIENT USING EXCEL Correlation Coefficient for correlation between two steams of data was calculated using the formula CovxySxSy as given above The data for variable x was entered in cells A67 to A71 Data for variable y was entered in cells B67 to B71 Calculations for square of x square of y product of x and y Xm Ym and covxy were made in columns C D E F and G respectively Other calculations were made as follows Cell A72 Sum of x SUMA67A71 Cell B72 Sum ofy SUMB67B71 Ce C72 Sum of square ofx SUMC67C71 Cell D72 Sum of square ofy SUMD67D71 Cell E72 Sum of product ofx and y SUME67E71 Cell F72 Mean ofx A725 where 5 is the number of observations Cell G72 Mean ofy B725 where 5 is the number of observations Ce F73 Sx SQRTC725F72F72 Ce G73 Sy SQRTD725G72G72 Cell H73 Covxy E725F72G72 Cell H74 Correlation coefficient H73F73G73 The above formulas are in line with formulas described earlier 215 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU TE Micreeeft Excel Lecture Elle Edit Eiew insert Fgrmat Innis gate window elp 39 3 3 in v E v E SLIM r X J 8 H3fF3G3 is E C n E F e H 54 CRRELPlTIN 55 as Y Y Km Ym a 2 00 4 3600 120 as 5 100 25 10000 500 59 4 70 10 4900 230 m 90 8100 540 in 80 6400 240 2 20 400 90 33000 1080 4 80 is Standard deviatin s 14 1414i ml 03 r H73ilF7339iiGT3 r5 39 CORREL Returns the correlation coefficient of the array1 and array2 cell ranges Use the correlation coefficient to determine the relationship between two properties For example you can examine the relationship between a location39s average temperature and the use of air conditioners Syntax CORRELarray1 array2 Array1 is a cell range of values Array2 is a second cell range of values Remarks 0 The arguments must be numbers or they must be names arrays or references that contain numbers o If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included o If array1 and array2 have a different number of data points CORREL returns the NA error value If either array1 or array2 is empty or ifs the standard deviation of their values equals zero CORREL returns the DlVO error value EXCEL calculation The X and Y arrays are in cells A79 to A83 and B79 to B83 respectively The formula for correlation coefficient was entered in cell D84 as CORREA79A83B79B83 The value or r 08 is shown in cell C86 216 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Lecture25 Eile Edit Eiew insert Fgrmat IDEIIS Eata indnw Help as La m 2 v SLIM v 2 4 CDRHELEA9AEEEHTBEEESJ F5 A E C D E F IE r5 CRRELarray1 arrEyZ F re 1 r9 2 an 5 100 31 4 32 E Era 34 r CRRELHBH33 3395 B9333 SE 18 2 17 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 30 Measures of Dispersion LINE FITTING PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 29 0 Line Fitting EXCEL SUMMARY OF SAMPLE STATISTICS For summary of sample statistics use Tools gt Data Analysis gt Descriptive Statistics For individual sample statistics use Insert gt Function gt Statistical and select the function you need EXCEL STATISTICAL ANALYSIS TOOL You can use EXCEL to perform a statistical analysis On the Tools menu click Data Analysis If Data Analysis is not available load the Analysis ToolPak In the Data Analysis dialog box click the name of the analysis tool you want to use and then click OK In the dialog box for the tool you selected set the analysis options you want You can use the Help button on the dialog box to get more information about the op ons LOAD THE ANALYSIS TOOIPAK You can load the EXCEL Analysis ToolPak as follows On the Tools menu click AddIns In the AddIns available list select the Analysis ToolPak box and then click OK If necessary follow the instructions in the setup program Copyright Virtual University of Pakistan 218 Business Mathematics amp Statistics MTH 302 VU Eile Eclit iew Lnsert Fgrmat leels Qata inclew Help Arial EB 139 r A E C D E I F G H 1 2 3 4 AddIns g clcl Ins a3939ailalle F 3939393939393939393939393939393939393939393939393939393939393939393939393939393939393939393939393939393939393939 39 l Analysis TeelPak 39u39ElFI H I CDI39IClithII39IEII Sum Wizard Cancel El IE Eure Currency Teels vquot Internet Assistant 39u39ElFI ll I LcncIltLIp Wizard 1 12 l Selver FiclclIn Agtumatiunl H l 1 3 1 4 1 5 1 E 1 Fquot 1 El 19 J 20 Analysis TcIcIIP39alt 21 Previcles Functicns ancl interfaces Fcr Financial ancl 22 scientific clata analysis 23 24 9F Add Iris Add Ins a3939aialle 1F Anal3939sis Tdle39alc 39 quot I Conditional Sum Wizard Cancel I EurI Curranty Tddls I Intarnat Assistant 3939EIA I Lacme Wizard l Salwar Add in EFDWSE Agtdrnatidn Analysis Tdleals 3939EiA 3939EIA Functidns Far Anal3939sis Tdle39alc 2 19 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Micrnft Excel Baum Eile Edit ew insert Fgrmat Innls Qatar indnw Help D E 1 g 3933 IE 3539 Queuing F m M l 3 Mal E3 u f1 quot 21 Error Checking 395 E C Share Wnrkhnnk 393 H 1 2 Erntectlnn Ir 3 Euro Conversion g Unline Cnllahnratinn Ir 5 Fnrmula auditing Ir I Tnnls nnthe Wag g Addins 1 gustnmize 11 gptlnns 12 13 gnnditinnal Sum 1411 Lnnkup 15 Data analysis IE v 1 Copyright Virtual University of Pakistan VU 220 Business Mathematics amp Statistics MTH 302 VU iata Analyeie analysis Tnnle Elli E39rrelatin Eaneel Cnrarianee Descriptive Statistics Eannnential Smnnthinn FTest TernSample lnr ilariancee Fnurier analysis Hietnnram liming average Flandnm Number Generatinn Help Ir 391 Data Analyeia analysis Tnnls m Hietnnram lrlnrrinn average gal3 Flandnm Number Eeneratinn Rank and Percentile iFaniaEliaIIIIIIIIIIIIIIIIXXIIIIIIIIIIIIIIIIE HE39F39 Samnlin tTeet Paired Tran Samnle lnr Means tTeet TwnSamnle assuming Enual ilarianees tTeet TwnSamnle assuming Unequal ilarianees aTest Tran Samnle lnr lleane quot39 221 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Regressiun I l npu K Input 1 Range I Cancel Input 5 Range Help M l lalIels l ICenstanl is gen l IZenfitlennze Level 3995 We C IuIu eptiens P Qutpul Range l F quoteI39I 393939erlshee Ely l P quoteI39I erkteelt Residuals l Residuals l Resi ual F39lets l Standardized Residuals l Line Fit F39lets Nermal Fretability l ermal Pratability F39lets SLOPE Returns the slope of the linear regression line through data points in knowny39s and knownx39s The slope is the vertical distance divided by the horisontal distance between any two points on the line which is the rate of change along the regression line Syntax SLOPEknowny39sknownx39s Knowny39s is an array or cell range of numeric dependent data points Knownx39s is the set of independent data points Remarks The arguments must be numbers or names arrays or references that contain numbers If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included If knowny39s and knownx39s are empty or have a different number of data points SLOPE returns the NA error value The equation for the slope of the regression line is 5112 tIthllZIl HZxEExli Example The known yvalues and xvalues were entered in cells A4 to A10 and B4 to B10 respectively The formula SLOPEA4A10B4B10 was entered in cell A11 The result 0305556 is the value of slope in cell B12 222 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E Micrusu Excel Lecture3 EiIE Edit Eiew insert Fgrrnat IncIs Eata irtjaw elp at a a m T E T E SLIM 1quot K J J SLDF39EIZMSA1EIEMSEHDII 395 I E 393 D E F 2 SLPEknwny39Sknwnx39S 3 ann y ann x 4 5 E 7 El 9 mum tween hhmH Imm lIII J SLiPE 12 Adlm l ems1 ml INTERCEPT Calculates the point at which a line will intersect the yaxis by using existing xvalues and yvalues The intercept point is based on a bestfit regression line plotted through the known xvalues and known yvalues Use the INTERCEPT function when you want to determine the value of the dependent variable when the independent variable is 0 zero For example you can use the INTERCEPT function to predict a metal39s electrical resistance at 0 C when your data points were taken at room temperature and higher Syntax INTERCEPTknowny39sknownx39s Knowny39s is the dependent set of observations or data Knownx39s is the independent set of observations or data Remarks The arguments should be either numbers or names arrays or references that contain numbers If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included If knowny39s and knownx39s contain a different number of data points or contain no data points INTERCEPT returns the NA error value The equation for the intercept of the regression line is 1 Iquot EzX where the slope is calculated as b HER EEIEF FEEit 2ij Example The data for yvalues was entered in cells A18 to A22 The data for xvalues was entered in cells B18 to B22 The formula NTERCEPTA18A22B18B22 was entered in cell A24 The answer 0048387 is shown in cell B25 223 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Lecture3 Eile Edit ew insert Fgrmalz Innls Eata indnw Help 3 IE E r a 1 SLIP391 T x 3 F NTERCEPTEA1Eamp22E1E522 A El 1 D E F 1a INTERCEPTImwny39EJImwnz395 1 anny ann x 1a 2 5 1g 5 23 Q 11 21 1 r 21 8 5 E 24 INTERCEP 25 313322 224 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 31 LINE FITTING PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 30 0 Line Fitting Types of Reqression Models There are different types of regression models The simplest is the Simple Linear Regression Model or a relationship between variables that can be represented by a straight line equation To determine whether a linear relationship exists a Scatter Diagram is developed first The Scatter iag ram Plot sofaquot mi vi pair quotr ll El i i 4 I El I I E I I i I i I i I 2 4 5 ln linear regression two types of models are considered The first one is the Population Linear Regression that represents the linear relationship between the variables of the entire population ie all the data It is quite customary to carry out sample surveys and determine linear relationship between two variables on the basis of sample data Such regression analysis is called Sample Linear Regression Copyright Virtual University of Pakistan 225 Business Mathematics amp Statistics MTH 302 VU TYPEE of Regressiun Models H atim liu l39l lIIIIT Linmr Nu Eeh nmlii IIIu39Jl39HSILEJIU EIII39JIIIC EIIIII Relationship between Variables is described by a Linear Function The change of one variable causes the other variable to change The relationship describes the dependency of one variable on the other If the relationship between the variables is exactly linear then the mathematical equation describing the liner relation is written as YabX Where Y represents the dependent variable X represents the independent variable a represents the Yintercept ie the value of Y when X is equal to zero b represents the slope of the line ie the value of the tan 9 where 9 represents the angle between the line and the horizontal axis 226 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Interpretation of a and b A very important point to note is that MANY lines can be drawn through the same scatter diagram In contrast to the above the liner relationship in some situations is not exact For this we add an unknown random error variable as Wab a We assume that the liner relationship between the dependent variable Y and the value X of the regressor X is Foouletion Lineer Regreeeion D Population Regreeeion Line le r Straight Line that IIIeeorihee The Dependence of The ovenme 1w rlue of ne lli erinlzile on The Either P animation Harmonquot Population 3 IDIE Error il 39 H IEo oient P71 n 81 r 39t 39 D reitlentf magma53 Pmrlle oly l llttl rentl t anel g Retiree e E itnlmetory line lur l39itille The slide below shows the graphical representation of the population regression equation It may be seen that the distance of the points from the regression line C Copyright Virtual University of Pakistan VU 227 Business Mathematics amp Statistics MTH 302 VU obtained by inserting values of X in the equation is the random error The intercept is shown on the Yaxis Flooulation Linear Regreaaion fwr nue E lr If a ll 381 z z ab rwd Value W o Iquot kaaaaa al C39i SEl 39Ed Value 15 Hi The slide below shows the regression equation for the sample Note that the intercept in this case has a notation Bo The slope is B1 The random error is e1 Different notations are used to distinguish between population regression and sample regression Sarnole Linear egreaaion Sample Fteg reeaion Line Prouitlee an Eatilnate of The Fionulation Reg region Line as well no a Pretliotetl Value of 1 Emma Saner Elmo lllta39ee t Era 5 ref Eoeiafieieli I a v F quot1 39 l39 I 1 bl bl l IE H Ha tllal quot r quot Ean39q e 15h ul wulee an eetllnate ot TE 4 lawman Line bl urouitlea an eatilnate of 31 REGRESSION EQUATION EXAMPLE Computer the least square regression equation of Y on X for the following data X 5 6 8 10 12 13 15 16 17 Y 16 19 23 28 36 41 44 45 50 Solution 228 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 The estimated regression line of Y on X is YabX In the previous lecture we learn that the equation for the slope of the regression line is b HZ 12012 HZ 3 Z x 3 And the equation Er the intercept of the regression line is 1IrquotEf X Now from the given data X Z 1029 1133 n Y n b nZXY 2ZXZY 22831 nZX ZX a i7 49 335628311183 147 Hence the desired estimated regression line is Y 2147 2831X REGRESSION EXAMPLE Regression Analysis can be carried out easily using EXCEL Regression Tool Let us see how it can be done We chose to carry out regression on data given in the slide below Yvalues are 60 100 70 90 and 80 Xvalues are 2 5 4 6 Copyright Virtual University of Pakistan VU 229 Business Mathematics amp Statistics MTH 302 and 3 E il ierosoft Excel Lecture31 Eile Edit Eiew Lnsert Format Iools gate inookI elp Firial D51 139 J15 1101 A El 393 a REGRESSION as EXAMPLE 1 39 4 1 JH 12 J13 I14 15 J15 We start the regression analysis by going to the Tools menu and selecting the Data Analysis menu as shown below Tools I Qata window Help 23 Sjelling F Share Workbook Erotection tr Euro lConversion Tools on the Wep adolns gate analysis h H quotis The Regression dialog box opens as shown in the following slide You click the Regression analysis tool and then OK Data Analysis analysis Tools K Histogram a Moving Fwerage Random Number lSeneration Ftank and Percentile Cancel Lil Help Sampling h tTest Paired Two Sample For Means tTest TwoSample assuming Equal 39u39ariances tTest TwoSample assuming Unequal 39u39ariances eTest Two Sample For Means quotquot 230 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 The regression dialog box opens as shown below In this dialog box Input range forX and Y is required One can specify labels confidence level and output etc Regressiu n Input DH Input 1 Range 3 Cancel I Input 3 Range 3 Hel Labels I IIenstant is gen l Cpn idence Level 3995 We lElutput pptipns Ii F gutput Range i lquot MelaI Wprksheet Ely I F MelaI prklapplt Residuals I Residuals F Resiglual Flats Standardized Residuals I Line Fit F39Ipts Nermal F39relzualzuilitsi l ermal F39relzualzuilitsi Plpts For the sample data the input Y range was selected by clicking in the text box for input y range data first and then selecting the Y range A85A89 The regression tool adds the sign in front of the column and row number to fix its location The input range for X was specified in a similar fashion No labels were chosen The default value of 95 confidence interval was accepted The output range was also selected in an arbitrary fashion All you need to do is to select a range of cells for the output tables and the graphs The range A91F124 was selected as output range by selecting cell A91 and then dragging the mouse in such a manner that the last cell selected on the right was F124 Copyright Virtual University of Pakistan 231 Business Mathematics amp Statistics MTH 302 VU iiigiiiii F39 77777 7 7739s Lquot E Criirriirrrrgi iriiiii lF39mrm ull ll runiiiiiiiira i I r Eile Edit EieliiI Insert Fgrmat Tools Data indpw elp v 539 it w v a a 91 v 5 A El C D E F El e1 REG RESSIDN RNRLYSIS e2 EXRMPLE 1 Regression 33 Input Y Input i Range a35pgg 2 84 l E Input 5 Range B35BEg E ES I l Labels r Constant is gerd il 35 1 l Con dence Level 3995 as er 70 ea 30 as 30 Output options l7 gutput Range R91F124 1 F New Worksheet Ely l NewI prkbppk Resduab SD M F EESldllals l RESlgU l F39lDtS 91 l I F Standardized Residuals l7 Line Fit Hats 92 I 93 Normal Probability 94 F mermal Probability F39Iots as I an The Regression dialog box with data is shown below for clarity Ragrassiun l I Input 1 Range nescies s 39 Cancel I Input 5 Range EIBEBBE lil Hal I l Lalzlels l IIdnstant is gen p l Confidence Level 3995 quot lClutput options 3 Qutput Range ln91l124l P New Wdrksheat Ely l P New mprkbpplct Residuals IF Residuals l Residual Pldts l Standardized Residuals I Line Fit F39ldts Narmal Pratability I drmal Pratability Plats When you click OK on the Regression tool box a detailed SUMMARY OUTPUT is generated by the Regression Tool This output is shown in parts below 232 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 SUMMARY iIUTPUT Ra urassidn Statistics Multiple R Square 054 Adjusted R Snare 052 Standard Err bser39vatizia 103541512 5 ANVA df Rarassin Rasiual Ttal 1 64B 64 361 12 4 10110 Coef cients Standard Error tStat Intercat Variable 1 48 1439593846 32559 8 3464101615 23119 Copyright Virtual University of Pakistan 233 Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Lecture31 E Eile Edit iew insert Format Innls Data window elp 1 539 Digggg g ngv m fmial 1nv f E IEE 1quot i5 E F G H 5 53 F Signi cance F e 0104088039 El LT I Ei BE 53 Pvalue Lewer Upper Lewer Upper 51 00409 1227738032 9477220 122773853 947722914 55 01041 3024327728 19024327 EE E Hicrnsnft Excel Lecture31 file Edit iew insert Fgrrnat Innls gala indnw Help v i Deal emexweeewe vlnv rua IEE v 7 B C D E 39 RESIDUAL UTPUT 7 1 Observatien Predicted Residuals i2 1 E4 4 73 2 88 12 74 10 75 4 90 6 FE 5 72 3 77 The regression Tool also generates a normal probability plot and Line Fit Plot 234 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Lecture31 Elle Edit iew lneert Fgrmet leele gate indew Help g rgEe l EfAriel r liDD 1 5 J H L M N CI F39 D H 3395 Nerrnel ereeaeility Plet BE EDD El 1DD EB D 2D 4D ED DD 1DD 59 Sample Pereent ile DD 91 t l v I lquot 92 I Verleble1 Llne Flt Plet 93 E14 EDD 95 nmnu u e f BE I Predicted 1 D 39 39 39 93 D 2 4 E 3 99 it U erielile 1 1DD 13911 13932 EXCEL REGRESSION TOOI OUTPUT In the regression Tool output there are a number of outputs for detailed analysis including Analysis of Variance ANOVA that is not part of this course The main points of our interest for simple linear regression are Multiple R Correlation Coefficient R Square Coefficient of determination STEMStandard Error of mean Standard deviation of populationsample size TStatistic sample slope population slope Standard error RSQ There is a separate function RSQ in EXCEL to calculate the coefficient of determination square of r Description of this function is as follows Returns the square of the Pearson product moment correlation coefficient through data points in knowny39s and knownx39s For more information see PEARSON The r squared value can be interpreted as the proportion of the variance in y attributable to the variance in x Syntax RSQknowny39sknownx39s Knowny39s is an array or range of data points Knownx39s is an array or range of data points Remarks The arguments must be either numbers or names arrays or references that contain numbers If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included If knowny39s and knownx39s are empty or have a different number of data points RSQ returns the NA error value The equation for the r value of the regression line is 23 5 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU r arm39139 mm 233 EXJEHHEFE Eff Example A B 1 Known y Known x 2 2 6 3 3 5 4 9 1 1 5 1 7 6 8 5 7 7 4 8 5 4 Formula Description Result RSQA2A8BZBS Square of the Pearson product moment correlation coefficient through data points above 005795 236 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PVALUE In the EXCEL regression Tool the PValue is defined as under Pvalue is the Probability of not getting a sample slope as high as the calculated value Smaller the value more significant the result In our example Pvalue0000133 It means that slope is very significantly different from zero Conclusion X and y are strongly associated SAMPHNG DISTRIBUTION IN r It is possible to construct a sampling distribution for r similar to those for sampling distributions for means and percentages Tables at the end of books give minimum values of r ignoring sign for a given sample size to demonstrate a significant nonzero correlation at various significance levels 01 005 002 001 and 0001 and degrees of freedom 1 to 100 It is to be noted that v degrees of freedom n 2 in all these calculations SAMPHNG DISTRIBUTION IN rEXAMPE Look at a sample size n 5 Null hypothesis r 0 Calculated coefficient 08 Test the significance at 5 confidence level m Look in the table at row with vn2 3 and column headed by 005 Pearson ProductMoment Correlation Coefficient Table of Critical Values df N2 Level of signi cance for twotailed test N number of pairs of data 10 05 02 01 1 988 997 9995 9999 2 900 950 980 990 3 805 878 934 959 4 729 811 882 917 5 669 754 833 874 6 622 707 789 834 7 582 666 750 798 8 549 632 716 765 9 521 602 685 735 237 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 10 497 576 658 708 11 476 553 634 684 12 458 532 612 661 13 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0878 Sample value of 095 ignoring sign is greater than 08783 Conclusion Correlation is significantly different from zero at 5 level Variables are strongly associated 267 254 239 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 32 TIME SERIES AND EXPONENTIAL SMOOTHING PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 31 0 Time Series and Exponential Smoothing SIMPLE LINEAR REGRESSION EQUATION EXAMPLE The slide below shows the data from 7 stores covering square ft and annual sales The question is whether there is a relationship between the area and the sale for these stores It is required to find the regression equation that best fits the data Simple Lineer Regreeeien Equetien Example utlulal Stere Siguare Sels Re Hit 391 39tJlJ 3631 quotlieu me te eten39ine the 2 them 33925 r atim tie linetweet the 3 Elm E 55553 aware feetege ef mettlee i Hera and their eliulel 4 555 ghee Senute late fer T 5 it 35313 anHE me el iinetl Fintl 5 33 55553 the t l ll ef the mi tt T t 3391 3 EJ alIl line that te the late Ire First of all a scatter diagram is prepared using EXCEL Chart Wizard as shown below The points on the scatter diagram clearly show a positive linear relationship between the annual sale and the area of store It means that it will make sense to proceed further with regression analysis The estimated regression line of Y on X is Y a bX In the previous lecture we learn that the equation for the slope of the regression line is b HZ tIlZIllZIl HZ x 3 Z x 3 And the equation for the intercept of the regression line is 240 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Stores X Y XY X2 1 1726 3681 6353406 2979076 2 1542 3395 5235090 2377764 3 2816 6653 18734848 7929856 4 5555 9543 53011365 30858025 5 1292 3318 4286856 1669264 6 2208 5563 12283104 4875264 7 1313 3760 4936880 1723969 2 ZX16452 ZY35913 ZXY 104841549 2X252413218 Now from the given data X X n M 164527 235029 Y Y Z 359137 5130429 n b quot2 XY EXXZY 14866 nZXZ ZX2 a 17 49 513043 14866235029 1636489 Hence the desired estimated regression line is Y 2 1636489 14866X Copyright Virtual University of Pakistan 241 Business Mathematics amp Statistics MTH 302 VU Seetter iegrem Eeemple III IIZI39III EIIIZI BIIIEI HZIIII39ZI SDIII EIIIZI quere Fe e1 Eeeel uent Using the EXCEL Regression Tool the regression equation was derived as given below The graph of the regression line was prepared using the regression Tool The result shows the data points regression line and text showing the equation As you see it is possible to carry out linear regression very easily using Excel s Regression Tool Greph ef the Semple Regreeeien Line IEIIIIEII E Item 39 elm EH EIIIIII m we E EIIIIII E I I I I I I I2 lIIiII EIIIIIIZI 3IIiII lJIIIZI SEIIIIZI EIZIIII Supera Feet lnterpretinq the Results The slide below gives the main points namely that for every increase of 1 sq ft there is a sale of 1487 units or 1407 Rs As each unit was equal to 1000 Now that the equation has been developed we can estimate sale of stores of other sizes using this equation 242 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Interpreting the Reeulte H e teeeete death The slope at 148 means that each hcreaee at me ahitlh A we prem39ctthe average at Fte increase t5 ah eathhded 148 hath The medeieathhatea that far each incna e ef1 mature feet hr the size at the attire the expected almaai39aaiea are predhaed te hrcreaee try Re t Interpreting the Reeulte er teeeete death The atepe at 148 mean that each hrcreaee at me ahitlh A we precb39ctthe average at l ta increase hf ah eathhded 143 quotme The medeleathhatea that far each incree e ef1 aware feet h the size at the attire the expected arhaal39aaiea are predhaed te increase tar Rm tl CHART WIZARD Let us look at how we can use the Chart Wizard We wish to study the problem shown in the slide below Copyright Virtual University of Pakistan 243 Business Mathematics amp Statistics MTH 302 VU Elmicrneuft Excel Baul Eile Edit iew insert Fgrmat Inels gate window Help lt35 gEv i fnrial TEUTIHE A2 1 f3 r ear A E t D E F Wl H I J K 1 SHLES RECRDS BRKEN DWN BY UARTERS 2 Year Quarter Ne50l 999 i 1993 Spring 14 i Summer 54 i Autumn 162 i Winter EDIE i 1994 Spring 139 i Summer 59 1 Autumn 1T4 E Winter 19 l 1995 Spring 126 3 Summer 42 3 Autumn 162 M Winter 16 f e e e we IIIIII quotE quot J ti 1L Ihert Wizardl You can start with the Chart Icon as shown on the right There are 4 steps in using the Chart wizard as shown below Step 1 244 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Chart Wizard Step 1 nf 4 Chart Type Standard Types Custem Types Chart type Chart subtype EIIIIZIlIJ rI39IFI E Eiar lg Line 3 Pie W Scatter I Ftrea Deughnut J Ftadar SurFace 3 Bubble Em Stcuclt j Clustered Celumn Ccumpares values acress categcuries Press and Held tcu ew Sample I Cancel Heste I Einish I Chart llo39J39izatrd Step 2 nf 4 Chart Snurce Beta Data Range 1 Series l Te create a chartJ click in the Data range hex ThenJ en the IllcurlsheetJ select the cells that centain the data and labels yeu want in the chart Qata range Series in C aws P CcILIn39ns l Cancel si atk eet I Einish 245 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ehart Wizard Step 3 f 4 Ehart ptine I hew legencl Placement F Etcuttcum r Cgrner lquot Inn f ight f LeFt lCancel ci aclt eet Einish Ehart Wizard Step 4 f 4 Ehart Leatin F39lace chart III P like new gheet lCl39IEII lZl ll 397 n5 glaject in Eitleetii IIIIIIIIIIIIIIIIIIIIIIIIIIIIIE i Cancel c ack x I I Einieh I The dialog boxes are selfexplanatory Let us look at the example above and see how Chart wizard was used First the data was selected on the worksheet Next the Chart Wizard was selected We chose Column Graph as the option as you can see in the slide below We clicked Next 246 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 ak am r We metal a a a m2 its net Ej U D a fr tin rmed liltlll lrirli nibartth M u lma 3 I ail Ed39sE i ange1amp2 n yme ezereame aI i You can see the selection of Column graph in the slide below Ehart Wizard Step 1 pf 4 Ehart Type Standard Types I Custprn Types I ghart type Chart subtype I III I U m Fl E Eiar IE Line 5 F39ie 31quotquot Scatter I Furea a Deughnut r Ftadar g SurFace 239 Bubble II t Stuck TI Clustered Celumn Iernpares yalues acress categeries Press and Held te gleny Sample I ICancel I ext I Einish I Under Step 2 the Chart Title Category X axis and value Y were entered as shown in the slide Then the button Next was clicked 247 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ehart Wizard Step 3 f 4 Ehart ptine Titles l Fixes l lErilzllines l Legend l Data Labels Data Table l ll39iarttitle l DDWN Bil QUpIR39IERS SALES HEEUHD EHDKEH DEVquot EquotIII HUAFITEFIE gategnrif it axis lQuarters H El Sari51 Eauemaxig g 15 H H ill Ill IEzrist 5 1quot ll ll EIEeriesS lsalesl 5 II II II II II II II II II mm 3 lllllllllllllllllllllll aeaEFMaEFWa EF ca 1333 1334 1335 l utters lCancel ack eet Einish Under the 4lll step the default values Chart1 and Sheet1 were selected Then the button Finish was clicked Ehart Wizard Step 4 f 4 hart Lcatin Plate thart lIlilSI39IEWEl39IEElI lCl39Iartl l3 Fi abject in Cancel Eat nish The result is shown below as a column graph 248 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 200 QUARTE 133 39 5 TEi ll i ll 1993 1994 uarters eelee I D Ser iesl I Series2 D Series D Series Spring Summer Winter Spring Summer Winter Wi rite r Quarter Spring eutumri eutumri Summer m m quotI re D U I I I Chart Wizard was used again to draw a Side by Side chart using the same data The result is shown below I I I SALES BRKEN BY UARTER D Seriesrl D SeriesS I SeriesZ D Seriesl LP 3 1 e 9 SALES I I I A line graph of the data was also prepared as shown below This graph shows the seasonal variations in the values of sales 249 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 539 m cu c1 TEJEJal tElEl i Quarter EXAMINATION OF GRAPH TREND The graph shows that there is a general upward or downward steady behaviour of figures There are Seasonal Variations also These are variations which repeat themselves regularly over short term less than a year There is also a random effect that is variations due to unpredictable situations There are cyclical variations which appear as alternation of upward and downward movement EXTRACTING THE TREND FROM DATA Look at the following data 170 140 230 176 152 233 182 161 242 There is no explanation regarding time periods What to do First step Plot figures on graph Horizontal as period 1 Vertical as period 2 Conclusion There is a marked pattern that repeats itself There is a well established method to extract trend with strong repeating pattern DATE 3m 252 r 3 2m 3 I 15E ii nw u w p E 1m I1 I i l I I I I 1 4 E 3 3 Funnels Copyright Virtual University of Pakistan 250 Business Mathematics amp Statistics MTH 302 VU MOVING AVERAGES Look at the data in the slide below There is sales data for morning afternoon and evening for day 1 2 and 3 We can calculate averages for each day as shown These are simple averages for each da Microsoft Excel Lecture32 Elle Edit view insert Fgrmal 112215 gate window Help 7 a neeeaa em ez aaarlevueee 812 c 1 E F L H I J a L 1111 122 AVERAGES 122 Data Meving Average Trend 121 Day 1 lll39lerning 17121 Afterneen 1412 122 Evening 23121 122 Average 1311 AVERAGEF124F123 122 Day 2 I ll39lerning 173 122 Afterneen 132 1212 Evening 233 121 Average 13 AVERAGEF123F1311 122 Day 3 Warning 132 122 Afterneen 131 121 Evening 242 122 Average 133 AVERAGEF132F134 122 Now let us look at the idea of moving averages First Avergqe DaJLl 170 140 2303 5403 180 Next Avergqe Morninq 140 230 1763 5463 182 Next AverageAfternoon 230 176 1523 186 Another method Drop 170 Add 176 l76l703 63 2 Last average 2 180 2 182 m You may make a mistake You saw how it is possible to start with the rst 3 values 170 140 and 230 for the rst day and work out the average 180 Next we dropped 170 and added 152 the morning value from day 2 This gave us an average of 182 Similarly the next value was calculated Look at the worksheet below for the complete calculation These averages are called moving averages You could have used the alternative method but you may make a mistake in mental arithmetic So let us only use EXCEL worksheets 251 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Micreeeft Excel Lecture Eile Edit view insert Fgrmat 1414 gate window elp 553g39Li ii v e x DEi cli 139 E39El i nmeuEEE 1 NH C D E F G H l J K L lvl 7 114 MUING AVEReGES 1351 Feried Date ll39leving Average Trend 1411 M511 llllerning 170 141 Afterneen 14H 39 13 AVERAGElF14lIlF142l 142 Evening 231 r 132 AVERAGElF1411F143l 143 Eley2 llllerning 17 39 AVERAGElF1412F1441l 144 Afterneen 152 r 13 AVERAGElF1413F145l 145 Evening r AVERAGElF1441F145l 1413 D3513 llllerning 132 r 192 AVERAGElF1415F14l 141 Afterneen 1E1 39 195 AVERAGElF145F143l 14e Evening 2412 1419 15D The moving averages were plotted as shown below You can see that the seasonal variation has disappeared Instead you see a clear trend of increase in sales This plot shows that moving averages can be used for forecasting purposes lu39l vilhliii AVE Ell g Eeriee2 339 15 quot Eerie53 39 1111i Eerieei Elli Ee1iee5 i r a eriee 1 e 4 e e 1139 11111 nEvieef Fe ed 14 ANALYSING SEASONAI VARIATIONS Let us find out how much each period differs from trend Calculate Actual trend for each period Day 1I Afternoon Actual 180 Trend 140 Actual Trend 140 180 40 Here 40 is the seasonal variation Similarly other seasonal variations can be worked out Copyright Virtual University of Pakistan 252 Business Mathematics amp Statistics MTH 302 VU LECTURE 33 TIME SERIES AND EXPONENTIAL SMOOTHING PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 32 0 Time Series and Exponential Smoothing TREND As discussed briefly in the handout for lecture 32 the trend is given by the moving average minus the actual data Look at the slide shown below The average of the morning afternoon and evening of the first day is 180 This value is written in cell I179 which is the middle value for first day The next moving average is written in cell I180 This means that the last moving average will be written in cell I185 as the moving average of the morning afternoon and evening of 3rd day will be written against the middle value in cell F185 Now that all the moving averages have been worked out we can calculate the trend as derence of moving average and actual value Micrusnft Excel Lecture Eile Edit view Insert Fgrmat eels gate window elp 139 E X De em eEvEIMEfIUvBueee Live w 4 e e D E F e H I J r L M 7 1e rCTUPIL MINUS TREND we lay F39eriezd Date Meving Average Trend we 1 Merning 1TH 1e Afterneen 143 133 43 F173l173 en Evening 233 132 43 131 2 Merning 173 133 1lIl 1e Afterneen 132 13 33 1e Evening 233 133 44 1e 3 Merning 132 132 1lIl es Afterneen 131 133 34 ea Evening 242 13 133 The actual trend figures are now written as shown in the slide below with M for morning A for afternoon and E for evening The titles Day 1 day 2 and Day 3 were written on the left hand side of the table Further Total for each column was calculated The total was divided by the nonzero values in the column For example in column M there are 2 nonzero values Hence the total 20 was divided by 2 to obtain the average 10 Similarly the averages in column A and E were calculated This data is the seasonal variation and can now be used for estimating trend and random variations 253 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E icreee Excel Lecture le Edit Eiew insert Fgrmat IDEIIS gate window Help v a D ll fen Ev l f1 v 3 A202 1 5 A I E C D E F G H 191 ACTUALTREND FIGURES TGETHER 192 M A E 194 Day 2 10 35 M 195 Day 1U 34 U 19 Ttal 20 1 09 92 193 Average 10 435 45 199 EDI EXTRACTING RANDOM VARIATIONS Day 1 Afternoon trend 180 Afternoon seasonal variation 36 Trend variation 180 36 144 Actual value 140 Random variation 140 144 4 Conclusion Expected Trend Seasonal Random Actual expected 254 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Efflimeeft Excel Lecture Elle Edit flew insert Fgrmet eels gate window Help E D lgul gr Er l fze 3 IEH 1 it B C D E F G H J K l EDS 2m ACTUAL 140 176 152 182 151 rue Expected 2etrenilseasnel 144 228 176 151 182 159 zerR llfllTl 2naectuelerpected 1 2 0 1 2 l 2 EDS Forecast for day 4 t Trend for afternoon of day 4 Seasonal adjustment for afternoon period Trend 180 to 195 6 intervals 156 25 per period Figure for evening of day 3 195 25 1975 Morning of day 4 1975 25 200 Afternoon of day 4 200 25 2025 After adjustment of seasonal variation 36 2025 36 1665 or 166 SEASONABLE VARIATIONS Seasonal Variations are regarded as constant amount added to or subtracted from the trends This is a reasonable assumption as seasonal peaks and troughs are roughly of constant size In practice Seasonal variations will not be constant These will themselves vary as trend increases or decreases Peaks and troughs can become less pronounced Seasonal variations as well as the trend are shown in the graph below You can see that the trend clearly shows a downward slide in values 255 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU I I I 3 EEAEDHAL H i Ti HE l f 39 l 3 I v I 331331 lt hi I 39 V 39 533332 g 3 I L 3 5 1 i 3 ii V i i i7 i7 i7 V V 123433233131112 1 1 333333 i ll In the following slide the actual values are for 4 quarters per year Here there is no middle value per year The moving averages were therefore summarised against the 3rel quarter As this does not reflect the correct position the average of the first two moving averages was calculated and written as centred moving average in column H The first centred moving average is the average of 141 and 138 or 1395 This is used as the trend and the value ActualTrend is the difference of Actual Centred Moving Average Here also the last row does not have a value as the moving average was shifted one position upwards Microsoft Excel Lecture33 Eile Edit ew lnsert Fgrmat 1333 gate indow Help v 3 D t lt3 E131 13BH E Eta3313 MES v r E C D E F G H J I42 L M 39 5E1 39 53 TREND AND SEASNAL VARIATINS 33 uarter Actual Mving Centred Actual 31 Average MAverage trend 32 391 142 33 2 54 34 3 132 141 1333 223 a 4 233 133 1333 333 I 3 1 133 13 1333 33 3 2 33 143 1333 333 3 3 124 133 1333 333 3 4 133 13 1333 323 m 1 123 133 1333 33 31 2 42 132 1333 333 32 3 132 123 The data from the previous slide was summarised as in the following slide using the approach described earlier It may be seen that the average seasonal variation for Spring Summer Autumn and Winter is 8 888 295 and 653 respectively 256 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnenft Excel Lecturej Elle Edit Eiew insert Fgrmat eels gata window elp v e x 05353 533 Er l fl v uEEEEHE 3 Lane 3 it D E F e H l J K 4 51 7 2395 5 uarterly Veriatins 1 Spring Summer Autumn Winter re 155539 0 0 225 55 re 1554 5 555 52 5 1995 me 5 e 0 H 51 52 Tetal 450 4725 550 1505 ea Average 0 255 553 54 as Rounded 30 55 BE The expected value now is the sum of centred movin averae and random Microsoft Excel Lecturejii Elle Edit eiew insert Fgrmat Iools gate window elp ifCarrelinTerkel v 5 X D2355 53957 Evelmemee E fe e mm v e V e o E F e H if at CMPLETE TABLE 55 tr Aetual Meeing Centred Aetual Expeeted Randem 89 Average MAeerage trend en 1 142 91 2 54 E52H35 C02G02 92 3 152 141 225 1505 5 C02G0 ea 4 205 133 135 2035 25 54 1 130 13 5 1305 05 95 2 50 140 1350 50 500 00 95 3 154 133 135 355 1555 55 at 4 105 13 1350 520 2020 40 98 1 125 135 1335 5 1255 05 99 2 42 132 1305 5 43 10 we 3 152 120 151 4 135 variation The random variation is the difference between the Actual and Expected value This gives us a complete table with all the values The values in this table were plotted using the EXCEL Chart Wizard as shown below You can see that the different components can now be seen clearly 257 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU I I I COMPLETE TALE FOR VARIATIONS 250 200 A 150 IIA r 39I39 CHI W Actual IUD 1quot ii E V V V Moving 50 3 Centred U II I I I I II I I I I 39 tual 1 3 5 r a 11 13 EEDEHEG 3950 n Random 100 1150 Pe ods I I I FORECASTING APPLE PIE SALES Forecast Sale steadily declined from 1390 to 1305 Over 4 quarters the sales declined by 1390 1305 85 Trend in Spring 1995 was 1335 We can assume annual decrease as on the basis of decline over the last 4 quarters 85 Trend in 1996 trend in 1995 less decline 1335 85 125 Seasonal variation as already worked out 8 Hence Final forecast 125 8 117 FORECASTING IN UNPRQICTABLE SITUATIONS Two methods were studied above Each one has certain features If there is steady increase in data and repeated seasonal variations there are many cases that do not conform to these patterns There may not be a trend There may not be a short term pattern Figures may hover around an average mark How to forecast under such conditions Data for sales over a period of 8 weeks is summarized and plotted in the slide below You may see that the values hover around an average value without any particular pattern This problem requires a different solution 258 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Lecture33 Elle Edit iew Insert Fgrmat Innls ghart window elp v El D tt rev flzv 01111110 11 1 31 0 0 0 E F 0 H 1110 1111 SALES 1111 Week N Sales LES 111 1 4500 5000 111 2 4000 HY H 4000 153 m 3000 Week ND 154 I g 2000 Sales I 111 5 4000 mg 111 E 4200 3 15 123450T8 vv Vteek 111 3 4100 I 150 100 FORECAST Let us assume that the forecast for week 2 is the same as the actual data for week 1 that is 4500 Week no Actual sales Forecast 1 4500 2 4000 4500 The Actual sale was 4000 Thus the Forecast is 500 too high Another approach would be to incorporate the proportion of error in the estimate as follows new forecast old forecast proportion of error on Or new forecast old forecast or x old actual old forecast This method is called Exponential Smoothing We shall learn more about this method in lecture 34 259 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 34 FACTORIALS PERMUTATIONS AND COMBINATIONS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 33 o Factorials o Permutations and Combinations Module 7 Module 7 covers the following Factorials Permutations and Combinations Lecture 34 Elementary Probability Lectures 3536 ChiSquare Lectures 37 Binomial Distribution Lectures 38 FORECAST Please refer to the Example discussed in Handout 33 Let OL 03 Then Forecast week 3 week 2 forecast 0c x week 2 actual sale week 2 forecast 4500 03 x 500 4350 Conclusion Overestimate is reduced by 30 of the error margin 500 The slide below shows the calculation for normal error as well as alpha x error You can s that the error is considerably reduced using this approach Microsoft Excel Lecture33 Elle Edit iew insert Formal Iools gate window elp viii iii39viii yiiquot 1 v 539 X D2355 om Ev l 1 v 2 33 3 5155 v 5 E III D E F G H T 15 Week No Sales Forecast Error alpha 1 Error 153 1 4500 151 2 4000 4500 500 150 155 300 4350 550 1 05 155 4 4000 4135 415 1245 15 5 4000 43005 2005 72 155 0 4200 43907 1 007 155 7 3000 43137 37 221 110 4100 41104 404 451 151 52 Forecast E104H104 ErrorD104E104 15 Error lpha 013 260 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The forecast is now calculated by adding alpha x Error to the actual sales The error is the difference between the actual sales and the forecast The first value is the same as the sale last week Use of alpha 03 is considered very common This method is called Exponential Smoothing and alpha is Smoothing Constant Rule for obtaining a forecait Let A Actual and F Forecast Then F t F t 1 0LA t1 F t 1 ocA t 1 1 on F t1 F t 1 0L A t2 1 on F t 2 Substituting F3 on A t1 1 0c ocA t 2 1 on F t 2 on A t1 1 on A t 2 1 0c quot2F t 2 Replacing F t 2 by a t3 1 on F t 3 F t OtAt1 1 on A t 2 1 0c quot2A t3 1 0LF t 3 WHERE TO APPLY EXPONENTIAL SMOOTHING What kinds of situations require the application of Exponential Smoothing What are good values Diet The accepted Criterion is Mean Square Error MSE You can find MSE for by squaring all and including the present one and dividing by the number of periods included Sign of good forecast is when MSE stabilizes Generally alpha between 01 and 03 performs best Example The slide below shows the calculation of MSE Detailed formulas can be seen in the Worksheet for Lecture 34 Micrnsnft Excel Lecturejil Eile Edit Eiew insert Fgrmat nulls Qatar window Help DEi i lt31 Evtl imvnu et g g EiEiE 1r 5 B C D F G H 122 Frecast E151H154 Er39rrD154E154 m ErrrAlpha 121 15 Week Actual FurecastErrnr 232Erranrrer12 MSE 135 22 22011 2221 l l l l 1 23 2400 2221 2GB 41110131 200m we 24 261111 2260 340 102 115601 5135 19 25 281211 2362 438 1314 191341 86361 13m 25 3000 24934 1521 2513544 122313 131 Forecast D1YQF1YQ WISE UM1H1FEH13W5 133 EXCEL EXPONENTIAL SMOOTHING TOOL 26 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU It is possible to use the Exponential Smoothing Tool included in the EXCEL Tools Beta Analysis analysis Tools limos39a TwoFactor Without Replication lCorrelation Cancel I j39 39 3939 392 quotI I l FTest TwoSample For 39u39ariances J Fourier Finalssis Histogram irio39ring Fwerage Ranclom Number Generation j Input UK input Range I Qamping Factor I Cal39IEEl l Labels Help I lElutput options gutput Range l Qhart lCilutput l tanclarcl Errors Different items in the Dialog Box are described below Input Range Enter the cell reference for the range of data you want to analyze The range must contain a single column or row with four or more cells of data Damping factor Enter the damping factor you want to use as the exponential smoothing constant The damping factor is a corrective factor that minimizes the instability of data collected across a population The default damping factor is 03 Note Values of 02 to 03 are reasonable smoothing constants These values indicate that the current forecast should be adjusted 20 to 30 percent for error in the prior forecast Larger constants yield a faster response but can produce erratic projections Smaller constants can result in long lags for forecast values Labels Select if the rst row and column of your input range contain labels Clear this check box if your input range has no labels Microsoft Excel generates appropriate data labels for the output table Output Range Enter the reference for the upperleft cell of the output table If you select the Standard Errors check box Excel generates a twocolumn output table with 262 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU standard error values in the right column If there are insuf cient historical values to project a forecast or calculate a standard error Excel returns the NA error value Note The output range must be on the same worksheet as the data used in the input range For this reason the New Worksheet Ply and New Workbook options are unavailable Chart Output Select to generate an embedded chart for the actual and forecast values in the output table Standard Errors Select if you want to include a column that contains standard error values in the output table Clear if you want a singlecolumn output table without standard error values Example Use of the Exponential Smoothing Tool is shown in the following slides First the Exponential Tool was selected Eiimmiiii first listiim iii quot31311 39 Eile Edit Eiew insert Fgrmet eels gate winders Help v E X DE i ll 5 il i Teii39ii 1 EEEIE 139 r C i E F G H J 7 es EKPUNENTIAL SMOOTHING USING EXCEL WIZARD 13E 220i NlA 1e 240i 22m eteinelieis 1ee 250i 234i nite iineee5ingeFetter x 139 iineee TweFecter With Fieplitetien am E entire TweFecterWitheutReplicetien 39 39 Cerreetien 19D 1 I Ceeeriente HE 191 Destritiee Statistics iEiiiEIIIIrIEFItlEil Sri39IIIIIIItl39IlnIJ 192 FTest TeeSample fer iierientes 193 Feurier Finelysis 194 Histegrem j 195 1 19 Next the Input and Output Range were specified Labels Chart Output and Standard Errors were ticked as options in check boxes 263 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E 111191msigisii i FTTTE i Qll a H i Qi Eli l ll lm Eile Edit Eiew lnsert Format Iools gate window elp Weee D gi v i 1de E209 1quot 13931 e e E F e H 195 EKFGNENTIAL smooTHiNG USING EXCEL s r 1e13 2200 7 13 Expo ne ntiel Smoothing Input v 11313 2000 I I lnputFianoe 7C196C191 1 v GK V 139 Eampingfacmr V Cancel l 1939 1 Labels V elp 1511 3100 output options 192 EUtF39UtRme isoslaesosm E39s 39 i l i i 1913 F shart Output 19E tsnsstssrtsrsi 197r 199 199 The output alon with standard raphs is shown on the followin slide Microsoft Excel Lecture33 Eile Edit ew insert Format eels gate window elp 1 islftTerkel 51 x Diasgit rev Evsi i lnvnueeesiiii C2135 v is C o E F G H I J 7 1135 EXF ONENTIAL SMOOTHING USING EXCEL WIZARD 1313 2200 11111 11111 131 2100 21m 1111A 11313 2000 251a 911A 1139 200 2122 11111 F 1511 3000 25111313 siesisem 191 3400 1512 193 Exponential Smoothing 191 195 ions 1 oi 3mm 19r 1 Actual E 2min 199 2 mm Ferecast 199 see 393 39 39 39 2m 1 2 3 4 5 292 EintnPoiIrt 203 2131 264 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU FACTORIAL Let us look at natural numbers Natural Numbers 1 2 3 Let us now define a factorial of natural numbers say factorial of 5 Five Factorial 5 54321 or 12345 Similarly factorial of 10 is Ten Factorial 12345678910 10987654321 In general n nn1n2321 or n nn1n2 nn1 FACTORIAL EXAMPLES 10 109876543213628800 85 8765 5 876 336 129 1211109l9 121110 1320 10895 109876595 10876 3360 WAYS lf operation A can be performed in m ways and B in n ways then the two operations can be performed together in mn ways Example A coin can be tossed in 2 ways A die can be thrown in 6 ways A coin and a die together can be thrown in 26 12 ways PERMUTATIONS An arrangement of all or some of a set of objects in a definite order is called permutation Example 1 There are 4 objects A B C and D Permutations of 2 object A amp B AB BA Permutations in three objects A B and C ABC ACB BCA BAC CAB CBA Example 2 Number of permutations of 3 objects taken 2 at a time 3P2 332 32 6 AB BA AC CA BC CB Number of permutations of n objects taken r at a time nPr nnr Example 3 Let39s say you and a friend love going to movies and you get a Saturday afternoon free so you can indulge yourselves You go to a multiplex that is showing 6 movies simultaneously each starting at 200 pm 400 pm and 600 pm after which you have to get back home How many different ways can you watch the most different movies Answer You have a choice of 6 movies so this is your set You can watch one movie at 265 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Answer You have a choice of 6 movies so this is your set You can watch one movie at 200 one at 400 and one at 600 therefore you can watch 3 movies and you39re looking for the number of 3permutations We have then P6 3 6 63 6 3 6543 3 654 120 There are 120 different ways for you to watch 3 of the 6 movies that Saturday afternoon Example Suppose there are 100 numbers to choose from 00 to 99 you must choose 5 numbers in a specific order and you can only choose a number once What are your chances of winning the grand prize There are 100 choices and we39re only picking 5 of those so we have P1005 100 100 5 9034502400 ways to pick 5 numbers in a specific order Since only one of those is the winning sequence your chances of winning are l in 9034502400 Example A government keeps some confidential information in a heavily guarded room that is locked with a 5card mechanism Eight different off1cials each carry a card and to get access the cards must be inserted in a speci c order The order changes daily and 3 of the 8 cards will not be used on any given day A novice spy needs to acquire some documents in this room He manages to acquire all eight cards and slip past the guards but doesn39t realize until he gets to the door that only five cards are used and they must be inserted in the correct order A wrong entry brings with it a mass of large mean guys with big guns What are his chances of getting the right sequence Solution The spy has a set of 8 cards to choose 5 from therefore n8 and F5 and P85 83 6720 Only one of those 6720 possibilities is correct so he has a l in 6720 chance or 16720 000015 0015 266 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PERMUTATIONS OF n OBJECTS Number of n permutations of n different objects taken n at a time is n nPn nnn n 0 n 1 n nn1n2321 Number of permutations of n objects of which n1 are alike of one kind n2 are alike of one kind and nk are alike Nin1in2ink1 Examgle 3 How many possible permutations can be formed from the word STATISTICS S3A1 T3 l 2 C1 m nPr nn1n2nk 103132110987654333l2l 50400 PERMUT EXCEL function PERMUT can be used to calculate number of permutations Returns the number of permutations for a given number of objects that can be selected from number objects A permutation is any set or subset of objects or events where internal order is significant Permutations are different from combinations for which the internal order is not significant Use this function for lotterystyle probability calculations Syntax PERMUTnumbernumberchosen Number is an integer that describes the number of objects Numberchosen is an integer that describes the number of objects in each permutation Remarks Both arguments are truncated to integers lf number or numberchosen is nonnumeric PERMUT returns the VALUE error value If number S 0 or if numberchosen lt 0 PERMUT returns the NUM error value If number lt numberchosen PERMUT returns the NUM error value The equation for thelnumber of permutations is 2 Fit a 3 Example Suppose you want to calculate the odds of selecting a winning lottery number Each lottery number contains three numbers each of which can be between 0 zero and 99 inclusive The following function calculates the number of possible permutations 267 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E3 Micrnsnft Excel Baum Eile Edit iew insert Fgrmat nulls ata window Help DEQJL r Ev i fl v BED v 5 A E r D E F 3 1 PERMUTU IUH IIJE UWbELChDSEI I 3 ata escripti n 4 Number if nbjects 5 Number if mbjects in each permutatien a PERMUTMA5 268 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 35 COMBINATIONS ELEMENTARY PROBABILITY PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 34 o Combinations 0 Elementary Probability COMBINATIONS Arrangements of objects without caring for the order in which they are arranged are called Combinations Number of n objects taken r at a time denoted by nCr or n given by r nCr nrnr Difference between Combination and Permutation Suppose we have to form a number of consisting of three digits using the digits 1234 To form this number the digits have to be arranged Different numbers will get formed depending upon the order in which we arrange the digits This is an example of Permutation Now suppose that we have to make a team of 11 players out of 20 players This is an example of combination because the order of players in the team will not result in a change in the team No matter in which order we list out the players the team will remain the same For a different team to be formed at least one player will have to be changed Example Number of combinations of 3 different objects A B C taken two at a time 3232 62 3 These combinations are AB AC and BC COMBINATIONS EXAMPLES Here are a few examples of combinations which are based on the above formula Example 3 ln how many ways a team of 11 players be chosen from a total of 15 players n15r11 15 1 141 1211 1 141 12 C 15 5 3 5 3 1365Ways 11 1115 11 114 4321 Example4 There are 5 white balls and 4 black balls ln how many ways can we select 3 white and 2 black balls 5 4 3X 2 35 3X24 2 Example 5 269 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU If a committee of 3 people is to be selected from among 5 married couples so that the committee does not include two people Who are married to each other how many such committees are possible Solution Total number of ways of picking 3 out 5 married couples or 10 people 10C3 120 A set can have any of the ve married couples gt 5 The third person can be any one of the remaining eight gt 8 one married couple is already part of the chosen set so total Number of ways in Which the set of three can have a married couple 58 40 number of combinations Which don39t have any of the married couples 10C3 58 80 RESULTS OF SOME COMBINATIONS Here are some important combinations that can simplify the process of calculations for Binomial Expansion nCO nCn 1 eg 400 404 1 nC1 nCn1 n eg 401 403 4 nCr nCnr eg 5C2 503 270 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU BINOMIAL EXPANSION An expression consisting of two terms joined by or sign is called a Binomial Expression Expressions such as ab ab xyquot2 are examples of Binomial Expressions We can verify that Xyquot1 X y xyquot2 xquot2 2xy yquot2 xyquot3 xquot3 3xquot2y 3xyquot2 yquot3 xyquot4 xquot4 4xquot3y 6xquot2yquot2 4xyquot3 yquot4 Expressions on the right hand side are called Binomial Expansions COEFFICIENTS OF BINOMIAL EXPANSION The coefficients of the binomial expansion for any binomial expression can be written in combinatorial notation xyquot5 5C0xquot5 5Clxquot4y 5CZxquot3yquot2 5C3xquot2yquot3 5C4xyquot4 5C5yquot5 Solving xquot5 5xquot4y 10xquot3yquot2 10xquot2yquot3 5xyquot4 yquot5 CALCULATION OF BINOMIAEXPANSION COEFFICIENTS Coefficient of first and last term is always 1 Coefficient of any other term coefficient of previous termpower of x from previous termnumber of that term Example First term xquot5 Last term yquot5 Second coefficient 51 8 Third coefficient 542 10 Fourth coefficient 1033 10 Fifth coefficient 1024 5 PROJECT DEVELOPMENT MANAGER S PROBLEM A toys manufacturer intends to start development of new product lines A new toy is to be developed Development of this toy is tied with a new TV series with the same name There is 40 chance of TV series The production in such a case is estimated at 12000 units The Profit per toy would be Rs 2 Without TV seriessale there may be demand for 2000 units Already 500000 Rs has been invested A rival may bring to the market a similar toy If so the sale may be 8000 units The chance of rival bringing this toy to the market is 50 M The company has two choices Abandon new product Risk new development How should the company tie it all to nancial results Sample space Event The set of collection of all possible outcomes of an experiment is called the sample space Each possible outcome of an experiment is called event Thus an event is a subset of the sample space For example all six faces of a die make a sample space By rolling the dice occurrence of number 1 is an event Probability 271 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Probability is the numerical measure of the chance that an uncertain event will occur The probability that the event A will occur is usually denoted by pA The probability of any event must be between zero and one inclusive For any event A 0 S pA S 1 pA 1 means certain pA 0 means impossible PROBABILITY EXAMPLE 1 How can we make assessment of chances Look at a simple example A worker out of 600 gets a prize by lottery What is the chance of any one individual say Rashid being selected m Chance of any one individual say Rashid being selected 1600 The probability of the event quotRashid is selectedquot is the probability of an event occurringpRashid 1600 This is a priori method of finding probability as we can assess the probability before the event occurred 272 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PROBABILITY EXAMPLE 2 When all outcomes are equally likely a priori probability is defined as pevent Number of ways that event can occurTotal number of possible outcomes If out of 600 persons 250 are women then the chance of a women being selected pwoman 250600 PROBABILITY EMPIRICAL APPROACH In many situations there is no prior knowledge to calculate probabilities What is the probability of a machine being defective Method 1 Monitor the machine over a period of time 2 Find out how many times it becomes defective This experimental or empirical approach EXPERIMENTAI AND THEORETICAL PROBABILITY pevent Number of times event occursTotal number of experiments Larger the number of experiments more accurate the estimate Experimental probability approaches theoretical probability as the number of experiments becomes very large m Consider two events A and B What is the probability of eitherA or B happening What is the probability ofA and B happening What is the number of possibilities Probability of A or B happening Number of ways A or B can happen Total number of possibilities Number of ways A can happen number of ways B can happen Total number of possibilities Or Number of ways A can happen Total number of possibilities Number of ways B can happen Total number of possibilities Probability ofA happening Probability of B happening Condition for Or Rule A and B must be mutually exclusive When A and B are mutually exclusive MA or B MN pB OR RULE EXAMPLE If a dice is thrown what is the chance of getting an even number or a number divisible by three peven 36 pdiv by 3 26 peven or div by 3 36 26 56 The number 6 is not mutually exclusive Hence Correct answer 46 AND RULE Probability ofA and B happening Probability ofA x Probability of B Example 273 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In a factory 40 workforce are women Twenty five percent females are in management grade Thirty percent males are in management grade What is the probability that a worker selected is a women from management grade m pwoman chosen 25 25 females management grade 30 of males management grade pwoman amp Management grade pwoman x pmanagement Assume that the total workforce 100 pwoman 04 p management 025 pwoman x p management 04 x 025 01 or 10 SET OF MUTUALLY EXCLUSIVE EVENTS To cover all possibilities between mutually exclusive events add up all the probabilities Probabilities of all these events together add up to 1 PW I003 I0C I0N 1 EXHAUSTIVE EVENTS A happens orA does not happen then A and B are Exhaustive Events pA happens A does not happen 1 The sum of the probabilities of all mutually exclusive and collectively exhaustive events is always equal to 1 That is pA pB pC 1 if A B C are mutually exclusive and collectively exhaustive events 274 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 1 pyou pass 09 pyou fail 1 09 01 EXAMPLE1 EXHAUSTIVE EVENTS A production line uses 3 machines The Chance that 1st machine breaks down in any week is 110 The Chance for 2nd machine is 120 Chance of 3rd machine is 140What is the chance that at least one machine breaks down in any week w pat least one not working pall three working 1 pat least one not working 1 pall three working pall three working p1st working x p2nd working x p3rd working p1st working 1 p1st not working 1110 910 p2nd working 1920 p3rd working 3940 pall working 910 x 1920 x 3940 66698000 pat least 1 working 1 66698000 13318000 APPLICATION OF RULES A firm has the following rules When a worker comes late there is A chance that he is caught First time he is given a warning Second time he is dismissed What is the probability that a worker is late three times is not dismissed m Let us use the denominations 1C Probability of being Caught first time 1NC Probability of being Not Caught first time 2C Probability of being Caught 2nd time 2NC Probability of being Not Caught 2nd time 3C Probability of being Caught 3rd time 3NC Probability of being Not Caught 3rel time Probabilities of different events can be calculated by applying the AND Rule 1C14 amp 2C14 Dismissed 1 116 464 1C14 amp 2NC34 amp 3C14Dismissed 2364 1C14 amp 2NC34 amp 3NC34Not dismissed 1964 1NC34 amp 2C14 amp 3C14Dismissed 3364 1NC34 amp 2C14 amp 3NC34Not dismissed 2964 1NC34 amp 2NC34 amp 3C14Not dismissed 3964 1NC34 amp 2NC34 amp 3NC34Not dismissed 42764 pcaught first time but not the second or third time A x x 964 pcaught only on second occasion x A x 964 plate three times but not dismissed pnot dismissed 1 pnot dismissed 2 pnot dismissed 3 pnot dismissed 4 964 964 964 2764 5464 guqht using OR Rule pcaught pdismissed 1 pdismissed 2 pdismissed 3 464 364 364 1064 guqht and pnot caught using rule about Exhaustive events pnot caught 1pnot caught 1 1064 5464 275 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 LECTURE 36 ELEMENTARY PROBABILITY PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 35 0 Elementary Probability PROBABILITY CONCEPTS REVIEW Most of the material on probability theory along with examples was included in the handout for lecture 35 You are advised to refer to handout 35 Some of the concepts and examples have been further elaborated in this handout Probability means making assessment of chances The simplest example was the probability of Rashid getting the lottery when he was one of 600 The probability of the event was 1600 PERMUT EXAMPLE ln handout for lecture 35 we looked at the function PERMUT that can be used for calculations of permutations An example is shown in the slide kE Micrueuft Excel HunkI Elle Edit iew Insert Fgrmat eels Data indew Help DEQJE rm Evil iH vHHEEE BED 139 i5 e e r e E r e 1 PERMUTI1LIITIIJEI39I lLll l lel hUSEl l 3 Date Description 4 l Number bf bbjeete e 13 Number bf bbjeete in each permutetien e r PERMUTMA5 below 9 OR RULE REVIEW When two events are mutually exclusive the probability of either one of those occurring is the sum of individual probabilities This is the OR Rule This is a very extensively used rule A and B must be mutually exclusive The formula for the OR rule is as under MA or B MN pB Copyright Virtual University of Pakistan 276 Business Mathematics amp Statistics MTH 302 Example If a dice is thrown what is the chance of getting an odd number or a number divisble by three Podd 36 pdiv by 3 26 podd or div by 3 36 26 56 The number 6 is not mutually exclusive Hence correct answer 56 AND RULE REVIEW The AND Rule requires that the events occur simultaneously Example 60 workforce are men pman chosen 35 25 females management grade 30 of males management grade What is the probability that a worker selected is a man from management grade Example pman amp management grade pman x pmanagement Total workforce 100 pman 06 p management 03 pman x p management 06 x 03 018 or 18 SET OF MUTUALLY EXCLUSIVE EVENTS REVIEW Between them they cover all possibilitiesProbabilities of all these events together add up to 1 Exhaustive Events are events that happen or do not happen pit rains 09 pit does not rain 1 09 01 Example In Handout for lecture 35 we studied the problem of the three machines A production line uses 3 machines Chance that 1St machine breaks down in any week was 110 Chance for 2nd machine was 120 Chance of 3rd machine was 140 What is the chance that at least one machine breaks down in any week What are the probabilities Probability that one or two or three machines are not working in other words at least one not working and that all three areworking add up to 1 as exhaustive events Pat least one not working paII three working 1 From the above the probability that at least one is not working is worked out Pat least one not working 1 paII three working Now to work out the probability that all three are working we need to think in terms of machine 1 and machine 2 and machine 3 working This means application of the AND Rule paII three working p1st working x p2nd working x p3rd working Now the probability of machine 1 working is not known The probability that machine 1 is not working is given These two events working and not working are exhaustive events and add up to 1 Thus the event that machine 1 is working p1st working can be calculated as 1 p1st not working 1 110 910 The calculations for the other machines are p2nd working 1120 1920 p3rd working 1 140 3940 277 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Now the combined probability of pall working is a product of their individual probabilities using the AND Rule 910 X 1920 X 3940 66698000 Finally Pat least 1 working or 1 66698000 13318000 278 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 37 PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 36 0 Patterns of Probability Binomial Poisson and Normal Distributions MODULE 7 Module 7 covers the following Factorials Permutations and Combinations Lecture 34 Elementary Probability Lectures 35 36 Patterns of probability Binomial Poisson and Normal Distributions Pan14 Lectures 37 40 MODULE 8 Module 8 covers the following Estimating from Samples Inference Lectures 41 42 Hypothesis Testing ChiSquare Distribution Lectures 43 44 Planning Production Levels Linear Programming Lecture 45 Assignment Module 7 8 EndTerm Examination EXAMPLE 1 We covered in the past two lectures Elementary Probability Most of the material was included in Handout 35 Some questions were discussed in detail in handout 36 In lecture 37 the example where the employee was warned on coming late and dismissed if late twice will be discussed The material for this example is given in handout 35 Here we shall cover the main points and the method A firm has the following rules When a worker comes late there is A chance that he is caught First time he is given a warning Second time he is dismissed What is the probability that a worker is late three times is not dismissed m How do we solve a problem of this nature The answer is to develop the different options first Let us see how it can be done First time There are two options Caught 1C Not Caught 1NC 2nd time Caught 2C Not Caught 2NC 3ml time Caught 3C Not Caught 3NC 279 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Look at combinations up to 239quot1 stage 1CampZC 1Camp 2NC 1NCamp 2C 1NCamp 2NC Look at combinations unto 3ml stage 1Camp20amp3C 1C amp 2NC amp 3C 1C amp 2NC amp 3NC 1NC amp 2C amp 3C 1NC amp 2C amp 3NC 1NC amp 2NC amp 3C 1NC amp 2NC amp 3NC You saw that the first case is 1C amp 2C Here the employee was caught twice and was dismissed He can not continue Hence this case was closed here In other cases the combinations were as given above Now the probability of being caught was A As an exhaustive event the probability of not being caught was 1 A Now the probabilities can be calculated as follows 1C amp 2C 14X14 116 1C amp 2NC amp 3C 14X 34X14 364 1C amp 2NC amp 3NC 14X34X34 964 1NC amp 2C amp 3C 34x14x14 364 1NC amp 2C amp 3NC 34x14X34 964 1NC amp 2NC amp 3C 34x34x14 964 1NC amp 2NC amp 3NC 34x34x34 2764 The probabilities for each combination of events are now summarized below First Caught Second Caught Dismissed 10 14 amp 20 14 Dismissed 1 116 464 First caught Second Not Caught 3rel Caught Dismissed 10 14 amp 2NC 34 amp 30 14 Dismissed 2 364 First caught Second Not Caught 3rel Not Caught Not Dismissed 10 14 amp 2NC 34 amp 3NC 34 Not dismissed 1 964 First Not Caught Second Caught 3rd Caught Dismissed 1NC 34 amp 20 14 amp 30 14 Dismissed 3 364 First Not caught Second Caught 3rd Not Caught Not Dismissed 1NC 34 amp 20 14 amp3NC 34 Not dismissed 2 964 First caught Second Not Caught 3rel Caught Not Dismissed 1NC 34 amp 2NC 34 amp 30 14 Not dismissed 3 964 First caught Second Not Caught 3rd Not Caught Not Dismissed 1NC 34 amp 2NC 34 amp 3NC 34 Not dismissed 4 2764 Probabilities pcaught The probability of being caught can be calculated by thinking that these are mutually events All situations where there was a dismissal can be considered Probabilitycaught pdismissed 1 pdismissed 2 pdismissed 3 464 364 364 1064 280 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU pnot caught Once we have the probability of being caught we can find out the probability of not being caught as an exhaustive event Thus pnot caught 1 pcaught 1 1064 5464 EXAMPLE 2 Two firms compete for contracts A has probability of 3A of obtaining one contract B has probability of A What is the probability that when they bid for two contracts firm A will obtain either the first or second contract m PA gets first or A gets second 64 Wrong Probability greater than 1 We ignored the restriction events must be mutually exclusive We are looking for probability that A gains the first or second or both We are not interested in B getting both the contracts pB gets first x pB gets both A x A 116 pA gets one or both 1 116 1516 Alternative Method Split quotA gets first or the second or bothquot into 3 parts A gets first but not second 3A x A 316 A does not get first but gets second A x 316 A gets both x 916 PA gets first or second or both 316 316 916 1516 EXAMPLE 3 In a factory 40 workforce is female 25 females belong to the management cadre 30 males are from management cadre lf management grade worker is selected what is the probability that it is a female Draw up a table first 281 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Male Female Total Management NonManagement Total 40 100 Calculate Total male 100 40 60 Management female 025 x 40 10 NonManagement female 40 10 30 Management male 03 x 60 18 NonManagement male 60 18 42 Management total 18 10 28 NonManagement total 42 30 72 m Male Female Total Management 18 10 28 NonManagement 42 30 72 Total 60 40 100 pmanagement grade worker is female 1028 EXAMPLE 4 A pie vendor has collected data over sale of pies This data is organized as follows No Pies sold lncomeX Daysf fX Rs 40 x 35 1400 20 28000 50 1750 20 35000 60 2100 30 63000 70 2450 20 49000 80 2800 10 28000 Total 100 203000 Meanday 203000100 2030 The selling price per pie was Rs 35 What was the mean sale per day Such a question can be solved by calculating the sale in each slab and then dividing the total sale by number of pies days is the probability lf multiplied with the income from each pie the expected sale from all pies can be calculated The overall expected value was 203000 When divided by the number of days 100 an average of 2030 Rs Per day was obtained as average sale per day EXPECTED VALUE EMV Z probability of outcome x financial result of outcome Example In an insurance company 80 of the policies have no claim In 15 cases the Claim is 5000 Rs For the remaining 5 the Claim is 50000 Rs What is the Expected value of claim per policy Applying the formula above EMV 08 x 0 015 x 5000 005 x 50000 0 750 2500 3250 Rs 282 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU TYPICAL PRODUCTION PROBEM In a factory producing biscuits the packing machine breaks 1 biscuit out of twenty p 120 005 What proportion of boxes will contain more than 3 broken biscuits This is a typical Binomial probability situation The individual biscuit is broken or not two possible outcomes Conditions for Binomial Situation 1 Either or situation 2 Number of trials n known and fixed 3 Probability for success on each trial p is known and fixed CUMULATIVE BINOMIAL PROBABILITIES The Cumulative Probability table gives the probability of r or more successes in n trials with the probability p of success in one trial In the table The total number of trials n 1 to 10 The number of successes r 1 to 10 The probability p 005 01 02 025 03 035 04 045 05 283 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 38 PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 37 0 Patterns of Probability Binomial Poisson and Normal Distributions CUMULATIVE BINOMIAL PROBABILITIES Probability of r or more successes in n trials with the probability of success in each trial Look in column for n Look in column for r Look at column for value of p005 to 05 In other words a cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range eg is greater than or equal to a stated lower limit and less than or equal to a stated upper limit For example we might be interested in the cumulative binomial probability of obtaining 45 or fewer heads in 100 tosses of a coin see Example 1 below This would be the sum of all these individual binomial probabilities bxg45 100 05 bx 0 100 05 bx 1 100 05 bx 44 100 05 bx 45 100 05 x n You can calculate these values by using the formula Px S c ij 1 P H 620 x Or directly from the table Example The probability that a student is accepted to a prestigious college is 03 If 5 students from the same school apply what is the probability that at most 2 are accepted m To solve this problem we compute 3 individual probabilities Using the binomial formula The sum of all these probabilities is the answer we seek Thus 284 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU bx 5 2 5 03 0837 Table Z t tttil ti Eirmltiial 111 0l111bliiii 55 17 139 H P 1139 1 11 E ii Lain 1x 13quot E 111121 111I1 171111 113131 11411 1511 111111 1111 113131 111211 H115 nl 11 12335112 11912121 11311112 LTtil LEEDS ISLEIIIIII ISL ltIZIII 1111quot ELEVII 111IIIII1 11115131 1 11312111 1111E 11211112 1111111 11211111 11111 1111111 1tIIIIIIII 1111111 lIIIIIIII1 1111111E 313 11 111111 1131121 111411 H410 13611 011111 III l III 11115 IEll 1111111 1111113 1 1311 119 tit1 DELI ISLE111 LEI ELIEll 11110 IIL1ISIII 111911 111193 1 11113111 111E 11213111 1112110 1111111 1111111 HitII 1tIIIIIIII 111110 lIIII111 11111 E 313 11 LEST 1511 H143 H116 H115 IltiISl 111111 ISLE2115 11IIII11 1311111131 1 113211 131911 1131 LTE4 III lE ISLEIZIIZI H151 11116 01114 111113 13111111 1 1121111 11932 111211 I 73 01115 0315 0134 1155 ISL135 11111 131143 3 11312111 1111E 11211112 1111111 11211111 11111 1111111 112IIIIII 1111111 lIIIIIIII1 1111111E i4 11 111127 111555 14111 Illll 11111 ISLE163 111115 111105 ISLE2111 1111011 1311111131 1 115136 131941 113111 01551 I143 H111 Il39lTE 1111134 ISLE1 111111 13111 1 1121111 1319 11111 01116 IILEEfl USES 17112 11143 IIL lE1 1111112 1311114 3 11313111 111E 03111 DEFl 03115 LETIII 11ISIII IILEE39III 11144 171135 4 11312111 1111E 11211112 1111111 11211111 11111 1111111 1tIIIIIIII 1111111 lIIIIIIII1 11111111 11 1114 1315 1313 H165 113113 011111 01110 111101 Eli2110 11III111 13111E 1 1531quot 131311quot 113 0515 I131quot 1155 ISLEIE 1111111 Eli211 1111011 1311111131 P 1133 11991 1941 ELSEE ISLEIIIIII I111 111161 ISLE155 11IIII15 11111111 3 11213111 111E 1111 01165 1911 LEE 0661 11411 01151 111151 111123 BINOMDIST Returns the individual term binomial distribution probability Use BINOMDIST in problems with a fixed number of tests or trials when the outcomes of any trial are only success or failure when trials are independent and when the probability of success is constant throughout the experiment For example BINOMDIST can calculate the probability that two of the next three babies born are male Syntax BlNOMDISTnumberstrialsprobabilityscumulative Numbers is the number of successes in trials Trials is the number of independent trials Probabilitys is the probability of success on each trial Cumulative is a logical value that determines the form of the function If cumulative is TRUE then BINOMDIST returns the cumulative distribution function which is the probability that there are at most numbers successes if FALSE it returns the probability mass function which is the probability that there are numbers successes Remarks Numbers and trials are truncated to integers lf numbers trials or probabilitys is nonnumeric BINOMDIST returns the VALUE error value If numbers lt 0 or numbers gt trials BINOMDIST returns the NUM error value If probabilitys lt 0 or probabilitys gt 1 BINOMDIST returns the NUM error value 285 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The binomial probability mass function is I I sass a s where as Js39 a The cumulativehbinomial distribution is Sines Zbis s 4 Iii39339 Micrusnft Excel Lecturej Eile Edit Eiew insert Fgrmat eels gate induw elp 3 Er 7 E Q i E sun v x a s slscuulsrrsssasssasa A E 0 II E 1 BINMDISTnumberstrialsprbabilityscumulatiue 2 Data Descriptin s BlNumber f successes in trials 4 10 Number f independent trials 5 0539Prbability f success n each trial BINMDISTlA3A4A5FALSE a Prbability f exactly 6 f 10 trials 9 being successful 0205078 1 Example In the above example the BINOMDIST function was used to calculate the probability of exact 6 out of 10 trials being successful Here the value of Cumulative was set as False The following example also shows a similar calculation 286 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EE Micrusuft Excel Lecture33 Eile Edit ew insert Fgrmat Innls gate indnw Help Eggs itav ma fluv uggss i ass 2 s a E c 12 EXAMPLE 22 Five sins are tssed simultaneusly 21 What is the chance f btaining heads 22 22 Data Descriptin 22 Number f successes in trials 22 5 Number f independent trials 22 05 Prbability f success n each trial 2 22 03125 22 Prlsalsility f exactly f 5 trials 22 being successful 03125 31 EXAMPLE 1 The probability of wet days is 60 Note that the figure 06 is beyond the maximum value 05 as given in the tables Let us first convert our problem to pdry 1 06 04 Now p5 or more wet days can be restated as p2 or less dry days The BINOMDIST function is for pr or more Let us convert p2 or less dry days to 1 p3 or more days Now the value of n 7 r 3 and p 04 Using BINOMDIST the answer is 04199 Note that the value of cumulative was TRUE 287 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E P icrusuft Excel Lecture3 Eile Edit Eiew insert Fgrmat eels gate indnw Help De ie g t e v v ezveememmw Ariel vIDvBIHEEEEE E gfdgfg D419 139 fa A El I3 35 EXAMPLE 344 Prebability ef wet deye in current menth 34 Prebebility ef er mere wet deye next week 34 p The teblee are fer p upte 39 Turn the queetien Pljdry 1 04 4D pf er mere wet days 7 er leee dry days 44 p er leee dry days 1 pj 3 er mere dry days 42 Date Deeeriptien 43 2 Number ef eueeeeeee in triele 44 739 Number 11quot independent trials 45 04 Prebebility ef eueeeee en each trial 45 BNOMDISTfA43A44A45TRUE 44 At meet eueeeeeee 1 EXAMPLE 2 In a transmission where 8 bit message is transmitted electronically there is 10 probability of one bit being transmitted erroneously What is the chance that entire message is transmitted correctly We can state that the probability required is for 0 successes errors in 8 trials bits pone bit transmitted erroneously 01 PX0 8 p01 0430 For exact binomial distribution n x n x 8 0 8 0 Pxn P1 P 0 011 01 0430 x 288 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU T0010 30111111011120 011101111101 15101301511003 Emmimed c 005 000 030 040 050 000 000 000 000 005 0003 0100 0050 0010 0004 0001 0000 0000 0000 0000 0043 0503 0255 0100 0035 0000 0001 0000 0000 0000 0004 0003 0000 0552 0315 0145 0050 0011 0001 0000 0000 1000 0005 0044 0000 0504 0503 01004 0050 0010 0000 0000 1000 1000 0000 0042 0020 0030 0400 0104 0050 0005 0000 5 1000 1000 0000 0000 0050 0055 0005 0440 0303 0030 0000 0 1000 1000 1000 0000 0001 0005 0004 0045 0400 0100 0050 0 1000 1000 1000 1000 0000 0000 0003 0043 0031 0500 0530 0 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 n 0 0 0030 03300 0134 0040 0010 0002 0000 0000 0000 0000 0000 1 0020 0005 0430 0100 0001 0000 0004 0000 0000 0000 0000 2 0002 0041 0030 0403 0233 0000 0025 0004 0000 0000 0000 5 0000 0002 0014 0030 0403 0254 0000 0035 0003 0000 0000 4 1000 0000 0000 0001 0033 0500 0200 0000 0000 0001 0000 5 1000 1000 0000 0005 0001 0040 0510 0000 0000 0000 0001 0 1000 1000 1000 0000 0005 0010 0000 0530 0302 0053 0000 0 1000 1000 1000 1000 0000 0000 0000 0004 0504 0205 0001 r7 050 1100 in i N a 1 20 ii 01 51 Using BINOMDIST The data was for 0 or more successes BINOMDIST function gives the value for at most r successes Hence the answer was obtained directly 289 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ejlicrnsnft Excel Lecturej Elle Edit Eiew insert Fgrrnat Innis gate window Help assesses rm gEv lii i jmnaT Mal 1D39BIHEEE T63E3 E ElEri v 5 A 51 EXHMPLE 52 Prbability f1 errneus bit 01 53 Prbability f 8 crrect bits l errneus hits a Data Descriptin 55 1 Number f successes in trials as 8 Number f independent trials 5 01 Prbability f success n each trial as 041305 BINMDISTBS B EB 7TRUE as M mst U successes 04305 El a E El EXAMPLE 3 A surgery is successful for 75 patients What is the probability of its success in at least 7 cases out of randomly selected 9 patients psuccess in at least 7 cases in randomly selected 9 patients Here n 9 psuccess 075 pat lease 7 cases p 075 is outside the table Let us invert the problem pfailure 1 075 025 Success at least 7 Failure 2 or less Pxgt7 n9p075 1 pxlt7 n9p075 1 pxlt6n9p075 1 03995 06005 60 290 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 i 003 0003 1 0943 3 0994 3 1000 4 1000 5 1000 0 1000 3 1000 3 1000 0030 1 0939 3 0992 3 0999 4 1000 5 1000 1000 3 1000 3 1000 010 0430 0313 0903 0993 1000 1000 1000 1000 1000 033 03 094 0993 0999 1000 1000 1000 1000 030 0103 0503 039 0944 0990 0999 1000 1000 1000 0134 0430 0333 0914 0930 099 1000 1000 1000 Qlculgtion usinq BINOMDIST Here the question was inverted We had to find 7 successes out of 9 The probability was 75 for success It becomes 030 003 3 0130 0532 0300 0942 0939 0999 1000 1000 0040 0190 0403 030 0901 093 0990 1000 1000 040 001 0100 0313 0394 0330 0950 0991 0999 1000 0010 001 0233 0433 033 0901 093 0990 1000 P 050 0004 0030 0140 0303 003 035quot 0903 0990 1000 0002 0030 0090 0354 0500 0340 0910 0930 0993 000 0001 0009 0050 0134 0400 0035 0394 0933 1000 25 for failure Now let us restate the problem in terms of failure We are interested in 7 or more successes It means 2 or less failures Now the BINOMDIST function gives us at most r successes In other words 2 or less Hence if we specify r 2 we get the answer 06007 directly 0000 0001 0011 0053 0194 04743 0343 0943 1000 0000 0000 0004 0033 0099 030 0000 0000 0001 0010 0030 0303 049 0333 1000 0300 090 0000 0000 0000 0000 0003 0033 013 0330 1000 0000 0000 0000 0000 0001 0003 01 31935 093 0000 0000 0000 0000 0000 0000 0105 033 1000 0000 0000 0000 0000 0000 0001 0003 001 030 291 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EJhicrusuft Excel Lecturej Elle Edit Eiew insert Fgrmat Innis Qatar induw Help 139 51 near sea a a m a sltl aamnro a Fll39ial vl v f ggg ga3 3 amp 234 v 5 A E C D E as EXAMPLE rr Prbability f success 075 Failure 025 ra Prbalaility f 7 successes ut f 9 r 2 r less failures F9 an Data Descriptin m 2 Number f successes in trials as 9 Number f independent trials as 025 Prbability f success n each trial a 16007 BINMDISTBB1BB2383TRUE as At mst 2 successes 06007 SE 292 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NEGATIVE BINOMIAL DISTRIBUTION A negative binomial experiment is a statistical experiment that has the following properties The experiment consists of X repeated trials Each trial can result in just two possible outcomes We call one of these outcomes a success and the other a failure The probability of success denoted by P is the same on every trial The trials are independent that is the outcome on one trial does not affect the outcome on other trials The experiment continues until rsuccesses are observed where r is specified in advance Consider the following statistical experiment You flip a coin repeatedly and count the number of times the coin lands on heads You continue flipping the coin until it has landed 5 times on heads This is a negative binomial experiment because The experiment consists of repeated trials We flip a coin repeatedly until it has landed 5 times on heads Each trial can result in just two possible outcomes heads or tails The probability of success is constant 05 on every trial The trials are independent that is getting heads on one trial does not affect whether we get heads on other trials The experiment continues until a fixed number of successes have occurred in this case 5 heads Negative Binomial Formula Suppose a negative binomial experiment consists of xtrials and results in r successes If the probability of success on an individual trial is P then the negative binomial probability is W6 r F x1cr1 Pr 1 P Example Bob is a high school basketball player He is a 70 free throw shooter That means his probability of making a free throw is 070 During the season What is the probability that Bob makes his first free throw on his fifth shot 30111 170quot This is an example of a geometric distribution which is a special case of a negative binomial distribution Therefore this problem can be solved using the negative binomial formula or the geometric formula We demonstrate each approach below beginning with the negative binomial formula The probability of success P is 070 the number of trials X is 5 and the number of successes r is 1 We enter these values into the negative binomial formula 293 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU bX r X1Cr1 Pr QXr b5 1 07 400 071 034 b5 3 07 000567 NEGBINOMDIST Returns the negative binomial distribution NEGBINOMDIST returns the probability that there will be numberf failures before the numbersth success when the constant probability of a success is probabilitys This function is similar to the binomial distribution except that the number of successes is fixed and the number of trials is variable Like the binomial trials are assumed to be independent For example you need to find 10 people with excellent reflexes and you know the probability that a candidate has these qualifications is 03 NEGBINOMDIST calculates the probability that you will interview a certain number of unqualified candidates before finding all 10 qualified candidates Syntax NEGBINOMDISTnumberfnumbersprobabilitys Numberf is the number of failures Numbers is the threshold number of successes Probabilitys is the probability of a success Remarks 0 Numberf and numbers are truncated to integers o If any argument is nonnumeric NEGBINOMDIST returns the VALUE error value o If probabilitys lt 0 or if probability gt 1 NEGBINOMDIST returns the NUM error value o If numberf numbers 1 S 0 NEGBINOMDIST returns the NUM error value 0 The equation for the negative binomial distribution is xr1 H I with 1 P 1 P where x is numberf r is numbers and p is probabilitys 294 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NEGBLNOMDIST EXAMPLE You need to find 10 people with excellent reflexes and you know the probability that a candidate has these qualifications is 03 NEGBINOMDIST calculates the probability that you will interview a certain number of unqualified candidates before finding all 10 qualified candidates lilicreeeft Excel Lecture Elle Edit iew insert Fgrmet eels gate window Help T D g rg r Evsl mmv ueeetset BEE 1r f e s c u E 11 NEGBINMDISTlnumberj numbersprbabilitys 12 Date Descriptin 3 10 Number f failures 4 5 Threshld number f successes 5 025 Prbebility f success a 1r Negative binmiel distributin fr the 1E terms shire 0055040 19 m CRITBINOM Returns the smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value Use this function for quality assurance applications For example use CRITBINOM to determine the greatest number of defective parts that are allowed to come off an assembly line run without rejecting the entire lot Syntax CRITBINOMtriaIsprobabilitysalpha Trials is the number of Bernoulli trials Probabilitys is the probability of a success on each trial Alpha is the criterion value Remarks If any argument is nonnumeric CRITBINOM returns the VALUE error value If trials is not an integer it is truncated lf trials lt 0 CRITBINOM returns the NUM error value If probabilitys is lt 0 or probabilitys gt 1 CRITBINOM returns the NUM error value If alpha lt 0 or alpha gt 1 CRITBINOM returns the NUM error value i i i i Example A B 1 Data Description 2 6 Number of Bernoulli trials 3 05 Probability of a 295 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 4 075 Formula CRITBNOMA2A3A4 success on each trial Criterion value Description Result Smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value 4 296 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 39 PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS PART 3 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 38 0 Patterns of Probability Binomial Poisson and Normal Distributions CRITBINOM EXAMPLE The example shown under CRITBINOM in Handout 38 is shown below Microsoft Excel Lecture3 Eile Edit Eiew losert Format Iools gate window Help v infcig 39 739 3 1 E SUM v X of f3 CRITEiNDM1 EB EEtAFDj A I El es CRITBINMtrialsprbabilitysalphaj EE 5 Data Description 53 Number of Bernoulli trials 05 Prbability f a success h each 5E trial a U75Criterin value Smallest value fr which the cumulative e CRITBIN binomial distribution is greater than a WAE or equal to a criterion value 4 e A69ATU as EXPECTED VALUE EXAMPLE A lottery has 100 Rs Payout on average 20 turns Is it worthwhile to buy the lottery if the ticket price is 10 Rs Expected win per turn pwinning x gain per win plosing x loss if you loose 120x 100 10 1920x 10 Rs 9020 19020 Rs 45955Rs So on an average you stand to loose 5 Rs DECISION TABLES Look at the data in the table below No of Pies demanded Occasions 25 10 30 20 35 25 40 20 45 15 50 10 297 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Price per pie Rs 15 Refund on return Rs 5 Sale price Rs 25 Profit per pie Rs 25 15 Rs 10 Loss on each return Rs 15 5 Rs 10 How many pies should be bought for best profit To solve such a problem a decision table is set up as shown below The values in the first column are number of pies to be purchased Figures in columns are the sale with share of sale within brackets If the number of pies bought is less than the number that can be sold the number of pies sold remains constant If the number of pies bought exceeds the number of pies sold then the remaining are returned This means a loss For every value the sum of profit for sale and loss for pies returned is calculated The average sale for each row is calculated by multiplying the profit for each sale with sale in the column An example calculation is given as a guide for 30 pies DECISION TABLES 2501 3002 35025 4002 45015 5001 EMV 25 250 250 250 250 250 250 250 30 200 300 300 300 300 300 290 35 150 250 350 350 350 350 310 Buy 40 100 200 300 400 400 400 305 45 50 150 250 350 450 450 280 0 100 200 300 400 500 240 Expected profit 30 Dies 01x 200 02 x 300 025 x 300 02 x 300 015 x 300 01x 300 206075604530 290 Rs Best Profit It may be noted that the best profit is for 35 Pies Rs 310 DECISION TREE TOY MANUFACTURING CASE The problem of the manufacturer intending to start manufacturing a new toy under the conditions that the TV series may or may not appear that the rival may or may not sell a similar toy is now solved below Here a Decision tree has been developed with the possible branches as shown below Each sequence represents an application of the AND rule 1A Abandon 18 Go ahead gt2A Series appears 60 gt28 No series 40 gt2Agt3A Rival markets 50 gt2Agt3B No Rival 50 Production Series no rival 12000 units Series rival 8000 units No series 2000 units Investment Rs 500000 Profit per unit Rs 200 Loss if abandon Rs 500000 What is the best course of action 298 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Decision Tree Profit if rival markets series appears 8000 x 200 500000 1600000 500000 1100000 Rs Profit if no rivals 12000 x 200 500000 2400000 500000 1900000 Rs ProfitLoss if no series 2000 x 200 500000 400000 500000 100000 Rs No series EMV Rival markets and no rivals 05 x 1100000 05 x 1900000 1500000 Series EMV 06 x 1500000 04 x 100000 900000 40000 860000 Rs Conclusion It is clear that in spite of the uncertainty there is a likelihood of a reasonable profit Hence the conclusion is Go ahead THE POISSON DISTRIBUTION The Poisson distribution is most commonly used to model the number of random occurrences of some phenomenon in a speci ed unit of space or time For example 0 The number of phone calls received by a telephone operator in a 10minute period 0 The number of aws in a bolt of fabric 0 The number of typos per page made by a secretary It has the following characteristics Either or situation No data on trials No data on successes Average or mean value of successes or failures This is a typical Poisson Situation Characteristics Eitheror situation Mean number of successes per unit m known and fixed p chance unknown but small event is unusual For a Poisson random variable the probability that X is some value X is given by the formula x PXx ue39 x0l X where J is the average number of occurrences in the speci ed interval For the Poisson distribution EX VarX Example 299 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The number of false re alarms in a suburb of Houston averages 21 per day Assuming that a Poisson distribution is appropriate the probability that 4 false alarms will occur on a given day is given by 2 146 21 PX 4 T 00992 THE POISSON TABLES OF PROBABILITIES Gives cumulative probability of r or more successes Knowledge of m is required Table gives the probability of that r or more random events are contained in an interval when the average number of events per interval is m Example 2 Attendance in a factory shows 7 absences What is the probability that on a given day there will be more than 8 people absent m Method 1 PX gt81 PX 8l PxlPx2Px3Px4Px5 Px6Px7Px8 7161 7262 7363 7464 7565 7666 7767 7868 1 l 2 3 4 5 l 7 7 l 00064002230052l009 l20 l2770 1490 1490 1304 02709 Method 2 PX gt 8 l PX S 8 l0729l 02709 3 00 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU p9 or more successes 02709 Iif3333333i3ti3te Pnieenn Dietrihntinn T3333 Table 31333333 ennntletit39e 13r3h3l3iiit fnnetiene 3f P3333333 Dietrihentien with 33333313 339 E33333 13qie t3 nd the 13333133hiiit33 if where here 3 Pnieeon Distribution with 339 2 13333 in r3w 3 3333 311333333 3 t3 nd if 33085 1 where i3 P33333332 3 33 3 33 3 3 33 3 33 3 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33 3 33 b 33 3 3 33 33 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 333 33 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 1 33333 33333 33333 333 33333 33333 33333 33333 33333 33333 53 33333 33333 33333 33 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33133 33333 33333 33333 33333 33333 3333 33 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 33 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 33 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 33 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 33 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 33 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 33 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 Example 3 An automatic production line breaks down every 2 hours Special production requires uninterrupted operation for 8 hours What is the probability that this can be achieved m a 82 4 x 0 no breakdown 0 0 p x0 00183 1 83 0 From Table 301 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Chitiiulatite Peleebn Dietributibh Table Table ahawe cumulative parbbabilit le tjti I lE bf Paiaaen Dietributieh with variants rt Exam ple tb nd the prebabilitgr PEX if where X haa a Paiaaen Dietributibh with rt 2 lablt in raw 4 and mlumn 4 tb nd if 3U85T1 where ia Pbaieabaii j Q at i la a a 35 4L at a Elu l II 3 1337 II 23 1 II 1 35 3 13213132 1 EIUIHQE II393III392 iEi1l 3 3 ll l l l 1 El IJ EIIE T EII E IZIIQE393 InT3rir3 II SHE 414113161 In33T3 b1991 411359 illfligll lufli ll ll lanai b9353 LB 1939 b3383 Ill T T In3433 b4333 Ill33EI398 233 l 1321733 El 12439 b9933 3931i 9344 Illt CiTll LT5713 IEIh LTE II 531313 b4335 1313433 Ell 2135i b9993 1319963 Kilt31314 419433 1313912 1313 153 II T354 1162233 b5331 creative 1El39llllililit II Q SELL II 9955 ID 983 4 II 958i lug lift 1 IIJEETE i 735 l 117132 9 DUB l i ll U39EIIIEIEIJ II 9 991339 II 139939 ll ID 93995 5 II 985 3 lug E363 5 3934 II i 3 3 l l El T Ilfi 3 3 ll IEIIIEIIIEIEIJ Il IIJIIJIIJII i 9998 ID 9 93 9 II 995 3 lug 38 ll il39ElT33 i Q43 9 LE1 Il U39 3 131313 a HL lJt a 39lll39ll7irimtt39 LLHR t mmm a mmn a FLA t nan t FEE45 Example 4 An automatic packing machine produces on an average one in 100 underweight bags What is the probability that 500 bags contain less than three underweight bags Solution m 1x500100 5 pxlt3 0 5 pxlt2 0 5 012471247 Chitiiulatitre Pbieebh Dietributibh Table Table ahbwa euhiulatiwe prbbabilit funetibha emf PtJIlSEDIH Dietribtitiah with rarieua rt Exam ple tb nd the probability if where haa a Peiaabrt Dietributibh with be 2 lbbl in raw 4 and ebluirih 4 ta nd if 3 85 l where ia Peieabh CI ml at i 15 a a 35 a 45 C 5 aetiaa aaara aaaai aiaaa aaaai amaa abate ataaa aaiii timer agree araaa aaara aaaaa aaara b1991 aiaaa abate aaaii errata agate b9113 aeriaa aerer aaaaa aaaaa aaaaa aaaai airaa aaaaa again b9344 man were were aaaee aaaaa aaaaa aaaaa aaaaa aaaea t19814 aaara aaaia aaiaa araaa aaaaa aaaai Haiti retire 4119an aaaaa aaaaa traaaa aaiei aaare arear artea aeiea retire aaaaa aaaai agate aaaaa aaaea aaaar aaaaa aaaii areaa inane their aaaaa aaeaa aaaaa b9381 aaraa aaaaa 1191134 aaaea were that iiith aaaaa aaaaa aaaaa again aaraa uaaar aaaia retire termini were three aaaar 399813 aaaar aaaia tiaaaa aaaaa 39E i i iim ri b gl aiei 302 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 40 PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS PART 4 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 39 0 Patterns of Probability Binomial Poisson and Normal Distributions Part 4 POISSON WORKSHET FUNCTION Returns the Poisson distribution A common application of the Poisson distribution is predicting the number of events over a specific time such as the number of cars arriving at a toll plaza in 1 minute Syntax POISSONxmeancumulative X is the number of events Mean is the expected numeric value Cumulative is a logical value that determines the form of the probability distribution returned If cumulative is TRUE POISSON returns the cumulative Poisson probability that the number of random events occurring will be between zero and x inclusive if FALSE it returns the Poisson probability mass function that the number of events occurring will be exactly x Remarks 0 Ifx is not an integer it is truncated Ifx or mean is nonnumeric POISSON returns the VALUEI error value If x S 0 POISSON returns the NUM error value If mean S 0 POISSON returns the NUM error value POISSON is calculated as follows For cumulative FALSE 1 xii PGLS SGN I For cumulative FJIALSE 1 E A I CUMPGISSGN Z irI If Example An application of the POISSON function is shown below In this slide the value of Cumulative was TRUE It means that the probability is for at the most case 3 03 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ejl icreseft Excel Lecture39 Eile Edit iew Lnsert Fgrmat eels gate indew Help if cit IE v r v s v 41 e SLIM v x a s F39DISSDHEA3MTRLJEJ A El 3 D E 1 PISSNxmeeneumuletiue 2 Date Deserintin 3 2 Number f events 4 5 Expected mean 5 s PIS Cumulative Pissn pirbeleility with Sm the terms slave 012d652 3 e3 1 M 12 In the slide below the Cumulative is FALSE which means that the probability is for ectl 2 events Ejiliiicrueeft Excel Lecture E Eile Edit iew insert Fgrmat eels gate window elp 29 it s m 2 v e SLIM 1 X J 52 PDISSDNWFA IEFALSEJ A e t D E 15 PISSNmmean umuletive 15 Date Descriptin 1 2 Number f events W SHExpected mean 19 SPISSM Cumulative Pissn prbebility with 21 MEMB the terms alive 0384224337 3 FALSE 24 3 04 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU THE PATTERN In Binomial and Poisson the situations are eitheror Number of times could be counted In the Candy problem with underweight boxes there is measurement of weight Binomial and Poisson are discrete probability distributions Candy problem is a Continuous probability distribution Such problems need a different treatment FREQUENCY BY W GHT Look at the frequency distribution of weight of sample bags Microsoft Excel Lecture Elle Edit iew lnsert Fgrrnal Iools Qatar window elp De neaavt lt3 azv irinaluuorove trial vIDvBIH tutitt F3r v 5 A E C D E F G H J 23 24 FREQUENCY BY WEIGHT No of bags 25 5013 but under 505 2 25 505 but under 507 12 2 507 but under 509 21 23 509 but under 511 29 29 511 but under 5113 an 513 but under 515 11 31 515 but under 517 2 32 Frequency distribution graph of the sample is shown below You may see a distinct shape in the graph It appears to be symmetrical I I I FEEUENC V DIETEIEUTIN El Series l I Series a El Series I I El Seriesdl E l E E E l SeriesE I 1 39U 1 E g g H E r g Ln EISenesE g E E E E E E lSeriesf L 3 E El Seriesa E m E E m Weight I I I 3 05 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The shape of the distribution is that of a Normal Distribution as shown as New distribution in the slide below On this slide you also see a Standard Normal Distribution with 0 mean and standard deviations 1 2 3 4 etc Standard distribution New distributidn 43543212D 12345l3 3 06 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NORMAL DISTRIBUTION The blue Curve is a typical Normal Distribution A standard normal distribution is a distribution with mean 0 and standard deviation 1 The Yaxis gives the probability values The Xaxis gives the 2 measurement values Each point on the curve corresponds to the probability p that a measurement will yield a particular 2 value value on the xaxis Probability is a number from 0 to 1 Percentage probabilities multiply p by 100 Area under the curve must be one Note how the probability is essentially zero for any value 2 that is greater than 3 standard deviations away from the mean on either side Mean gives the peak of the curve Standard deviation gives the spread Weiqht distribution Gag Mean 510 g StDev 25 g What proportion of bags weighs more than 515 g Proportion of area under the curve to the right of 515 g gives this probability 3 07 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EA UNER THE STANDARD NOFIMAI CURVE The normal distribution table gives the area under one tail only zvalue Ranges between 0 and 4 in first column Ranges between 0 and 009 in other columns Example Find area under one tail for zvalue of 205 Look in column 1 Find 20 Look in column 005 and go to intersection of 20 and 005 32233 2323 222 33302 Ell033 31313 ELI Els 313329 33333 33333 33333 33333 333333333 3333 33333 32 ll 303323 32512 0552 3063935 2323 0325 2234 2253 332 32322 223 0932 2222 223 1033 E Hilfi 342 til3 322 1222 1255 3223 2332 2353 4326 l342 3423 314 1554 152 3322 l r W 39 2236 11222 322 2244 1229 35 1915 3250 3225 239 2053 2032 2 22 2 3 52 2 323339 2223l 33 2252 2221 2323 2352 2322 2322 2354 2423 25 2 2542 122 2523 2331 l 2332 2323 2 2233 2223 2234 2233 2223 2352 322 2221 2912 2922 2932 2225 33222 225 33222 2223 3 353 2 2122 2136 2212 32 22 3254 2222 3215 2235 232939 33 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 33 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 33 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 L3 3333 3333 3333 3333 3333 3313 3333 3333 3333 3333 33 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 33 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 33 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 lllJ39 3553 4564 4523 3522 342 3522 4203 1133 3 4625 3333 333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 12 3213 4219 3226 4222 4232 4243 42 533 3253 3263 3262 3333 333 3333 3333 3333 3333 3333 3333 333 393 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3911 quot339 l39ill 32332 E d 39 39l l 333339 33121 31E di dl d You can nd the probability 04798 at the intersection of 20 and 005 ie 25 which is corresponding zvalue As the probability at right from the center of the curve is 05 and also at left is 05 E33333 3333333 EH333333 3333333333 333quot3333333333 3333 3331333 33333333 3333333 33333 373333 3 3 33 2 3 0 8 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Now if we want to nd the probability corresponding to 2 value greater than equal to 25 then we have to subtract the value of probability correspoing to given 2 value from 05 EMMU LawFE HEREAIL FREQUENCY quotEETREUTlDH area unde standard normal curve from E the Z 31 0504798 00202 202 309 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU CALCULATING Z VALUES 2 Value x MeanStDev Process of calculating z from x is called Standardization 2 indicates how many standard deviations the point is from the mean W Find proportion of bags which have weight in excess of 515 g Mean 510 StDev 25 g Solution 2 515 51025 2 From tables probability corresponding to 2 value is 04772 J hrLE 555 552 552 525 255 5525 22 252 522 2 55255 52555 5522 55552 55155 55225 55222 5555 52552 51 5555 5555 5555 5555 5555 5555 5555 5555 5555 5555 52 5555 55 52 555 5 55 55 5555 5555 5525 139555 5 555 5 155 53 1555 1215 5255 5253 1355 5255 5555 1453 5555 555 55 1555 l 555 5 525 5555 5 53955 5 555 5 552 5555 5 555 5555 5 2555 525 1525 2252 2555 2525 2525 2552 2552 2225 55 2255 2255 2525 2555 2555 2522 2555 2555 2515 2555 55 2555 255 5 2552 2555 2555 2555 2555 2555 2525 2552 55 2555 255 2555 2555 2555 5525 5555 5555 5555 5555 55 5555 5555 5252 5255 5255 5255 5555 5555 5555 5555 15 5515 3435 3555 5555 5555 555 3555 5555 255 3625 55 5555 5555 5555 5555 5525 5555 5555 5555 5555 5555 52 5555 5555 5 55 5555 5525 5555 5552 5555 5555 5555 5 5552 5555 5552 5555 55 5 5 5555 5 55 5 552 55 55 15 5152 5252 5225 5252 5255 5255 5252 5555 5222 5 5 5552 5555 5555 5555 5552 5555 5555 5555 5525 5555 5 5 5552 5555 5555 5555 5555 4555 55 5 5525 5535 5555 5 5 55 55 5555 5555 5552 5555 5555 555 555 5 5525 5555 5 5 5555 5555 5555 5555 555 5 5555 5555 5555 5555 5555 55 5555 55 5 5525 5552 5555 5555 5555 5555 5555 5555 5222 5255 5222 5252 5255 5552 5252 522 5555 39l 5525 5525 5535 5535 5555 5552 555 5555 5555 5555 22 5555 5554 5555 5555 5555 5 5555 55 5 5555 5555 5555 25 5 5225 5555 55 5555 5555 55 551 5 555 5 5555 l w l 5 l 391 05 04772 00228 3 10 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 2 What percentage of bags filled by the machine will weigh less than 5075 9 Mean 510 g StDev 25 9 Solution 2 5075 51025 1 Look at value of 2 1 2111111111111119111111 91111111111111111 2111111111111 1111 11111111 111111111 111111111 111111 111m 9 11 E 11 By table 05 03413 01587 1111quot 111 111 111 111 1111 11111 111 111 99 999199 111 991099 111111211 11111111 911199 99219 99219 911119 119359 I11 9399 9499 9419 115 11 9551 99 9929 9915 91 14 9159 2 9191 11992 9911 119 19 9949 9991 1929 1964 1 1111 1 141 113 1119 1211 1255 1293 1311 1193 1999 1443 1491 151 114 1554 1591 1129 1114 1199 1199 1112 1199 1944 1219 1 1 1111 1111 1111 1111 1151 1111 1111 1111 1111 1111 I1 1 2251 2291 2324 2151 2199 2422 2494 2491 2511 2549 11 1 2599 2511 2942 2911 213994 2194 2114 2194 2923 2952 11 2 2991 2919l 2999 2991 2995 9923 3951 3919 3191 9131 11 9 3159 3191 2212 1213 3219 1229 3315 3141 1165 1399 1411 1411 1111 1111 1114 1111 1199 3111 1i 1626 1199 9129 3149 21111 11911 219 111 3939 39 I 3999 3991 3925 3944 1962 1999 9991 4915 13 4992 9949 4919 4992 4999 41 1 1 4131 4141 4112 4111 14 4192 4291 4222 4231 4251 4295 4219 4292 4196 4319 15 4332 4349 43 51 4219 4392 4394 4499 441 9 4429 4441 11 4151 1111 4111 1114 1415 4511 4115 4111 4131 1145 111 45 13 i 3 quot 22111 1 4625 4633 1 9 4941 4149 4159 4114 4911 4119 4991 4692 4699 4199 19 4112 4119 4129 4122 4129 4144 4159 4159 4161 4111 1 11 1111 1111 1113 4111 1111 4191 1111 4111 4111 111 139 2 1 4221 4929 4939 4234 9999 9242 4941 4959 4994 4951 1 1 1111 1114 1111 4111 1111 1111 411 1114 1111 4191 2 9 4993 4991 4999 4991 4994 4991 4999 4911 4913 4919 2 4 4919 4929 4922 4925 4921 4929 4931 139 4932 49 94 4916 3 1 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU W What is the probability that a bag filled by the machine weighs less than 512 g z 512 51025 08 m 22222 2243 22224 2135 3336 22222 2133 2223 2222 23232232 21W 42202322 22222222 2222622 423124 321234 62223 2222322 22352 121 223 6433 4423 215 2 2 4552 4536 23636 1262 5 42 24 4233 0233 23 32 2232 2 223 223 3342 4332 22226 1664 2 2223 2 142 23 1223 1212 2253 2293 1332 2363 2446 2443 2432 2522 44 2554 2542 2622 2664 2244 2236 2222 2443 2244 2323 2925 2254 1425 2222 2254 2443 2223 2252 2244 2224 2252 222 2 2324 2 352 2344 2422 2454 2426 2512 2 544 2524 262 2 2642 2623 2244 2234 2264 2 23934 2223 2252 7 222 i 29 222 2339 2362 2445 34223 365 2 3623 3 2 236 3233 3253 3236 3222 3233 3264 32562 3325 3346 3365 3333 22 3413 3432 3462 3425 3543 3532 3354 3522 3339 3622 2 2 3643 3665 3636 3243 3222 3243 32212 32222 33 222 3632 t 2 3443 3363 3334 3362 342 5 3344 3962 3226 3292 4222 5 2 3 4232 4443 4232 42233 42 2 5 4232 4 42 4262 42 2393 24 4232 4242 4222 4236 4252 4265 4222 4232 4346 4323 2 5 4332 4345 43 32 4324 4332 4394 4446 442 3 4423 4442 26 4452 4463 4434 4434 4493 4305 39 39 4335 4343 22 45544564 4544 4542 4542 4544 4625 4633 2 2 4642 4642 4656 4664 4622 4622 4629 4326 EMMULATWENURMM 2666621464 DETREUTMN 426a M d fi 42464464 6642662 411646 2mm 23 66 E 0502881 07881 3 12 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 4 What percentage of bags weigh between 512 and 515 21 512 51025 08 Solution Area 1 02119 22 515 51025 2 Area 2 002275 pbags weighs between 512 and 515 Area 1 Area 2 02119 002275 018915 189 Required Area 39 iumu LATEWE Nammt FREQUENCY quotESTREMTtDH area underi Standard i t t curve ff m it m E 11 Areal 050288102119 CUMULATWE NURMAL FREQUENCY l ETREMTtnw area funded standard nurmat curve firam E be 3 2413 I2 l 115 US39 Area 2 05i 47720022s ampd 39 5 Ea ain manr 39CNJMUMTWE NDRMAL FREQUENCY 3quotESTREUTtDN area Handed standard nmmal curve from El he E zna i2 l I15 05quot 3 13 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 0211900228 01891 1891 So we obtain CMMLII LATWE HEREEL FEEQUEHC quotESTREUTJIDH area runderi standard marina Eu t tram in E F quot quot215 22 n5 05quot 314 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 41 ESTIMATING FROM SAMPLES INFERENCE PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 40 0 Estimating from Samples Inference NORMDIST Returns the normal distribution for the specified mean and standard deviation This function has a very wide range of applications in statistics including hypothesis testing Syntax NORMDISTxmeanstandarddevcumulative X is the value for which you want the distribution Mean is the arithmetic mean of the distribution Standarddev is the standard deviation of the distribution Cumulative is a logical value that determines the form of the function If cumulative is TRUE NORMDIST returns the cumulative distribution function if FALSE it returns the probability mass function Remarks o If mean or standarddev is nonnumeric NORMDIST returns the VALUE error value o If standarddev S 0 NORMDIST returns the NUM error value o If mean 0 standarddev 1 and cumulative TRUE NORMDIST returns the standard normal distribution NORMSDIST o The equation for the normal density function cumulative FALSE is ix ixf fEItquot 435 1 e I t 0 When cumulative TRUE the formula is the integral from negative infinity to x of the given formula Example In the slide the x value is 42 Arithmetic mean is 40 Standard deviation is 15 The cumulative distribution is 09 32 315 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E icrnsn Excel Lecture41 Elle Edit Eiew insert Fgrmat Innis gate window elp D al v Evai fl v DE v a a E t i NRMDISTWmeanstandartLdevcumulative 2 Data Descriptin 3 42 Value fr which yu want the distributin 1 4i arithmetic mean f the distributin 5 15 Standard deaiatin f the distributin j r Cumulative diatributian function for 39I39I39I lll til a the terms above 03079 9 W Returns the standard normal cumulative distribution function The distribution has a mean of 0 zero and a standard deviation of one Use this function in place of a table of standard normal curve areas Syntax NORMSDISTz z is the value for which you want the distribution Remarks lf 2 is nonnumeric NORMSDIST returns the VALUE error value The equation for the standard normal density function is Example The input to the NORMSDIST function is the zvalue The output is the cumulative probability distribution In the example 2 1333333 The normal cumulative probability function is 0908789 3 16 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnsnft Excel Lecturejl Elle Edit iew insert Fgrmat Innls Qatar indnw Help H t a v n v E v E SLIM r X J f3 NDRMSDISTHEEEEHE E A 11 NURMSDISTCZ 12 A B 13 Frmula Descriptin Result NORMINV Returns the inverse of the normal cumulative distribution for the specified mean and standard deviation Syntax NORMINVprobabilitymeanstandarddev Probability is a probability corresponding to the normal distribution Mean is the arithmetic mean of the distribution Standarddev is the standard deviation of the distribution Remarks If any argument is nonnumeric NORMINV returns the VALUE error value If probability lt 0 or if probability gt 1 NORMINV returns the NUM error value If standarddev S 0 NORMINV returns the NUM error value If mean 0 and standarddev 1 NORMINV uses the standard normal distribution see NORMSINV NORMINV uses an iterative technique for calculating the function Given a probability value NORMINV iterates until the result is accurate to within 1 3x10quot7 f NORMINV does not converge after 100 iterations the function returns the NA error value Example Here the probability value arithmetic mean and standard deviation are given The answer is the xvalue 3 17 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EE ilicrusuft Excel Lecture41 Eile Edit ew lnaert Fgrmat IDDIS Qatar window elp cit E v E E IE 3 SLIM r K inquot f NDRMINW i23 24 25j 4 E 2i NRMINUfprhahility meanatandarddeu 21 H B 22 Data Descriptin Prhahility crrea nding 22 t the nrmal distributin 10 i irithmetic mean f the 22 distribution 15 Standard deviatin f the 22 distr39ihutin 25 iNRMI NVA23 Z l 2 lnuerae f the nrmal cumulative 25 diatr39ihutin ferquot the terms ahdue 12 W Returns the inverse of the standard normal cumulative distribution The distribution has a mean of zero and a standard deviation of one Syntax NORMSINVprobability Probability is a probability corresponding to the normal distribution Remarks 0 If probability is nonnumeric NORMSINV returns the VALUE error value 0 If probability lt 0 or if probability gt 1 NORMSINV returns the NUM error value NORMSINV uses an iterative technique for calculating the function Given a probability value NORMSINV iterates until the result is accurate to within 1 3x10quot7 If NORMSINV does not converge after 100 iterations the function returns the NA error value Example In this case the input is the zvalue The corresponding cumulative distribution is calculated 3 18 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU lilicrnsnft Excel Lecturele Elle Edit Eleni Insert Fgrmat Innis Qatar intlnnl Help ll Ev W E r E sun r X J 8 NDRMSDIST1333333 A El 11 NURMSDISUZ 12 A B 13 Frmula Descriptin Result 14 N1RMSDIST1 15 Nrmal cumulative distributin functin lEi at 0908789 17 SAMPHNG VARIATIONS Electronic components are despatched by a manufacturer in boxes of 500 A small number of faulty components are unavoidable Customers have agreed to a defect rate of 2 One customer recently found 25 faulty components 5 in a box Was this box representative of production as a whole The box represents a sample from the whole output In such a case sampling variations are expected lf overall proportion of defective items has not increased just how likely is it that a box of 500 with 25 defective components will occur SAMPLING VARIATIONS EXAMPLE 1 In a section of a residential colony there are 6 households say Household A B C D E and F A survey is to be carried out to determine of households who use corn flakes of in breakfast Survey data exists and the following information is available Households A B C and D Use corn flakes Households E and F Do not It was decided to take random samples of 3 households The first task is to list all possible samples and find of each sample using corn flakes Possible Samples mple of users mple of users ABC 100 BCD 100 ABD 100 BCE 67 ABE 67 BCF 67 ABF 67 BDE 67 ACD 100 BDF 67 ACE 67 BEF 33 ACF 67 CDE 67 ADE 67 CDF 67 ADF 67 CEF 33 AEF 33 DEF 33 Percentage In Sample 3 19 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Out of 20 samples 4 contain 100 of users 12 contain 67 of users 4 contain 33 of users with required characteristic If the samples are selected randomly then each sample is likely to arise The probability of getting a sample with 100 of users is 420 or 02 with 67 1220 or 06 with 33 420 or 02 This is a Sampling Distribution SAMPLING DISTRIBUTION The sampling distribution of percentages is the distribution obtained by taking all possible samples of fixed size n from a population noting the percentage in each sample with a certain characteristic and classifying these into percentages Mean of the Sampling Distribution Using the above data Mean 100 x 02 67 x 06 33 x 02 67 Mean of the sampling distribution is the true percentage for the population as a whole You must make allowance for variability in samples Conditions For Sample Selection Number of items in the sample n is fixed and known in advance Each item either has or has not the desired characteristic The probability of selecting an item with the characteristic remains constant and is known to be P percent If n is large gt30 then the distribution can be approximated to a normal distribution STANDARD ERROR OF PERCENTAGES Standard deviation of the sampling distribution tells us how the sample values differ from the mean P It gives us an idea of error we might make if we were to use a sample value instead of the population value For this reason it is called STandard Error of Percentages or STEP STEP The sampling distribution of percentages in samples of n items ngt30 taken at random from an infinite population in which P percent of items have characteristic X will be A Normal Distribution with mean P and standard deviation STEP P100Pnquot12 The mean and StDev of the sampling distribution of percentages will also be percentages 320 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 42 Estimating from Samples Inference Part 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 41 0 Estimating from Samples Inference EXAMPLE 1 In a factory 25 workforce is women How likely is it that a random sample of 80 workers contains 25 or more women m standard deviation STEP P100Pnquot12 Mean P 25 N 80 STEP 25100 2580quot12 25 x 7580quot12 484 women in sample 2580 x 100 3125 2 3125 25484 129 Look for p against 2 129 in the table you will get 04015 39 z 333 331 333 333 333 333 33 333 00 000 030040 00000 01 I 30 001013 010100 00330 00030 003 39339 001350 0 1 0300 0430 0410 115 1 1 055 139 30595 0435 053 5 00 14 0053 01 0093 0033 Ii011 0310 0040 0031 1033 1054 1 103 1 141 03 1113 1211 1355 1303 1331 1300 1405 1443 1400 151Ir 04 1554 1 501 1333 1354 1100 39 333 1113 1003 1044 130311 05 1015 105I 1005 2010 2054 3000 3123 0153 3100 2234 015 3351 3201 3334 2 3511 3333 2433 2454 3403 3511 3 543 01 2500 351 1 3543 350 3 3104 3134 3104 3134 3023 2052 030 3301 39 0 3030 3331 3305 3033 3051 3035 3 1 03 31 3 3 00 31 5039 3133 3213 331 33 3304 3300 331 5 3340 3 335 3303 3 3313 3333 3331 3333 3333 3331 3333 3313 1 3333 3333 3333 3333 3333 3133 3313 3133 3333 3333 3333 3331 3333 3333 3333 3333 39 3 3333 3333 3333 3333 3333 311 3 3131 3133 1 3133 3333 3333 3333 3331 3333 3333 3333 15 4332 4345 4351 4330 4300 4304 441 3 4430 4441 10 4453 4453 4474 4404 4495 4505 3951 5 4535 4535 4545 11 4 JEEJ Atrial JEEP Ii l39 JEH39I mm Il39lf Ji39 Al l Al 39 Since we want to nd psample contains 25 or more women so 3 2 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU CMMULATIWE Notmm FREQUENCY 3quotESTRIEJIJTllDN area untied Standard normal curve from I lie 2 11 balms w 39 z1EE I I15 LEM psample contains 25 or more women 05 04015 00985 or about 10 APPLICATIONS OF STEP Some important issues are 0 What is the probability that such a sample will arise o How to estimate the percentage P from information obtained from a single sample 0 How large a sample will be required in order to estimate a population percentage with a given degree of accuracy 0 To obtain answers to these questions let us solve some typical problems CONFIDENCE LIMITS A market researcher wishes to conduct a survey to determine consumers buying the company s products He selects a sample of 400 consumers at random He finds that 280 of these 70 are purchasers of the product What can he conclude about of all consumers buying the product First let us decide some limits It is common to use 95 confidence limits These will be symmetrically placed around the 70 buyers In a normal sampling distribution 25 corresponds to a zvalue of 196 on either side of 70 With 95 con dence limit we have 5 chance of errors level of signi cance That mean 25 on each side of curve 25 0025 050025 0475 We have subtracted 0025 from 05 because we nd out the probability of acceptance region as the total probability under the curve is always 1 that mean 05 on right side of mean and 05 on left side of mean And 025 is rejection region so 0475 is acceptance region So 0475 is the probability of acceptance region 322 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Corresponding 2 value is196 z Ml lint an tam m are ill um acute tiltmm u 39 uizlleg 39 EEQix39 J ELM39939 Lil35939 1 i AME Illle i39 39 9135 ll 7 1355 l 0596 IDEISE 126 5 JDquot 14 JETS EISLE WQ39E39 332 IEliE39Tlt it til 0943 1393 lll l Jill64 li m3 it Hit 13 1111quot 121 llfl fi JEQS 133l 1563 1 1443 MEI ll l 04 1554 1591 3625 1664 H tt l 315 l39it39itl l l t l JET ti l 939 5 WEI 1935 33 t9 2354 3133 El 23 2 t 5 2 ll Eel 2224 LE 2257 229 2324 235 2339 2122 2154 2436 251 2549 a E i l 39 E ll l 2642 2623 ET EM 2234 all Ewell 2323 2352 ELIE z t E l 2939 295 2995 123 3tt5l Ellis jll 3l33 19 3159 EIEIE SZIZ 3233 3264 EaEEtt 331155 33 3365 3339 LG Jude13a 3433 3435 35 3531 355 35 3599 362l Lll 35 3665 E l EliTZEI39 EM NW BTQHU 33 Ml HESEl LE 3349 3369 3910 3925 3944 3962 3930 399 0115 3 4332 4349 4133 All itl 39l 3 All EM 414 4 ME tl I TI a 4192 AM 4236 4251 4265 AZ A292 4336 4319 4332 4345 43 5 ALETl ABE2 4394 sl ll 4429 Ala le MSE H l quot 4525 535 4545 A d a u 1211 quot AME Al 4633 AME 39 4 I 46 4699 was ATM 4229 4126 4 13932 4333 s 4 l A316 41 Now the sample percentage of 70 can be used as an approximation for population percentage P Hence STEP 70100 70400quot12 229 Confidence gmits Estimate for population percentage 70 196 x STEP Or 70 196 x 229 65515 and 7449 as the two limits for 95 confidence interval We can round off 196 to 2 323 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Then with 95 confidence we estimate the population percentage with that characteristic as lying in the interval P 2 x STEP EXAMPLE 2 A sample of 60 students contains 12 20 who are left handed Find the range with 95 confidence in which the entire left handed students fall Range 20 2 x STEP 20 2 x 20 x 10020l60quot12 967 and 3033 ESTIMATING PROCESS SUMMARY 1 Identify n and P the sample size and percentage in the sample 2 Calculate STEP using these values The 95 confidence interval is approximately P 2 STEP 99 confidence For 99 confidence limits zvalue 258 With 99 con dence limit we have 1 chance of errors level of signi cance That mean 05 on each side of curve 05 0005 050005 0495 We have subtracted 0005 from 05 because we nd out the probability of acceptance region as the total probability under the curve is always 1 that mean 05 on right side of mean and 05 on left side of mean And 005 is rejection region so 0495 is acceptance region So 0495 is the probability of acceptance region Corresponding 2 value is 258 from table as we did in example stated above 324 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU FINDING A SAMPLE SE To satisfy 95 confidence 2 x STEP 5 STEP 25 Pilot survey value of P 30 STEP 30 x 70nquot12 25 Mg n 336 We must interview 336 persons to be 95 confident that our estimate is within 5 of the true answer DISTRIBUTION OF SAMPLE MEANS The standard deviation of the Sampling Distribution of means is called STandard Error of the Mean STEM STEM E J sd denotes standard deviation of the population n is the size of the sample EXAMPLE 3 What is the probability that if we take a random sample of 64 children from a population whose mean IQ is 100 with a StDev of 15 the mean IQ of the sample will be below 95 m s 15 n 64 population mean 100 STEM 1564quot12 1564quot12 158 1875 2 100 95 STEM 51875 267 This gives a probability of 00038 So the chance that the average IQ of the sample is below 95 is very small 325 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 43 HYPOTHESIS TESTING CHISQUARE DISTRIBUTION PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 42 o Hypothesis testing ChiSquare Distribution EXAMPLE 1 An inspector took a sample of 100 tins of beans The sample weight is 225 g Standard deviation is 5 g Calculate with 95 confidence the range of the population mean m STEM E J sd is not known Use sd of sample as an approximation STEM i 05 100 95 confidence interval 225 2 x 05 or From 224 to 226 g PROBLEM OF FAULTY COMPONENTS REVISITED Box of 500 components may have 25 or 5 faulty components Overall faulty items 2 P 2 n 500 STEP 2 x 98500quot12 0626 To find the probability that the sample percentage is 5 or over 2 5 2STEP 30626 479 Area against 2 479 is negligible Chance of such a sample is very small FINITE POPULATION CORRECTION FACTOR lf population is very large compared to the sample then multiply STEM and STEP by the Finite Population Correction Factor 1 nNA12 Where N Size of the population n Size of the sample n less than 01 N TRAINING MANAGER S PROBLEM New refresher course for training of workers was completed The Training Manager would like to assess the effect of retraining if any Particular questions ls quality of product better than produced before retraining Has the speed of machines increased 39Do some classes of workers respond better to retraining than others Training Manager hopes to 0 Compare the new position with established 326 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 0 Test a theory or hypothesis about the course Case Study Before the course Worker X produced 4 rejects After the course Out of 400 items 14 were defective 35 An improvement The 35 figure may not demonstrate overall improvement It does not follow that every single sample of 400 items contains exactly 4 rejects To draw a sound conclusion Sampling variations must be taken into account We do not begin by assuming what we are trying to prove NULL HYPOTHESIS We must begin with the assumption that there is no change at all This initial assumption is called NULL HYPOTHESIS Implication of Null Hypothesis That the sample of 400 items taken after the course was drawn from a population in which the percentage of reject items is still 4 NULL HYPOTHESIS EXAMPLE Data P 4 n 400 STEP P100 Pnquot12 4100 4400quot12 098 At 95 confidence limit Range 4 2 x 098 204 to 596 Conclusion Sample with 35 rejects is not inconsistent No ground to assume that rejects has changed at all On the strength of sample there were no grounds for rejecting Null Hypothesis ANOTHER EXAMPLE Before the course 5 rejects After the course 25 rejects 10 out of 400 P 5 STEP 5100 5400quot12 5 x 95400quot12 109 Range at 95 Confidence Limits 5 2 x 109 282 to 718 Conclusion Doubt about Null Hypothesis most of the time Null hypothesis to be rejected PROCEDURE FOR CARRYING OUT HYPOTHESIS TE 1 Formulate null hypothesis 2 Calculate STEP amp P 2 x STEP 3 Compare the sample with this interval to see whether it is inside or outside If the sample falls outside the interval reject the null hypothesis sample differs significantly from the population If the sample falls inside the interval 327 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU do not reject the null hypothesis sample does not differ significantly from the population at 5 level HOW THE RUE WORKS Bigger the difference between the sample and population percentages less likely it is that the population percentages will be applicable 0 When the difference is so big that the sample falls outside the 95 interval then the population percentages cannot be applied Null Hypothesis must be rejected o If sample belongs to majority and it falls within 95 interval then there are no grounds for doubting the Null Hypothesis FURTHER POINTS ABOUT HYPOTHESIS TESTING 99 interval requires 258 x STEP Interval becomes wider It is less likely to conclude that something is significant A We might conclude there is a significant difference when there is none Chance of error 5 type 1 Error B We might decide that there is no significant difference when there is one Type 2 Error 328 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 44 HYPOTHESIS TESTING CHISQUARE DISTRIBUTION PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 43 o Hypothesis Testing ChiSquare Distribution FURTHER POINTS ABOUT HYPOTHESIS TESTING This is a continuation of the points covered under Handout 43 3 We cannot draw any conclusion regarding the direction the difference is in A Possible to do 1tailed test Null Hypotheis P gt 4 against the alternative Pgt 4 z 164 for 5 significance level Range P 164 x STEP 098 Example Range 4 164 x 098 239 New figure 35 Hence There is no reason to conclude that things have improved 4 We cannot draw any conclusion regarding the direction the difference is in B Possible to do 2tailed test Null Hypothesis P gt 4 against the alternative Pgt 4 z 196 for 5 significance level Range P 196 x STEP 098 Example Range 4 2 X196 x 098 208 to 592 New figure 35 There is no reason to conclude that things have improved HYPOTHESES ABOUT MEANS Let us go back to the problem of retraining course discussed earlier Before the course Worker X took 25 minutes to produce 1 item StDev 05 min After the course Foe a sample of 64 items mean time 258 min Null hypothesis No change after the course STEM sdnquot12 0564 12 00625 Range 25 2 x 00625 2375 to 2625 min Conclusion No grounds for rejecting the Null Hypothesis There is no change significant at 5 level 329 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU ALTERNATIVE HYPOTHESIS TESTING USING ZVALUE z sample percentage population meanSTEP 35 4098 051 Compare it with zvalue which would be needed to ensure that our sample falls in the 5 tails of distribution 196 or about 2 z is much less than 2 We conclude that the probability of getting by random chance a sample which differs from the mean of 4 or more is quite high Certainly it is greater than the 5 significance level Sample is quite consistent with null hypothesis Null hypothesis should not be rejected PROCESS SUMMARY 1 State Null Hypothesis 1tailed or 2tailed 2 Decide on a significance level and find corresponding critical value ofz 3 Calculate sample zsample value population value divided by STEP or STEM as appropriate 4 Compare sample 2 with critical value of z 5 If sample 2 is smaller do not reject the Null Hypothesis 6 If sample 2 is greater than critical value of 2 sample provides ground for rejecting the Null Hypothesis TESTING HYPOTHESES ABOUT SMALL SAMPLES Whatever the form of the underlying distribution the means of large samples will be normally distributed This does not apply to small samples We can carry out hypothesis testing using the methods discussed only if the underlying distribution is normal If we only know the Standard Deviation of sample and have to approximate population Standard Deviation then we use Student s tdistribution STUDENT S tDISTRIBUTION Student s TDistribution is very much like normal distribution In fact it is a whole family of tdistributions As n gets bigger tdistribution approximates to normal distribution tdistribution is wider than normal distribution 95 confidence interval reflects greater degree of uncertainty in having to approximate the population Standard Deviation by that of the sample EXAMPLE Mean training time for population 10 days Sample mean for 8 women 9 days Sample Standard Deviation 2 days To approximate population Standard Deviation by a sample divide the sum of squares by n 1 STEM 28quot12 071 Null Hypothesis There is no difference in overall training time between men and women tvalue sample mean population meanSTEM 9 10071 141 Forn8v8 1 7 For 505 significance level looking at 0025 2tailed t 2365 Calculated table value m Do not reject the Null Hypothesis 3 30 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU SUMMARY I If underlying population is normal and we know the Standard Deviation Then Distribution of sample means is normal with Standard Deviation STEM population sdn3912 and we can use a ztest SUMMARY II If underlying population is unknown but the sample is large Then Distribution of sample means is approximately normal With StDev STEM population sdn3912 and again we can use a ztest SUMMARY Ill lf underlying population is normal but we do not know its StDev and the sample is small Then We can use the sample sd to approximate that of the population with n 1 divisor in the calculation of sd Distribution of sample means is a tdistribution with n 1 degrees of freedom With Standard Deviation STEM sample sdn 12 And we can use a ttest SUMMARY IV If underlying population is not normal and we have a small sample Then none of the hypothesis testing procedures can be safely used TESTING DIFFERENCE BETWEEN TWO SAMPLE MEANS A group of 30 from production has a mean wage of 120 Rs per day with Standard Deviation Rs 10 50 Workers from Maintenance had a mean of Rs 130 with Standard Deviation 12 Is there a difference in wages between workers Difference of two sample means s1n1 1n23912 s n1s1quot2 n2s2quot2 n1 n2quot12 N1 30 n2 50 s1 10 s2 12 s 30 x 100 50 x14430 50 12 1129 Standard Error of Difference in Sample Means STEDM 1129130 150quot12 260 z difference in sample means 0STEDM 120 130260 385 This is well outside the critical 2 for 5 significance There are grounds for rejecting Null Hypothesis There is difference in the two samples PROCEDURE SUMMARY 1 State Null Hypothesis and decide significance level 2 Identify information no of samples large or small mean or proportion and decide what standard error and what distribution are required 3 Calculate standard error 4 Calculate z or t as difference between sample and population values divided by standard error 5 Compare your 2 or t with critical value from tables for the selected significance level if z ort is greater than critical value reject the Null Hypothesis 331 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MORE THAN ONE PROPORTION Look at a problem where after the course some in different age groups shows improvement while others did not Let us assume that the expected improvement was uniform An improvement of 40 if applied to 21 24 and 15 would give 14 16 and 10 respectively who improved Let us write these values within brackets Subtracting 14 16 and 10 from the totals 21 24 and 15 gives us 7 8 and 5 respectively who did not improve This is the estimate if every person was affected in a uniform manner Let us write the observations as O in one line 17 17 6 4 7 9 Let us write down the expected as E in the next line as 14 16 10 7 8 8 Calculate OE Next calculate OEquot2 Now standardize OEquot2 by dividing by E Calculate the total and call it x2 Age Improved Did not improve Total Under 35 1714 47 21 35 50 1716 78 24 Over 50 610 95 15 Total 40 20 60 O 17 17 6 4 7 9 E 14 16 10 7 8 8 OE 3 1 4 3 1 4 OEquot2 9 1 16 9 1 16 OEquot2E 0643 00625 16 1286 0125 32 692 Measurement of disagreement Sum OEquot2E is known as Chisquared x2 Degrees of freedom v r1xc1 3121 2 There are tables that give Critical value of chisquared at different confidence limits and degrees of freedom v columns1 x rows1 In the above case v 21 x 31 2 In the present case the Critical value of chisquared at 5 and v 2 5991 The value 692 is greater than 5991 This means that the Sample falls outside of 95 interval Null hypothesis should be rejected CHISQUARED SUMMARY Formulate null hypothesis no association form Calculate expected frequencies Calculate x2 Calculate degrees of freedom rows minus 1 x columns minus 1 look up the critical x2 under the selected significance level 5 Compare the calculated value of x2 from the sample with value from the table if the sample x2 is smaller within the interval don t reject the null hypothesis if it is bigger outside reject the null hypothesis Example Look at the data in the slide below bF Nf 332 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EE IHIicrosoft Excel Ernp Elle Edit iew Lnsert Fgrmat Iools gate window Help D a vs 39 v EvsIMi fmv ae F1 1 a 111 B I D E F I I3 39 1 STATUS SEE AGE 2 I I 3 I a I 1 3 39Jquot 4 I 1 3 I 5 I 1 4 I a 1 I 3 IS a I 1 3 I a I 1 4 39Jquot a 1 I 3 3 1o 1 I 3 I 11 I 1 4 S 12 1 I 4 4 13 I 1 3 1 It is possible to carry out ttests using EXCEL Data Analysis tool Data Analysis analysis Tools Histogram Moving Fwerage I Ftantlom Number Generation 3 Flank and Percentile Regression Hall I Sampling tTest Paired Two Sample For Means itIastI39SamolaassnmionquotLreonal39ss39at39iaolas eTest Two Sample For Means h quotquot When you select the tool and press OK the ttest dialog box is opened as below 333 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU tTest TwinSample Assuming Unequal Variances Input 39u39arialzule 1 Range 39u39arialzule a Range Cal39IEEl I Hypethesisecl Mean Difference I ElIII I l Labels alpha CLUE Clutput eptiens F Qutput Range I F New Werksheet Ely I F New erkheelt The ranges for the two variables labels and output options are specified For the above data the output was as follows Micrnsuft Excel Emp Eile gait Eiew insert Fgrmat leels gata window elp v a x DEH 5 v Ev m WBH E s 2 D15 139 f3 A e c D t 1 t Teet Ten Sample Assuming Unequal ariane 2 3 33 J 4 Mean 3913 351 5 Eariance 3356 5252 s Uheervatinne 23 35 Hypntheeized Mean Di 0 3 df SE a t Stat 1TEE m PTt ne tail H43 5 n t Critical ne tail 1625 n Pthtl ten tail n3511 u t Critical ten tail 2 32 1 11 CHITEST Returns the test for independence CHITEST returns the value from the chisquared y2 distribution for the statistic and the appropriate degrees of freedom You can use v2 tests to determine whether hypothesized results are verified by an experiment Syntax CHlTESTactualrangeexpectedrange Actualrange is the range of data that contains observations to test against expected values Expectedrange is the range of data that contains the ratio of the product of row totals and column totals to the grand total Remarks 3 3 4 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU o If actuarange and expectedrange have a different number of data points CHITEST returns the NA error value 0 The v2 test first calculates a v2 statistic and then sums the differences of actual values from the expected values The equation for this function is CHITESTp Xgty2 where and where Aij actual frequency in the ith row jth column Eij expected frequency in the ith row jth column r number or rows c number of columns CHITEST returns the probability for a v2 statistic and degrees of freedom df where df r 1c 1 Exam le Microsoft Excel Lecture44 Elle Edit Eiew insert Fgrmat Tools Data window elp v E X a as v n v 2 v E e a SUM v X J f3 CHITESTA3EEEABQ E C T M Women Actual Description 2 Actual 3 58 35 Agree 1 11 25 Neutral 5 10 23 Disagree E Exgeeged Women Expected Description a 39 4535 4765 Agree 8 1756 1344 Neutral 9 1609 1691 Disagree a 35M The V2 statistic for the data above is 1515957 11 as with 2 degrees of freedom 0000303 The above example shows two different groups The calculation shows that the probability for chisquared 1616957 with 2 degrees of freedom was 0000308 which is negligible 3 3 5 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 45 Planning Production Levels Linear Programming OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 44 0 Planning Production Levels Linear Programming INTRODUCTION TO LINEAR PROGRAMMING A Linear Programming model seeks to maximize or minimize a linear function subject to a set of linear constraints The linear model consists of the following components A set of decision variables X An objective function ch X A set of constraints 2 ailx 5 bi THE FORMAT FOR AN LP MODEL Maximize or minimize chxj c1x1 c2x2 cnxn Subject to ainjE bi 1m Nonnegativity conditions all x 0j 1 n Here n is the number of decision variables Here m is the number of constraints There is no relation between n and m THE METHODOLOGY OF LINEAR PROGRAMMING Define decision variables Handwrite objective Formulate math model of objective function Handwrite each constraint Formulate math model for each constraint Add nonnegativity conditions THE IMPORTANCE OF LINEAR PROGRAMMING Many real world problems lend themselves to linear programming modeling Many real world problems can be approximated by linear models There are wellknown successful applications in 0 Operations 0 Marketing 0 Finance investment 0 Advertising 0 Agriculture There are efficient solution techniques that solve linear programming models The output generated from linear programming packages provides useful what ifquot analysis ASSUMPTIONS OF THE LINEAR PROGRAMMING MODEL 1 The parameter values are known with certainty 2 The objective function and constraints exhibit constant returns to scale 3 There are no interactions between the decision variables the additivity assumption The Continuity assumption Variables can take on any value within a given feasible range A PRODUCTION PRO M A PROTOTYPEXAMPE A company manufactures two toy doll models Doll A mmewwe 336 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Doll B Resources are limited to 1000 kg of special plastic 40 hours of production time per week mirketinq requirement Total production cannot exceed 700 dozens Number of dozens of Model A cannot exceed number of dozens of Model B by more than 350 The current production plan calls for o Producing as much as possible of the more profitable product Model A Rs 800 profit per dozen 0 Use resources left over to produce Model B Rs 500 profit per dozen while remaining within the marketing guidelines Manaqement is seekinq a production schedule that will increase the company s profit A linear programming model can provide an insight and an intelligent solution to this problem Decisions variables X1 Weekly production level of Model A in dozens X2 Weekly production level of Model B in dozens Obiective Function Weekly profit to be maximized Maximize 800X1 500X2 Weekly profit subject to 2X1 1X2 1000 lt Plastic 3X1 4X2 2400 lt Production Time X1 X2 700 5 Total production X1 X2 350 5 Mix Xjgt 0 j 12 Nonnegativity ANOTHER EXAMPLE A dentist is faced with deciding how best to split his practice between the two services he offers general dentistry and pedodontics children s dental care Given his resources how much of each service should he provide to maximize his profits The dentist employs three assistants and uses two operatories Each pedodontic service requires 75 hours of operatory time 15 hours of an assistant s time and 25 hours of the dentist s time A general dentistry service requires 75 hours of an operatory 1 hour of an assistant s time and 5 hours of the dentist s time Net profit for each service is Rs 1000 for each pedodontic service and Rs 750 for each general dental service Time each day is eight hours of dentist s 16 hours of operatory time and 24 hours of assistants time 337 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU THE GRAPHICAL ANALYSIS OF LINEAR PROGRAMMING Using a graphical presentation we can represent all the constraints the objective function and the three types of feasible points GRAPHICAL ANALYSIS THE FEASIBLE REGION The slide shows how a feasible region is defined with nonnegativity constraints GRAPHIEAL ANAL I EIS THE FEASIBLE 3393 The nunnegativity constraints I 1quot THE SEARCH FOR AN OPTIMAL SOLUTION The figure shows how different constraints can be represented by straight lines to define a feasible region There is an area outside the feasible region that is infeasible THE FEASIBLE REEIDN The Plaitin mr tmi rt f Emug5 mm 1131quot Itdint ion Iu rat mirt angina redundant Inteaaihle F39rczluctizu n Tima 3x4 3mm fiIJIII lm It may be seen that each of the constraints is a straight line The constraints intersect to form a point that represents the optimal solution This is the point that results in maximum profit of 436000 Rs As shown in the slide below The procedure is to start with a point that is the starting point say 200000 Rs Then move the line upwards till the last point on the feasible region is reached This region is bounded by the lines representing the constraints 338 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 THE SDLUTIDN 35131315 arhinrym say un t HEM IIIIIIIIII A Then immaee te unlit tipnestle and cumint ur i hum eas e m Pro t FL5 lllll 5M fw n it SUMMARY OF THE OPTIMAL SOLUTION Model A 320 dozen Model B 360 dozen Profit Rs 436000 This solution utilizes all the plastic and all the production hours Total production is only 680 not 700 Model a production does not exceed Model B production at all EXTREME POINTS AND OPTIMAL SOLUTIONS If a linear programming problem has an optimal solution an extreme point is optimal EXT R EM E PIN s PTIMAL TIMItlgyi If a quotnew nrtr rnmming prelrleln has an liTil39l39l l E lllti ll an e reme mint i5 nptimnl Copyright Virtual University of Pakistan VU 339 Business Mathematics amp Statistics MTH 302 MULTIPLE OPTIMAL SOLUTIONS There may be more than one optimal solutions However the condition is that the objective function must be parallel to one of the constraints If a weightage average of different optimal solutions is obtained it is also an optimal solution MULTIPLE in w if QLUTI 1N3 For multiple optimal solutions to exist the olijeetiue funetion must he pamllelto one oftite eo nstmints i Inn IHeighten auemge of optimal solutions is also an optimal solution Jr Copyright Virtual University of Pakistan VU 340 Business Mathematics amp Statistics MTH 302 VU AMORDEGRC AMORLINC AVERAGE AVE RAG EA BINOMDIST CHITEST COMBIN CORREL COUNT COUNTA COUNTBLANK COUNTIF COVAR CRITBINOM CUMIPMT CUMPRINC DAVE RAG E DB DDB DPRODUCT DSUM Some useful functions of Excel Returns the depreciation of an asset for each accounting period by using depreciation coefficient French accounting system Returns the depreciation of an asset for each accounting period French accounting system Returns the average of its arguments Returns the average of its arguments including numbers text and logical values Returns the Binomial Distribution Probability Returns the test for independence CHITEST returns the value from the chisquared x2 distribution for the statistic and the appropriate degrees of freedom Returns Number of Combinations for a Given Number of Items Returns the correlation coefficient between two data sets Counts how many numbers are in the list of arguments Counts the number of cells that are not empty and the values within the list of arguments Counts the number of blank cells within a range Counts the number of nonblank cells within a range that meet the given criteria Returns covariance Returns smallest value for which the Cumulative Binomial Distribution is less than or equal to a criterion value Returns cumulative interest paid between two periods Returns cumulative principal paid on a loan between two periods Averages the values that match specified conditions Returns depreciation of an Asset for a specified period using fixed declining balance method Returns the depreciation of an asset for a specified period by using the doubledeclining balance method or some other method that you specify Multiplies the values in a list that match the specified condition Adds the numbers in a list that match specified conditions 341 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EFFECT EVEN EXP FACT FORECAST FREQUENCY FV FVSCHEDULE GEOMEAN HARMEAN INT INTERCEPT IPMT IRR ISPMT LN LOG LOG10 MEDIAN MDETERM MINVERSE MIRR MMULT MODE NEGBINOMDIST NOMINAL NORM DIST NORMSINV Returns effective annual interest rate Rounds Up to the Nearest Even Integer e Raised to the Power of a Given Number Returns factorial of a Number Prediction by Trend Returns a frequency distribution as a vertical array Returns future Value of an Investment Returns Future value of an initial principal with variable interest rate Returns Geometric Mean of Positive Numeric Data Returns Harmonic Mean of Positive Numbers Rounds to the Nearest Integer Calculates Point Where Line Will Intersect Y Axis Returns the interest payment for an investment for a given period Returns the internal rate of return for a series of cash flows Calculates the interest paid during a specific period of an investment Returns Natural Logarithm of a number Returns Logarithm of a Number to a Specified Base Returns Base 10 Logarithm of a number Gives Median or Number in Middle Matrix Determinant of an Array Gives Inverse Matrix for the Matrix Stored in an Array Returns the internal rate of return where positive and negative cash flows are financed at different rates Matrix Product of Two Arrays Returns Most Frequent Value of an Array Returns Negative Binomial Distribution Returns annual nominal interest rate Returns Normal Cumulative Distribution Returns Inverse of Normal Cumulative Distribution 342 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NPER NPV ODD PERCENTILE PERM UT PMT POISSON POWER PPMT PRODUCT PV QUARTILE RATE ROUND ROUNDDOWN ROUNDUP RSQ SLN SLOPE SQRT STDEV STDEVA STDEVP SUBTOTAL SUM SUMIF SUMPRODUCT Returns Number of Periods for an Investment Returns the net present value of an investment based on a series of periodic cash flows and a discount rate Rounds Number Up to the Nearest Odd Integer Returns K th Percentile of Values in a Range Returns Number of Permutations for a Given Number of Objects Returns the periodic payment for an annuity Returns Poisson Distribution Returns the result of a Number Raised to a Power Returns the payment on the principal for an investment for a given period Multiplies All Numbers Returns Present Value of an Investment Returns Specified Quartile of Data Set Returns Interest Rate of a Loan or Annuity Rounds Number to a Specific Number of Digits Rounds Number Down Towards Zero Rounds Number Up Away From Zero Returns Square of Pearson Product Moment Correlation Coefficient Returns the straightline depreciation of an asset for one period Returns Slope of a Linear Regression Line Returns Square Root of a Number Estimates standard deviation based on a sample Estimates standard deviation based on a sample including numbers text and logical values Calculates standard deviation based on the entire population Returns a Subtotal in a List Add all numbers in a range of cells Adds Specified Cells Multiplies Array and Gives Sum 343 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU SUMSQ SYD TREND TRUNC VAR VARP VDB XIRR XNPV Square Numbers and Then Add Returns the sumof years39 digits depreciation of an asset for a specified penod Return Values Along Linear Trend Truncates to an Integer by Removing Fractional Part Estimate Variance from Sample Calculates Variance from entire Population Returns the depreciation of an asset for a specified or partial period by using a declining balance method VDB stands for variable declining balance Returns internal rate of return for a schedule of cash flows that is not necessarily periodic Returns net present value for a schedule of cash flows that is not necessarily periodic 344 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Area under Standard Normal Curve from 0 to Z 000 001 002 003 004 005 006 007 008 009 00 00000 00040 00080 00120 00159 00199 00239 00279 00319 00359 01 00398 00438 00478 00517 00557 00596 00636 00675 00714 00753 02 00793 00832 00871 00910 00948 00987 01026 01064 01103 01141 03 01179 01217 01255 01293 01331 01368 01406 01443 01480 01517 04 01554 01591 01628 01664 01700 01736 01772 01808 01844 01879 05 01915 01950 01985 02019 02054 02083 02123 02157 02190 02224 06 02257 02291 02324 02357 02380 02422 02454 02486 02518 02549 07 02580 02611 02642 02673 02704 02734 02764 02794 02823 02852 08 02881 02910 02939 02967 02995 03023 03051 03078 03106 03133 09 03159 03186 03212 03238 03264 03289 03315 03340 03365 03389 10 03413 03438 03461 03485 03508 03531 03554 03577 03599 03621 11 03643 03665 03686 03708 03729 03749 03770 03790 03810 03880 12 03849 03869 03888 03907 03925 03944 03962 03990 03997 04015 13 04032 04049 04066 04082 04099 04115 04131 04147 04162 04177 14 04192 04207 04222 04236 04251 04265 04279 04292 04306 04319 15 04332 04345 04357 04370 04382 04394 04406 04418 04430 04441 16 04452 04463 04474 04485 04495 04505 04515 04525 04535 04545 17 04554 04564 04573 04582 04591 04599 04608 04616 04625 04633 18 04641 04649 04656 04664 04671 04678 04686 04693 04690 04706 19 04713 04719 04726 04732 04738 04744 04750 04758 04762 04767 20 04772 04778 04783 04788 04793 04798 04803 04808 04812 04817 21 04821 04826 04830 04834 04838 04842 04846 04850 04854 04857 22 04861 04865 04868 04871 04875 04878 04881 04884 04887 04890 23 04893 04896 04898 04901 04904 04906 04909 04911 04913 04916 24 04918 04920 04922 04925 04927 04929 04931 04932 04934 04936 25 04938 04940 04941 04943 04945 04946 04948 04949 04951 04952 26 04953 04955 04956 04957 04959 04960 04961 04962 04963 04964 27 04965 04966 04967 04968 04969 04970 04971 04972 04973 04974 28 04974 04975 04976 04977 04977 04978 04979 04980 04980 04981 29 04981 04982 04983 04983 04984 04984 04985 04985 04986 04986 30 049865 04987 04987 04988 04988 04989 04989 04989 04990 04990 31 049903 04991 04991 04991 04992 04992 04992 04992 04993 04993 345 Copyright Virtual University of Pakistan Introduction to Cultural Anthropology SOC401 VU Lesson 01 WHAT IS ANTHROPOLOGY Anthropology can be best de ned as the study of the various facets of what it means to be human Anthropology is a multidimensional subject in which various components are studied individually and as a whole to develop a better understanding of human eXistence In this lecture we will not only be developing an understanding of the definition of anthropology we will also be looking at what an anthropologist does In addition to this we will also be looking at the various branches of anthropology with a focus on cultural anthropology Definition of Anthropology Anthropology is derived from the Greek words am bmpos for human and logos for study so if we take its literal meaning it would mean the study of humans In one sense this is an accurate description to the extent that it raises a wide variety of questions about the human eXistence However this literal definition isn t as accurate as it should be since a number of other disciplines such as sociology history psychology economics and many others also study human beings What sets anthropology apart from all these other subjects Anthropology is the study of people their origins their development and variations wherever and whenever they have been found on the face of the earth Of all the subjects that deal with the study of humans anthropology is by far the broadest in its scope In short anthropology aims to describe in the broadest sense what it means to be human Activities of an Anthropologist As we already know anthropology is the study of what it means to be human So the study of the in uences that make us human is the focus of anthropologists Anthropologists study the various components of what its means to be human Branches of Anthropology A Physical Anthropology Is the study of humans from a biological perspective Essentially this involves two broad areas of investigation a Human paleontology this sub branch deals with re constructing the evolutionary record of the human eXistence and how humans evolved up to the present times b Human variation The second area deals with how why the physical traits of contemporary human populations vary across the world B Archeology study of lives of people from the past by examining the material culture they have left behind C Anthropological Linguistics the study of human speech and language D Cultural Anthropology the study of cultural differences and similarities around the world Now that we have brie y defined the various branches of anthropology lets us now take an in depth view of cultural anthropology Copyright Virtual University of Pakistan 1 Introduction to Cultural Anthropology SOC401 VU Cultural Anthropology As we have discerned above cultural anthropology concerns itself with the study of cultural differences as well as the similarities around the world On a deeper level the branch of anthropology that deals with the study of specific contemporary cultures elbmgmpJy and the more general underlying patterns of human culture derived through cultural comparisons elbmogy is called cultural anthropology Before cultural anthropologists can examine cultural differences and similarities throughout the world they must first describe the features of specific cultures in as much detail as possible These detailed descriptions elbmgmpbz39es are the result of extensive field studies in which the anthropologists observes talks to and lives with the people under study On the other hand ethnology is the comparative study of contemporary cultures wherever they are found The primary objective of ethnology is to uncover general cultural principals rules that govern human behavior Areas of Specialization in Cultural Anthropology I Urban Anthropology studies impact of urbanization on rural societies and the dynamics of life within cities H Medical Anthropology studies biological and socio cultural factors that effect health or prevalence of illness or disease in human societies HI Educational Anthropology studies processes of learning of both formal education institutions and informal systems which can use story telling or experiential learning IV Economic Anthropology studies how goods and services are produced distributed and consumed within different cultural contexts V Psychological Anthropology studies relationship between cultures and the psychological makeup of individuals belonging to them Holistic and Integrative Approach Cultural anthropologists consider in uences of nature and nurture across all locations and across different periods of time When various specialties of the discipline are viewed together they provide a comprehensive view of the human condition Common Responses to Cultural Difference A Ethnocentrism a belief that one s own culture is not only the most desirable but also superior to that of others B Cultural relativism looks at the inherent logic behind different cultures and practices in the attempt to understand them Relevance of Cultural Anthropology Cultural anthropology enhances understanding of differences and prevents oversimplified generalizations It increases self knowledge about our own thinking values and behavior and helps develop cognitive complexity through integration interconnectedness and differentiation different aspects of a singular entity Cultural anthropology is also useful in facilitating meaningful interaction with other cultures and sub cultures Copyright Virtual University of Pakistan 2 Introduction to Cultural Anthropology SOC401 VU Useful Terms Components parts Paleontology specialized branch of physical anthropology that analyses the emergence and subsequent evolution of human physiology Variation degree of difference Archeology sub field of anthropology that focuses on the study of pre historic and historic cultures through the excavation of material remains Contemporary current Urban city based Ethnocentrism the practice of viewing the customs of other societies in terms of one s own Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 1 in Cultural Ablbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 73 23971 blbropology by Ember cmd Peggrz39be Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture Which provide useful and interesting information Copyright Virtual University of Pakistan 3 Introduction to Cultural Anthropology SOC401 VU Lesson 02 THE CONCEPT OF CULTURE AND THE APPLICATION OF CULTURAL ANTHROPOLOGY Examining Culture We began this course by defining anthropology and its various branches We also looked at the chief duties of an anthropologist In this session we will be taking a more detailed look at cultural anthropology and its application We will also be dissecting the phenomena of culture and looking at the special functions of applied anthropology Last but not least as we all know all human occupations have their own set on ethical implications in this lecture we will be analyzing what an anthropologist owes to their profession and to society at large Before we take a more in depth look into cultural anthropology we must take a moment to first define what exactly is meant by culture In a non scientific way culture refers to such personal refinements as classical music the fine arts cuisine and philosophy So an example of this theory a person is considered more cultured if he listens to Bach rather than Ricky Martin or to make this example more nationalistic a person is said to be cultured if he listens to Nusrat Fateh Ali rather than Abrar ul Haq However anthropologists use this term in a much broader term than the average man Anthropologists don t differentiate between the cultured people and un cultured people All people have culture according to the anthropological definition We will define culture as every thing people have think and do as members of a society This definition can be most useful since the three verbs correspond to the three major components of culture That is everything people have refers to material possessions everything people think refers to the things they carry around in their heads such as ideas values and attitudes and everything people do refers too behavior patterns Thus all cultures compromise material objects ideas values and attitudes and patterned ways of behaving Just to give you better understanding of culture let us look at some of its main attributes 0 Culture includes everything that people have think and do as members of a society 0 All people have a culture 0 Culture comprises material objects ideas values and attitudes and patterned ways of behaving 0 Culture is a shared phenomenon For a thing behavior or idea to be classified as being cultural its must have a meaning shared by most people in a society Because people share a common culture they are able to predict with in limits how others will think and behave Cultural influences are reinterpreted and thus do not yield uniform effects Culture is learned One very important factor to remember about culture is that it s learned If we stop to think about it a loot of what we do during our waking hours is learned Brushing our teeth eating three times a day attending school tying our show laces these are all actions that we had to learn and yet they are an integral part of our culture While humans do have instincts culture is not transmitted genetically The process of learning culture is called enculturation which is similar in process but differs in terms of content Culture is necessary for our survival and effects how we think and act People from the same culture can predict how others will react due to cultural conditioning Copyright Virtual University of Pakistan 4 Introduction to Cultural Anthropology SOC401 VU Cultural Universals Cultural universals include economic systems systems of marriage and family education systems social control systems and systems of communication Some cultural systems are seemingly invisible such as insurance in the form of family based social safety nets many people in the developing world do not have insurance instead they rely on their families for support While it seems that these people have no one to help them in times of need they in fact do have social safety nets in the form of family support The versatility of cultural systems illustrates how exible and adaptable humans are Adaptive and Maladaptive Features of Culture Human beings rely more on cultural than biological adaptation to adjust to different types of environments including deserts and very cold areas The clothing habits of Eskimos in the North Pole allows them to live in a place which is naturally very inhospitable Biologically they are the same as us but they have learned to wear more appropriate clothing with lots of fur to keep the cold out These items of clothing have become a cultural trade mark with them Whenever we think of Eskimos we think of them laden with furs Humans can now even live in outer space or under water for limited periods of time Maladaptive or dysfunctional aspects of culture such as pollution can threaten or damage human environments The consumption of leaded petrol is bad for the environment yet given our reliance on automobiles it is difficult to do without them So what started of as an adaptive aspect allowing us to travel great distances has no become a maladaptive aspect of culture due to the sheer number of cars to be found around the world Integrative Aspects of Culture Cultures are logical and coherent systems shaped by particular contexts Various parts of culture are interconnected Yet culture is more than a sum of its parts Culture and the Individual Although culture in uences on the thoughts actions and behavior of individuals it does not determine them exclusively There is a diverse range of individuality to be found within one culture Most cultures are also comprised of subcultures for example artists in most societies have a slightly different way of dressing talking and thinking that mainstream people in their communities Applied versus Pure Anthropology Pure anthropology is concerned refining methods and theories to obtain increasingly accurate and valid anthropological data On the other hand applied anthropologists aims to understand and recommend changes in human behavior to alleviate contemporary problems ProblemOriented Research Anthropologists can apply anthropological data concepts and strategies to the solution of socio economic political problems facing different cultures Anthropologists can focus on development research or advocacy to help improve the human condition Specialized Functions for Applied Anthropologists a Policy Researcher provides cultural data to policy makers to facilitate informed decisions b Evaluator use research skills to determine how well a policy or program has succeeded in its objectives Copyright Virtual University of Pakistan 5 Introduction to Cultural Anthropology SOC401 VU C Impact Assessor measuring or assessing the effect of a particular project or policy 1 Needs Assessor use research skills to determine particular needs of a community of people e Trainer impart cultural knowledge about certain populations to different groups Ethical Implications Responsibility to the People Studied Anthropologists have an ethical responsibility to the people they are studying they need to present their finding in an unbiased way so that the true picture of their culture way of life can be presented Responsibility to the discipline The chief concern of all anthropologists should be to their discipline They must conduct their research in such a way that their findings play an integral part in consolidating their discipline Responsibility to Sponsors Most research that is done in the field is sponsored by one organization or another or in some cases some individuals are carrying out the burden of sponsorship the anthropologists must ensure that he carries out his duties with the utmost sense of responsibility Responsibility to Own and Host government Most researchers conduct research internationally where they have to respect the laws of their own country and that of the host country Useful Terms Implications results Dissection to take apart Enculturation the process by which human infants learn their culture Versatile different having a varying range Ethical moral Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapters 2 and 3 in Cuz umAm bropoogy A Applied Pmpecz z39ve 7y Fermrro cmdor Cbapz er 73 23971 H Zbropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following website for this lecture Applied Anthropology htt wwwindianaedu wanthro a liedhtm Copyright Virtual University of Pakistan 6 Introduction to Cultural Anthropology SOC401 VU Lesson 03 MAJOR THEORIES IN CULTURAL ANTHROPOLOGY What is a theory A theory suggests a relationship between different phenomenons Theories allow us to reduce the complexity of reality into an abstract set of principles which serve as models to compare and contrasts different types of realities Theories are based on hypotheses which provide a proposition that needs to be tested through empirical investigations If what is found is consistent with what was expected the theory will be strengthened if not the theory will be either abandoned or some more time will be spent on it to revise it Anthropological theory changes constantly as new data comes forth Anthropological theories attempt to answer such questions as why do people behave the way they do And how do we account for human diversity These questions guided the early nineteenth attempts to theorize and continue to be relevant today We will explore the in chorological order the major theoretical schools of cultural anthropology that have developed since the mid nineteenth century Some of the earlier theoretical orientations such as diffusionism no longer attract much attention however others such as evolutionism have been modified and re worked into something new It is easy in hindsight to demonstrate the inherit flaws in some of the early theoretical orientations However we should keep in mind however that contempary anthropological theories that may appear plausible today were built on what we learnt from those older theories Cultural Evolutionism According to this theory all cultures undergo the same development stages in the same order To develop a better understanding of these various development stages it is important to brie y review these various stages and their sub stages Savagery barbarism and civilization were three classifications that classical anthropologists used to divide culture However in 1877 Lewis Henry Morgan wrote a book titled A dem 06769 in it the three stages of cultural anthropology were further classified into 7 stages which are as follows 0 Lower Savagery From the earliest forms of humanity subsisting on fruits and nuts 0 Middle Savagery Began with the discovery of fishing technology and the use of fire 0 Upper Savagery Began with the invention of bow and arrow 0 Lower Barbarism Began with the art of pottery making 0 Middle Barbarism Began with the domestication of plants and animals in the old world and irrigation cultivation in the new world 0 Upper Barbarism Began with the smelting of iron and the use of iron tools 0 Civilization Began with the invention of the phonetic alphabet and writing 1877212 Evolution is unidirectional and leads to higher levels of culture A deductive approach used to apply a general theory to specific cases Evolutionists were often ethnocentric as they put their own societies on top of the evolutionary ladder Yet it did explain human behavior by rational instead of supernatural causes Diffusionism Like evolutionism diffusionism was deductive and rather theoretical lacking evidence from the field It maintained that all societies change as a result of cultural borrowing from one another The theory highlighted the need to consider interaction between cultures but overemphasized the essentially valid idea of diffusion Copyright Virtual University of Pakistan 7 Introduction to Cultural Anthropology SOC401 VU Historicism Any culture is partially composed of traits diffused from other cultures but this does not explain the existing complexity of different cultures Collection of ethnographic facts must precede development of cultural theories inductive approach Direct eldwork is considered essential which has provided the approach a solid methodological base emphasizing the need for empirical evidence Each culture is to some degree unique So ethnographers should try to get the view of those being studies not only rely on their own views Historicists emphasized the need for training female anthropologists to gain access to information about female behavior in traditional societies Their anti theoretical stance is criticized for retarding growth of the anthropological discipline Psychological Anthropology Anthropologists need to explore the relationships between psychological and cultural variables according to this theory Personality is largely seen to be the result of learning culture Universal temperaments associated with males and females do not exist in practice based on research conducted by psychological anthropologists for example it was noticed that there are no universally consistent personality traits like being hard working on the basis of being a male or a female Functionalism Like historicism functionalism focused on understanding culture from the viewpoint of the native It stated that empirical fieldwork is absolutely essential Functionalists stressed that anthropologists should seek to understand how different parts of contemporary cultures work for the well being of the individual and the society instead of focusing on how these parts evolved Society was thought to be like a biological organism with all of the parts interconnected The theory argued that change in one part of the system brings a change in another part of the system as well Existing institutional structures of any society are thought to perform indispensable functions without which the society could not continue NeoEvolutionism Neo Evolution states that culture evolves in direct proportion to their capacity to harness energy The theory states that culture evolves as the amount of energy harnessed per capita per year increases or as the efficiency of the means of putting energy to work increases Leslie White1900 1975 Culture I Energy x Technology Culture is said to be shaped by environmental and technological conditions Therefore people facing similar environmental challenges are thought to develop similar technological solutions and parallel social and political institutions Cultures evolve when people are able to increase the amount of energy under their control according to this theory Given this emphasis on energy the role of values ideas and beliefs is de emphasized Useful Terms Theory a general statement about how two or more facts are related to one another Hypotheses an educated hunch as to the relationship among certain variables that guides a research project Evolutionism the 19th century school of cultural anthropology represented by Morgan and Tyler that attempted to explain variations in cultures by the single deductive theory that they all pass through a series of evolutionary stages Copyright Virtual University of Pakistan 8 Introduction to Cultural Anthropology SOC401 VU Savagery the rst amongst the three basic stages savagery barbarism and civilization of cultural evolution Barbarism the middle of the three basic stages of the 19th century theory developed by Lewis Morgan that all cultures evolve from simple to complex systems Civilization a term used by anthropologists to describe any society with cities Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 4 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermrro and or Chapter 14 in Am bropoogy 7y Ember cmd Peggrz39 e Internet Resources Anthropological Theories http wwwasuaeduantFacultymurphy436anthroshtm Use the hyperlinks on the above website to read up on the following theories for today s lecture Social Evolutionism Diffusionism and Acculturation Historicism Functionalism American Materialism Cultural Materialism Copyright Virtual University of Pakistan 9 Introduction to Cultural Anthropology SOC401 VU Lesson 04 GROWTH OF ANTHROPOLOGICAL THEORY continued French Structuralism French Structuralism focused on identifying the mental structures that underpin social behavior drawing heavily on the science of linguistics Structuralism thought that cognition based on inherent mental codes is responsible for culture Structuralism focused on underlying principles that supposedly generate behavior at the unconscious level rather than observable empirical behavior itself It focused more on repetitive structures rather than considering reasons for cultural change or variation Cultural alterations and variation are explained by reference to external environmental and historical influences Structuralism is criticized for being overly theoretical and not easily verifiable through empirical evidence EthnoScience Ethno science describes a culture using categories of the people under study emic approach rather than by imposing categories from the ethnographer s culture etic approach This theory tires to minimize bias and make ethnographic descriptions more accurate by focusing on underlying principles and rules of a given context Due to the time consuming nature of this methodology ethno science is confined to describing very small segments of given cultures It is difficult to compare native data collected by ethno scientists since there is no common basis for comparison Despite its impracticality the theory draws attention to the relativity of culture and its principles are useful for other theorists as well Cultural Materialism Cultural materialists rely on supposedly scienti c empirical and the etic approach of an anthropologist rather than relying on the viewpoints of the native informant Cultural materialists argue that material conditions and modes of production determine human thoughts and behavior Material constraints that arise from the need to meet basic needs are viewed as the primary reason for cultural variations For cultural materialist the importance of political activity ideology and ideas is considered secondary since it can only retard or accelerate change not be the cause for it Post Modernism Post modernism refutes the generalizing tendency in anthropology and does not believe that anthropologists can provide a grand theory of human behavior Instead it considers each culture as being unique Post modernism is in uenced by both cultural relativism and ethno science Post modernists want anthropology to stop making cultural generalizations and focus on description and interpretation of different cultures They consider cultural anthropology to be a humanistic not a scientific discipline Post modernists argue that ethnographies should be written collaboratively so that the voice of the anthropologist co exists alongside that of local people Interpretive Anthropology Emerging out of post modernism interpretive anthropology focuses on examining how local people themselves interpret their own values and behaviors Using an emic approach interpretive anthropologists focus on the complexities and living qualities of human nature Useful Terms Structural functionalism a school of cultural anthropology that examines how parts of a culture function for the well being of society Copyright Virtual University of Pakistan 10 Introduction to Cultural Anthropology SOC401 VU Confined limited Cultural materialism a contemporary orientation in anthropology that holds that cultural systems are most in uenced by such material things as natural resources and technology EtiC Relying on the views of the researcher or the cultural anthropologists Emic Relying on views of local people Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 4 in Cultural A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 74 23971 Lil z bropoogy by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following website for this lecture Anthropological Theories http wwwasuaeduantFacultymurphy436anthroshtm Use the hyperlinks on the above website to read up on the following theories for today s lecture Ecological Anthropology Cognitive Anthropology Structuralism Symbolic amp Interpretive Anthropologies Postmodernism amp Its Critics Copyright Virtual University of Pakistan 11 Introduction to Cultural Anthropology SOC401 VU Lesson 05 METHODS IN CULTURAL ANTHROPOLOGY Fieldwork A distinctive feature of Cultural Anthropology is its reliance on experiential eldwork as a primary way of conducting research Cultural Anthropologists collect cultural data and test their hypothesis by carrying out eldwork in different parts of the world The areas where this eldwork is conducted can include both urban and rural areas in highly industrialized rich countries or poor developing nations of the world Detailed anthropological studies have been undertaken to study the way in which people belonging to different cultures and sub cultures think and behave Comments on Fieldwork Since the credibility of ethnographic studies rests on their methods of research often termed the methodology so cultural anthropologists have begun focusing on how to conduct fieldwork While every fieldwork situation is unique there are a number of issues in common like the need to prepare for fieldwork or to obtain permission from the country s government where this research is to be conducted Even if a researcher is doing research within his her own country often permission from the concerned level of the local government is required particularly if thee research is considering how government structures institutions like schools or health clinics for example effect the lives and behavior of a particular group of people Stages of Fieldwork Selecting a research problem Formulating a research design Collecting the data Analyzing the data Interpreting the data Selecting a Research Problem 991 91 Cultural Anthropologists have moved away from general ethnographies to research that is focused specific and problem oriented The problem oriented approach involves formulation of a hypothesis which is then tested in a fieldwork setting Formulating a Research Design The independent variable is capable of effecting change in the dependent variable The dependent variable is the one that we wish to explain whereas the independent variable is the hypothesized explanation If we want to look at the effect of urbanization on family interactions the independent variable will be urbanization Defining Dependent Variables Dependent variables must be defined specifically so they can be measured quantitatively To ascertain family interaction the following issues deserve attention 0 Residence Patterns 0 Visitation Patterns 0 Mutual Assistance Copyright Virtual University of Pakistan 12 Introduction to Cultural Anthropology SOC401 VU 0 Formal Family Gatherings 0 Collecting and Analyzing Data Once the hypothesis is made concrete the data is collected through an appropriate data collection technique Once collected the data is coded to facilitate analysis For example if a questionnaire is being used to get views of 100 people in a given community all those people who say yes to a given question could be identified using a code to obtain a statistical number Then a similar questionnaire in another community could identify people responding positively to the same question In this way a researcher could compare how many people in both communities responded positively to the same question In addition to surveys other research techniques can also be coded even ethnographies can be coded to enable comparison of peoples attitudes and behavior in different communities Interpreting the Data Interpretation is the most difficult step in research which involves explaining the ndings to refute or accept the hypothesis A researcher could hypothesize that there is a link between urbanization and increasing poverty and then go into a community to see if increasing poverty is responsible for more people shifting into the city based on these findings the hypothesis could either be rejected or accepted Findings of a particular study can be compared to similar studies in other areas to get more extensive information about a particular problem or how different communities with different cultures deal with similar problems The problem of poverty and how different people react to this problem is a good example of a research problem that can be examined by different researchers and their findings compared to see how different cultures respond when they are faced by poverty Need for Flexibility A technique originally mentioned in the research proposal can prove to be impractical in the field Cultural anthropologists need some options and remain exible in choosing an appropriate technique given surrounding circumstances Difficulties in Fieldwork Research in remote locations carries risks such as exposure to diseases or different forms of social violence Researchers can encounter psychological disorientation commonly termed mz me Mac when they have to live and deal with circumstances completely alien to their own surrounds Researchers must also try to find a balance between subjectivity and objectivity if they want to assure the quality of their research and to prevent its criticism on the basis of being biased by the researcher s own viewpoints Many anthropological studies have been criticized for being biased or ethnocentric in their attempt to look at how other people live Useful Terms Ethnography detailed anthropological study of a culture undertaken by a researcher Ethnocentric the view that one s own cultural is superior Data collection of facts Biased prejudiced holding an unfair view Culture shock psychological disorientation brought on due to cultural difference Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Copyright Virtual University of Pakistan 13 Introduction to Cultural Anthropology SOC401 VU Chapter 5 in Cuz umAm bmpoogy A Applied Pmpecz z39be by Fermer cmdor Cbapz er 74 cmd 28 23971 H lbropology by Ember cmd Peggrz39be Internet Resources In addition to reading from the textbook please visit the following website for this lecture Cultural Anthropology Methods htt www vctccommnetedu brian methodshtml Use the hyperlinks on the above website to read up on the following Methods of Research in Cultural Anthropology for today s lecture Participant observation Survey research Interviews Document Analysis Archival research Media analysis Historical analysis Copyright Virtual University of Pakistan 14 Introduction to Cultural Anthropology SOC401 VU Lesson 06 METHODS IN CULTURAL ANTHROPOLOGY continued Participant Observation Anthropologists use this technique more extensively and frequently than other social scientists Participant Observation means becoming involved in the culture under study while making systematic observations about what goes on in it Guidelines for Participant Observation Fieldwork Before approaching the field it is advisable to obtain clearance from all appropriate levels of the political administrative hierarchy Local people at the grassroots level know their own culture better than anyone else and their views need to be given due respect Advantages of Participant Observation It allows distinguishing between what people say they do and what they actually do The greater the cultural immersion is the greater is the authenticity of cultural data It allows observation of non verbal behavior as well Disadvantages of Participant Observation There are problems of recording observations while using this technique The technique has an intrusive effect on subject of study Also a smaller sample size is obtained through this technique than through other techniques and the data obtained is hard to code or categorize making standardized comparisons difficult Interviewing Enables collection of information on what people think or feel allitudim data as well as what they do bebcwz39om dam Ethnographic interviews are often used alongside other data gathering techniques Structured and Unstructured Interviews In structured interviews interviewers ask respondents exactly the same set of questions in the same sequence Unstructured interviews involve a minimum of control with the subject answering open ended questions in their own words Guidelines for Researchers To minimize distortions in collected data researchers can check the validity of their findings by either asking cross check information given by respondents or repeat the same question at a later time It is important to frame the questions neutrally Instead of asking You don t smoke do you ask Do you smoke Census Taking Collecting basic demographic data at the initial stages of fieldwork is the least intrusive manner to begin investigating the state of a given community Copyright Virtual University of Pakistan 15 Introduction to Cultural Anthropology SOC401 VU Document Analysis Documentary analysis of administrative records newspapers and even popular culture like song lyrics or nursery rhymes is often surprisingly revealing about the circumstances aspirations and values of different people Genealogies Mapping relations of informants particularly in small scale societies is very revealing since they tend to interact more closely with their families than people in more complex societies which have a greater number of institutions and professionals Photography Cameras and video recorders allow researchers to see without fatigue without being selective and provide a lasting record of cultural events and physical surroundings Some local communities however can object to the use of cameras due to their conservative values or they consider it an intrusion on their privacy Choosing a Technique Choice of technique depends on the problem being studied Choice of a technique also depends on the receptiveness of the community in question to a particular technique For example if a given community does not allow the anthropologist undertaking research to use cameras the researcher will have to respect the wishes of the community in question and document descriptions of relevant events instead of being able to take a photograph by which this information could have been captured more easily Undertaking CrossCultural Comparisons For undertaking such comparisons particular with the help of statistics the following issues deserve attention 0 Quality of data being compared must be consistent and based on the same methodology information based on interviews conducted in one culture cannot be compared with information obtained from questionnaires in another culture 0 Units of analysis must be comparable it s not possible to compare different levels of social systems a village cannot be compared to a city for example 0 Contrasting cultural traits out of context from their remaining culture is problematic but useful in identifying similarities across different cultures which is an important objective for cultural anthropology Useful Terms Attitudinal based on how people think or feel about something Receptiveness response to a particular action Participation being a part of something Perspective point of view Cultural traits particular features of a culture Crosscultural comparison of differences between cultures Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Copyright Virtual University of Pakistan 16 Introduction to Cultural Anthropology SOC401 VU Chapter 5 in Cuz umAm bmpoogy A Applied Pmpecz z39be by Fermer cmdor Cbapz er 74 cmd 28 23971 H lbropology by Ember cmd Peggrz39be Internet Resources In addition to reading from the textbook please visit the following website for this lecture Cultural Anthropology Methods htt www vctccommnetedu brian methodshtml Use the hyperlinks on the above website to read up on the following Methods of Research in Cultural Anthropology for today s lecture Participant observation Survey research Interviews Document Analysis Archival research Media analysis Historical analysis Copyright Virtual University of Pakistan 17 Introduction to Cultural Anthropology SOC401 VU Lesson 07 COMPARATIVE STUDY OF PRODUCTION DISTRIBUTION AND CONSUMPTION IN DIFFERENT PARTS OF THE WORLD What is Economic Anthropology Economic Anthropology involves examining how different cultures and societies produce distribute and consume the things they need to survive All cultures need to be able to manage these processes in accordance with their given circumstances to ensure the survival of their people Differentiating Economics from Economic Anthropology While economists assume that people are preoccupied by the need to maximize profits and this is the basic impulse due to which they produce goods and services Economic anthropologists do not believe profit maximization is equally important for all cultures They point out that there are several other processes besides profit maximization which exist in different cultures of the world by the allocation distribution of resources need to produce goods and services and the distribution of the goods and services takes place For example these economic anthropologists look at how different cultures distribute land which is an important resource needed for production of agricultural goods and have noticed that different cultures have different ways in which this distribution takes place However economic anthropologists realize that like economists they too must answer some basic questions concerning basic economic needs of human beings which all cultures around the world face given that some human needs are universal and must be met no matter what type of culture people belong to Economic Universals Economic anthropologists have to consider the following economic universals which are of vital importance to human beings no matter what their cultural systems are like a Regulation of Resources How land water and other natural resources like minerals are controlled and allocated b Production How material resources sugarcane are converted into usable commodities sugar c Exchange How the commodities once produced are distributed among the people of a society Examining the Issue of Land Rights Free access to land is found in environments where water and pasturage is scarce Land rights are more rigidly controlled among horticulturalists and agriculturalists than among foragers and pastoralists Division of Labor Durkbez39m the famous sociologist responsible for establishing this branch of study in the early twentieth century had distinguished between two types of societies those based on mecbam39m solidarz and others based on organic solidarigl Societies with a minimum specialization of labor are held together by mechanical solidarity based on commonality of interest In these societies people are more self reliant therefore they need other people to a lesser degree than people in societies where people focus on production of a very specific good or service and then rely on others to provide them other necessities of life in exchange for their specialized product Copyright Virtual University of Pakistan 18 Introduction to Cultural Anthropology SOC401 VU Highly specialized societies are held together by organic solidarity based on mutual interdependence Such societies emphasize the need for specialization and people depend on other people in order to obtain the different things that they need Gender Roles and Age Specialization Generally many cultures allocate specific responsibilities on the basis of age and gender Ole people and those very young are given lighter tasks in most cases where circumstances permit cases of extreme poverty child labor can also take place Similarly women are usually allocated tasks which allow them to maintain exible timings so that they can look after their homes as well There are exceptions to this rule however since many educated women do work as long as men often leaving their children to the care of day centers In many countries around the world the process of urbanization has led men to move away to the cities in order to earn more cash often leaving women behind to undertake agricultural work which was previously done by men Circumstances also compel poor women to take on heavy work burdens like their men folk to ensure the survival of their families Moreover the same type of activity weaving may be associated with the opposite gender in different cultures the division of labor by gender is seen as being arbitrary Is Nepotism Always Bad In many societies people relate to each other based on the principle of particularism family and kinship ties rather than on universalistic terms using standardized exams interviews Nepotism is not necessarily a sign of corruption since consideration of ground realities like kinship ties can often help determine how people will adjust to specific work environments Useful Terms Allocation of resources the distribution of resources Barter the direct exchange of commodities between people that does not involve a standardized currency Division of Labor the set of rules found in all societies dictating how the day to day tasks are assigned to the various members of a society Reciprocity the practice of giving a gift with an expected return Globalization the world wide process dating back to the demise of the Berlin wall which involves a revolution in information technology opening of markets and the privatization of social services Labor specialization a form of having command over one activity Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 8 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 77 23971 Lil z bropoogy by Ember cmd Peggrz39 e Copyright Virtual University of Pakistan 19 Introduction to Cultural Anthropology SOC401 VU Internet Resources In addition to reading from the textbook please visit the following website for this lecture Economic Anthropology http enwikipediaorg wiki Economic anthropology Use the hyperlinks on the above website to read up on the following aspects of Economic Anthropology for today s lecture Anthropological theories of value The Anthropological view of Wealth Copyright Virtual University of Pakistan 20 Introduction to Cultural Anthropology SOC401 VU Lesson 08 ECONOMIC ANTHROPOLOGY continued THE DISTRIBUTION OF GOODS AND SERVICES Modes of Distribution Economic Anthropologists categorize the distribution of goods and services in three modes reciprocity redistribution and market exchange Based on these three forms of exchange cultures around the world distribute the goods and services produced by them in order to ensure the survival of the various people which belong to that particular culture 1 Reciprocity implies exchange of goods and services of almost equal value between two trading partners 2 Redistribution most common in societies with political bureaucracies is a form of exchange where goods and services are given by a central authority and then reallocated to create new patterns of distribution 3 Market exchange systems involve the use of standardized currencies to buy and sell goods and services Types of Reciprocity The idea of reciprocity can be divided into the following distinct types of practices evident in cultures around the world 1 Generalized reciprocity involves giving gifts without any expectation of immediate return For example the parents look after their children and these children when they grow older look after their aging parents This is an unsaid rule or obligation towards one family which people undertake willingly out of love and concern and without any external compulsion or the idea of getting something back in return for their caring attitudes 2 Balanced reciprocity involves the exchange of goods and services with the expectation that the equivalent value will be returned within a specific period of time For example if a neighbor s son or daughter is getting married the neighbors will take gifts to the wedding and then expect the same courtesy when their own child s wedding The notion of birthday gifts is even more time specific and thus serves as a good example of balanced reciprocity 3 Negative reciprocity involves the exchange of goods and services between equals in which the parties try to gain an advantage in order to maximize their own profit even if it requires hard bargaining or exploiting the other person Redistribution Whereas reciprocity is the exchange of goods and services between two parties redistribution involves a social centre from which goods are redistributed Often this redistribution takes place through a political or bureaucratic agency eg the revenue collection or tax department which is found in most countries or even the meal system in Pakistan based on a religious ideology which is meant to redistribute wealth to those who are destitute Market Exchange Market exchange is based on use of standardized currencies or through the barter exchange of goods and services This system of exchange is much less personal than either reciprocity or redistribution People Copyright Virtual University of Pakistan 21 Introduction to Cultural Anthropology SOC401 VU trade in a marketplace to maximize their pro ts The greater the specialization of labor that exists in a society the more complex is the system of market exchange to be found in that society Globalization Globalization involves the spread of the free market economies to all parts of the world based on the assumption that more growth will take place when free trade and competition becomes a universal phenomenon Globalization has begun to show visible impacts on the cultures and lives of people around the world There are people who favor globalization thinking it will help remove poverty across much of the world but they are also those who think that globalization will do the exact opposite Useful Terms Organic Solidarity a type of social integration based on mutual inter dependence Particularism the propensity to be able to deal with people according to one s particular relationship to them rather than according to a universal standard Production a process where by goods are taken from the natural environment and then altered to become consumable goods for society Property Rights western concept of individual ownership Standardized Currency a medium of exchange with well defined and an understood value Universalism the notion of awarding people on the basis of some universally applied set of standards Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 8 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 77 23971 Lil z bropoogy by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following website for this lecture Economic Anthropology http enwikipediaorg wiki Economic anthropology Use the hyperlinks on the above website to read up on the following aspects of Economic Anthropology for today s lecture Non market economics Copyright Virtual University of Pakistan 22 Introduction to Cultural Anthropology SOC401 VU Lesson 09 FOCUSING ON LANGUAGE An Anthropological Perspective Language is a unique phenomenon which allows human beings to communicate meaning to others and express our thoughts and feelings to other people Perhaps the most distinctive feature of being human is our capacity to create and use language Many anthropological linguists would agree that without language human culture could not exist beyond a very basic level The Nature of Language The meaning we give to language is arbitrary random It is due to this arbitrary nature of language there is such a diversity of languages Languages of the World Almost 95 percent of people speak fewer than 100 languages of the approximately 6000 languages that are currently found in the world Due to this many languages face the threat of extinction with an increasingly small number of people who know the language This evident dying out of rarely spoken languages is an issue of concern to cultural anthropologists since the extinction of a language also means the death of a way of thinking and expressing human thought Of the more widely spoken languages Mandarin Chinese dialect is spoken by almost 1 in 5 people in the world Hindi is also spoken by multitudes of people Yet English is the most popular second language spoken by people all around the world Communication Human versus Nonhuman Humans are not the only species that communicate Animals use calls to mate find food and signal danger Human communication amongst humans is however much more complex than that of animals We can combine words in unique ways to express our innermost feelings or even very complicated ideas which can be understood by others who can speak the same language Open and Closed Communication Systems Animal sounds are mutually exclusive Closed Communication systems they cannot be combined to express new meanings A warning sound of an animal is always the same and this sound is used to convey the same message always it cannot be combined with other sounds to convey different types of meaning Only humans can put different meanings together through using of an Open Communication system which is the language they speak This categorization of Open and Closed communication systems has been questioned by anthropological linguists based on research conducted using sign language A chimpanzee for example can in fact combine two words to create a third word Researchers have trained a chimpanzee to learn the sign language for water and for bird but not shown it how to say duck using sign language This chimpanzee has however been able to create the two known words water and bird to refer to a duck indicating that other species could also use open communication systems like humans However no linguist has yet made the claim that any animal species has evolved language to a degree which can express the complexities of meaning that human beings can Copyright Virtual University of Pakistan 23 Introduction to Cultural Anthropology SOC401 VU Displacement Humans can speak of purely hypothetical and abstract things of things which happened in the past or may happen in the future Whereas animals only communicate in the present about things concerning their immediate surroundings animals cannot express abstract thoughts Learning to Communicate Imitating adult speech is partially responsible for acquisition of language Linguists like Noam Chomsky at the Massachusetts Institute of Technology think that children are born with a universal grammatical blueprint which helps them pick up the rules of the language being spoken around them so quickly and that this is a biological gift that only the human species seems to possess since no other species has such compleX communication abilities Structure of Language All languages have logical structures or rules which are followed by all those who can speak read and write that particular language 0 Phonology provides the sound structure to a language so it can be commonly understood when spoken 0 Morphemes the smallest units of speech that convey meaning art ist s by standing alone or being bound to other words 0 Grammar provides the unique rules of a language which help give a logical structure to a language Grammar also provides rules by which words are arranged into sentences AgWax Consider the words Adam apples likes eating which make no sense since the verb eating and the adjective likes are not in their grammatically correct position Correcting the mistake will make the sentence clear Adam likes eating apples The underlying structure of sentences which enables us to correct such a mistake and speak in a clear manner is due to the grammatical rules of syntaX The fact that we can even say this sentence is due to phonology and morphemes help us create a sentence by providing us with different meanings in smaller words eat ing like s Useful Terms Displacement the ability that humans have to talk about things remote in time and space Free Morphemes morphemes that appear in a language without being attached to other morphemes Grammar the systematic way in which sounds are combined in any given language to send and receive meaning full utterances Phonology the study of language s sound system Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 6 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 75 23971 Lil z bropoogy by Ember cmd Peggrz39 e Copyright Virtual University of Pakistan 24 Introduction to Cultural Anthropology SOC401 VU Internet Resources In addition to reading from the textbook please Visit the following website for this lecture Copyright Virtual University of Pakistan 25 Introduction to Cultural Anthropology SOC401 VU Lesson 10 FOCUS ON LANGUAGE continued Changes in Language Language evolves over time Linguists can undertake a Jymbmm39c mag3239s to understand language structures and its underlying rules at a given point in time Undertaking a diacbmm39c mag5239s however means looking at how a given language changes over time Language Families Language families include languages derived from a proto language Linguists began clustering languages upon finding similarities between Sanskrit and classical Latin and Greek in the 18805 From the perspective of language families Germanic is mother tongue of English French and Spanish are its sister languages They all belong to the Indo European language family All languages have internal dialects as well as sharing features with other languages as well particularly with those belonging to the same language family as them Levels of Complexity Linguists have proven that languages of less technological societies are as capable of communicating abstract ideas as advanced societies For example the Navaho do not have singular and plural nouns like English does but their verbs contain much more information than English Instead of merely saying going the Navaho say how they are going if they are going on a horse they must further indicate how fast the horse is going which is a lot more information than a phrase in English which just mentions I am going Cultural Emphasis The vocabulary of languages emphasizes significant words in a given culture This is known as a cultural emphasis Technologically related words show emphasis on technology in highly industrialized countries There are numerous new words to describe computer technology in various languages which did not even exist a few decades ago Language and Culture Some ethno linguists suggest that language is more than a symbolic inventory of experiences from our physical world experiences According to them language even shapes our thoughts and provides a standardized way to react to experiences The SapirWhorf Hypothesis According to this hypothesis different cultures see the world differently due to their different languages Language in uences and channels our perceptions and thus shapes our resulting behavior as well The hypothesis has conducted several tests in the attempt to validate its claim Linking Language to Culture It is difficult to establish causation to prove either that language determines culture or that culture in uences language Language does mirror values of a culture consider for example the emphasis on self in individualistic societies On the other hand in more traditional societies like Japan the use of collective words like we is much more evident Copyright Virtual University of Pakistan 26 Introduction to Cultural Anthropology SOC401 VU SocioLinguistics Socio linguistics examines links between languages and social structures While earlier cultural linguists focused on language structures there is now greater focus on the situational use of language ie how the same language is used to speak in different manners depending on the context of the conversation Diglossia Often two varieties of the same language are spoken in different social situations High forms are associated with literacy and education and the elite whereas the lower forms for example Pidgin are considered to be less sophisticated Language and Nationalism Language has important implications for ethnic identities To forge national unity political leaders have often suppressed use of local languages in favor of standardized national languages to provide a sense of unity to the nation and to develop a common means of communication Useful Terms Evolve Develop Synchronic Analysis the analysis of cultural data at a single point in time rather than through time Diachronic Analysis the analysis of socio cultural data through time rather than at a single point in time Derived taken from Abstract Not clear or vague Emphasis To lay importance on Perceptions Viewpoints Suppressed concealed or covered up Dialect form of speech peculiar to particular region Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 6 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 75 23971 Lil z bropoogy by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following website for this lecture Copyright Virtual University of Pakistan 27 Introduction to Cultural Anthropology SOC401 VU Lesson 11 OBTAINING FOOD IN DIFFERENT CULTURES Strategies for Obtaining Food Food obtaining strategies vary from culture to culture Food obtaining strategies are developed in response to particular environments There are five major food obtaining strategies found in different cultures of the world These five forms of obtaining food are not mutually exclusive and within each category there are evident variations due to technological and environmental differences Therefore often one form of obtaining food predominates within a given culture While food obtaining strategies vary widely around the world none is necessarily superior then another Major Food Obtaining Strategies Food Collection collecting wild vegetation hunting and fishing Horticulture cultivation using simple tools and small and shifting plots of land Pastoralism keeping livestock and using its produce for food Agriculture cultivation using animals irrigation and mechanical implements Industrialization producing food using complex machinery Most developed countries and an increasing amount of developing countries rely on industrialized processes to obtain food Food Environment and Technology Some environments enable a number of modes of food acquisition while others permit limited number of adaptations Technology provides the advantage of adaptation to a given environment It can be said that specific food obtaining modes are in uenced by the interaction of a people s technological and environmental conditions The extent to which any society can procure food depends on sophistication of tools used and the abundance of plant and animal life in a given area Productivity of agriculturalists not only depends on technology but also availability of natural resources like water and fertile soil Anthropologists agree that while the environment does not set limits on food obtaining patterns it does place a limit on the adaptations possible and on the ultimate productivity of an area People with simple technologies also cope well with their environments and are intelligent given their circumstances and surroundings The environmental capacity of a given area is referred to as carrying capacity The natural consequence of exceeding carrying capacity is to harm the environment Optimal Foraging Many foraging societies spend extra time and effort to obtain a particular food Ethnographic studies of the Ache in Paraguay for example have revealed that this is not irrational behavior but due to caloric returns of these food sources despite the energy expended in killing collecting and preparing it This reveals that optimal foraging is a calculated strategy not a irrational whim Useful Terms Foraging collecting or gathering Copyright Virtual University of Pakistan 28 Introduction to Cultural Anthropology SOC401 VU Optimal best or most feasible Expended spent Environmental capacity carrying capacity of the environment ie the amount of productive pressure the air water and soil can take Without being damaged Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 7 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermrro cmdor Cbapz er 76 23971 Lil z bropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Anthropology of Food http WWWarchaeolinkcom anthropology of food general reshtm Copyright Virtual University of Pakistan 29 Introduction to Cultural Anthropology SOC401 VU Lesson 12 FOOD AND CULTURE continued Food Collection Food collection involves systematic exploration of natural plants and animals available in given natural environments People have been foragers for an overwhelming majority of time and have only developed other options to secure food in the last 10000 years or so Food Collectors Most societies prefer to produce food but half a million people in different cultures live by foraging even today There are considerable variations in the life patterns of current foragers but it is possible to make some generalizations about them Contemporary Food CollectorsForagers Food collecting societies have low population density They are usually nomadic or semi nomadic rather than sedentary since their prey often migrates The basic social unit amongst food collectors is a family or a band a loose federation of families Contemporary food collectors occupy remote and marginal habitats due to pressure from food processing people with their dominating technology and thirst for more land While food collectors hunt as well as collect wild plants vegetation provides almost 80 of their food intake Food collecting people live in a wide variety of environments including deserts tropical forests mountains and the polar regions of the Artic and Antarctic circles Unlike food producers food collectors possess inbuilt mechanisms low population and little use of technology which prevents it from becoming too efficient and completely destroying their own source of food Do Foragers Live Well Despite inhabiting the most unproductive parts of Earth foragers are well off and dubbed the original af uent society by anthropologists They enjoy leisure time have enough food and use remarkable intelligence and ingenuity in securing their food Most contemporary foraging societies remain small scale unspecialized egalitarian and non centralized The Khung in the Kalahari Desert in Namibia and the Inuit in the Artic region provide good examples of hunting and gathering peoples today Food Production About 10000 years ago humans made a transition from collecting to producing food by cultivating crops and keeping herds of animals The earliest cultivation occurred in the Fertile Crescent of the Middle East Archeologists think this transition was due to demographic and environmental pressures Early farmers paid a high price for this new food strategy They did not switch convinced by the superiority of agriculture which was more monotonous less secure and required more labor and time Evidence reveals early cultivators also experienced a decline in nutritional and health standards because they had to shift from collecting to growing food Changes Resulting From Neolithic Revolution Food production resulted in the first population explosion Fertility rates also increased since children could make an economic contribution People became sedentary and civilizations began to develop As farming became more efficient people had more free time and began making farm implements and pottery leading to the division of labor and specialization The egalitarianism of foraging societies was replaced by social inequalities and the thirst for private ownership Copyright Virtual University of Pakistan 30 Introduction to Cultural Anthropology SOC401 VU Useful Terms Population number of people in a given area Sedentary settlement or settled down in one place Strategy a thought out method to obtain some objective or goal Monotonous boring Nutritional value amount of energy Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 7 in Cuz um Ambropoogy A Applied Pmpm z39ve 7y Fermrro cmdor Cbapz er 76 23971 Lil z bmpoogy 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Anthropology of Food http2 WWWarchaeolinkcom anthropology of food general reshtm Copyright Virtual University of Pakistan 31 Introduction to Cultural Anthropology SOC401 VU Lesson 13 OBTAINING FOOD IN DIFFERENT CULTURE continued Horticulture Horticulture is the simplest form of farming using basic tools no fertilizer or irrigation and relying on human power Horticulturists use shifting cultivation techniques also referred to as slash and burn cultivation Horticultural Crops Crops growth by horticulturists can be divided into three categories tree crops seed crops and root crops Common tree crops include bananas figs dates and coconuts Major seed crops are high in protein Wheat barley rice millet oats and sorghum are all seed crops Major root crops are high in starch and carbohydrates Yams sweet potatoes potatoes are all root crops The Lacondon Maya of Chiapas Mexico are more productive than mono crop agriculturalists They achieve three levels of production from the same land and do so by maintaining by imitating the dispersal patterns found within ecological systems of tropical rainforest rather than displacing them Slash and Burn Technique In unused areas of vast land slash and burn can be a reasonably efficient form of production Ash fertilized soil resulting from slashing and burning wild vegetation must lie fallow to restore fertility Under drought conditions of Al Nino during the 1990s horticulturists were severely criticized for destroying large tract of grasslands and forests in Madagascar Brazil and Indonesia since the fires they lit for clearing land often raged out of control Pastoralism Keeping domesticated livestock as a source of food is widely practiced in areas where cultivation is not possible Pastoralism involves a nomadic or semi nomadic lifestyle within small family based communities Pastoralists also maintain regular contact with cultivators to help supplement their diets Agriculture More recent than horticulture agriculture uses technologies like irrigation fertilizers and mechanical equipment to produce high yield and large populations Agriculture is associated with permanent settlements and high levels of labor specialization Intensive agriculture leads to even further specialization and use of technological inputs It also leads to social stratification political hierarchies and administrative structures Industrialization Since several centuries people have used industrialized food getting strategies There is increasing amounts of mechanical power available for the purpose of obtaining storing and processing food Industrialization also uses a mobile labor force and a complex system of markets which has led to the increasing commercialization of food Therefore food is grown not only for consumption but also for exporting to other countries of the world Copyright Virtual University of Pakistan 32 Introduction to Cultural Anthropology SOC401 VU Biotechnology provides a current example of industrialized food getting as does laser leveling or use of GPS transmitters on grain harvesters All these technological innovations have been incorporated into the food production process and helped to increase food output Yet there are environmental costs resulting from exceeding the carrying capacity of land and from overuse of technological innovations such as pesticides and fertilizers The use of biotechnology in food production is also a much debated topic Useful Terms Horticulture A form of small scale crop cultivation characterized by the use of simple technology and the absence of irrigation Carbohydrates energy source found in particular types of food group Tropical humid Drought lack of rainfall Criticize disapprove of Supplement add on Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 7 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 76 23971 Lil z bropoogy by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Anthropology of Food http WWWarchaeolinkcom anthropology of food general reshtm Copyright Virtual University of Pakistan 33 Introduction to Cultural Anthropology SOC401 VU Lesson 14 RELEVANCE OF KINSHIP AND DESCENT Kinship Defined Kinship is the single most important social structure in all societies Kinship is based on both consanguineal blood and affinal marriage relations or even fictive ties adoption godparents Functions of Kinship Vertical Function Kinship systems provide social continuity by binding together a number of generations Horizontal Function Kinships provide social solidarity and continuity Within the same generation as well Cultural Rules Regarding Kinship Kinship systems group relatives into certain categories and call them by the same name and behave With them in a similar manner Yet how particular cultures categorize relatives varies according to different principles of classification Kinship Criteria Different societies use different rules in formulating kinship ties Some of these are Generation uncles are in one generation cousins in another Gender cousins do not occupy gender determined kin categories Lineality kin of a single line ie son father grandfather Consanguineality kin through a linking relative Wife s brother Relative Age one kinship term for father s older brother another for his younger brother eg m and chacha Gender of Connecting Relative using different kinship terms for the father s brother s daughter his sister s daughter Social Conditions different kinship terms for a married or an unmarried bother Side of the Family different kinship terms for father s and mother s sides of the family eg phupho and khala Rules of Descent Rules of descent enable the affiliation of people with different sets of kin for example Patrilineal descent affiliates a person With the kin of the father Matrilineal descent affiliates a person With the kin of the mother Ambilineal descent permits an individual to affiliate With either parent s kin group Consanguineal versus Affinal Kin Copyright Virtual University of Pakistan 34 Introduction to Cultural Anthropology SOC401 VU Some societies make a distinction in kinship categories based on whether people are related by blood mmcmgm39 ea km or through marriage 7161 km For example take the difference between a sister and a sister in law or a brother and a brother in law Comparing Descent Groups Patrilineal descent groups are most common around the world The relations between man and wife tend to be more fragile in matrilineal societies Useful Terms Unilineal descent tracing descent through a single line such as matrilineal or patrilineal as compared to both sides bilateral decent Bilateral able to accommodate two sides simultaneously Matrineally mother s side of the family Patrineally father s side of the family Prevalent common amongst many Kinship relationship Merging integration Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 10 in Cuz um A lbropology A Applied Pmpecz z39be by Fermer andor Cbapz er 27 23971 Lil z bropology by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture which provide useful and interesting information Kinship Terms wwwmnsuedu emuseumz culturalz kinship 1 terms html Copyright Virtual University of Pakistan 35 Introduction to Cultural Anthropology SOC401 VU Lesson 15 KINSHIP AND DESCENT continued Tracing Descent In societies that trace their descent unilineally people recognize that they belong to a particular unilineal descent group or series of groups Sixty percent of cultures in the world are unilineal Unilineal groups are adaptive and clear cut social units based on birthright which in turn in uence inheritance marriage and prestige issues Kinship Organization Kinship is organized on the basis of different groups of varying sizes Lineages are based on a set of kin who can trace their ancestry back through known links Clans are unilineal groups which claim descent but they are unable to trace all their genealogical links Phratries are groups of related clan Moieties are two halves of a society related by descent Bilateral Descent A person is related equally to both sides of the family on the basis of bilateral descent This form of descent is prevalent in foraging and industrialized societies Bilateral systems are symmetrical and result in the formation of kindred which are loose kinship networks rather than being permanent corporate functioning groups Double Descent A double unilineal descent system is one where descendents are traced matrineally and patrineally As a result both sides of the family have a useful social function such as enabling inheritance Under this system it is possible for moveable property such as livestock or agricultural produce to be inherited from the mother s side whereas non moveable property land may be inherited from the father s side This system is found in only 5 of world cultures for eg the Yako in Nigeria Primary Kinship Systems There are siX basic types of kinship systems used to define how cultures distinguish between different categories of relatives Eskimo are found in one tenth of the world societies this system involves bilateral descent focusing on nuclear relations and lumping external relatives cousins uncles and aunts Hawaiian are found in a third of world societies this system uses the same term for all relatives of the same gender and generation so the term walker is used not only for the mother but also for her sisters and the father s sisters Cousins are termed brothers and sisters This is an ambivalent system which submerges the nuclear family into a larger kinship group Iroquois are a less prevalent system which emphasizes the importance of unilineal descent groups by distinguishing between members of one s own lineage and those belonging to other lineages Sudanese are the system is named after the country where the system was first identified It is the most pluralistic system since it makes the most terminological distinctions Copyright Virtual University of Pakistan 36 Introduction to Cultural Anthropology SOC401 VU Omaha emphasizes patrilineal descent the father and his brothers are referred to by the same term and the paternal cousins are called siblings but cross cousins are referred to by separate terms On the mother s side there is a merging of generations all her male relatives are called mother s brother Crow is the exact opposite of the Omaha system as it emphasizes maternal relations which are all important for determining the descent group of children Kinship Diagrams Cultural Anthropologists often use kinship diagrams to help explain family structures which use simple symbols for males and females and to indicate what their relationships are to each other The diagram below depicts a married couple and their two children a son and a daughter Useful Terms 0 j Descent established linage Ambivalent unclear In uence power Social function the particular purpose served for society A G Prestige Social honor or respect within a society Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 10 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro andor Cbapz er 27 23971 Lil z bropology by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture which provide useful and interesting information Descent Terms and Concepts http referenceallrefercom encyclopedia D descenthtml Copyright Virtual University of Pakistan 37 Introduction to Cultural Anthropology SOC401 VU Lesson 16 THE ROLE OF FAMILY AND MARRIAGE IN CULTURE Family and Marriage The family is a social unit in which its members cooperate economically manage reproduction and child rearing and most often live together Families can be based on lineage and marital ties Marriage the process by which families are formed is a socially approved union between male and female adults Marriage is based on the assumption that it is a permanent contract Yet there is a discrepancy between real and expected behavior within marriages given the high rates of divorce in many countries of the modern world Functions of Families Families reduce competition for spouses They also regulate the division of labor on the basis of gender Families also meet the material educational and emotional needs of children Marriage Restrictions Cultures restrict the choice of marriage patterns by exogamy which means marrying outside a given group E dogamy on the other hand implies marrying within a given group Conservative Hindus are mostly endogamous as are Rwandans in Central African It is important to note that endogamous groupings can be based on lineage or even ethnic or economic similarities Moreover it is possible to simultaneously have an endogamous marriage within an ethic group that is also exogamous outside one s lineage Types of Marriage Monogamy a marriage arrangement that implies having one spouse at one time Polygamy a marriage arrangement that implies a man marrying more than one woman at one time Polyandry a marriage arrangement that implies a woman marrying more than one husband at one time Economic Aspect of Marriage Marriages involve transfer of some type of economic consideration in exchange for rights of union legal rights over children and rights to each other s property There are many cultures in the world which consider marriage as more than a union of man and wife but instead an alliance between two families Types of Marriage Transactions Bridewealth transfer of wealth from a groom s family to that of the bride s approximately 47 Bride service labor in exchange for a wife common in small scale societies lacking material wealth approximately 17 Dowry transfer of wealth from a bride s family to that of the groom s This practice was popular in medieval Europe and may still prevail in several parts of Northern India approximately 3 Copyright Virtual University of Pakistan 38 Introduction to Cultural Anthropology SOC401 VU Woman Exchange two men exchanging sister s as wives This practice is limited to a small number of societies approximately 3 in Africa and the Paci c region Reciprocal Exchange a roughly equal exchange of gifts between bride and groom families Found amongst traditional Native Americans and islands in the Paci c region approximately 6 Note These above statistics are not very recent and should not be taken literally but rather as an indication of the popularity of the above types of transactions Useful Terms Discrepancy difference Reciprocal equal Groom husband Reproduction process of giving birth to children Transaction exchange Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 9 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 20 23971 Lil z bropoogy by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture which provide useful and interesting information Family enwikipediaorg wiki Family Copyright Virtual University of Pakistan 39 Introduction to Cultural Anthropology SOC401 VU Lesson 1quot ROLE OF FAMILY AND MARRIAGE IN CULTURE continued Residence Patterns Residence patterns are in uenced by kinship systems For eg patrilocal residence is common in patrilineal cultures Residence patterns can be disrupted due to events such as droughts famines wars or even due to economic hardship The most common types of residence patterns evidenced around the world are Patrilocal the couple can live with or near the relatives of the husband s father most prevalent Matrilocal the couple can live with or near the relatives of the wife s father Avunculocal the couple can live with or near the husband s mother s brother Ambilocal or bilocal the couple can live with or near the relatives of either the wife or the husband Neolocal Where economic circumstances permit the couple can also establish a completely new residence of their own Residence patterns are not static The Great Depression in America during the 19305 for example compelled neolocal residents to shift back to living with one of their parents again due to economic reasons Similar circumstances keep recurring in different societies of the world and result in changing residence patterns In many traditional societies joint family systems are also very common The dynamics of a joint family system differ from widely from living independently implying a shared responsibility for household responsibilities often under the charge of the oldest member of the household Family Structures Cultural Anthropologists distinguish between two types of family structures the nuclear family and the extended family Nuclear families are based on marital ties whereas the extended family is a much larger social unit based on blood ties among three or more generations Nuclear Family A two generation family formed around the marital union While a part of bigger family structures nuclear families remain autonomous and independent Nuclear families are often found in societies with greatest amount of geographic mobility Nuclear family patterns were encouraged by industrialization and technology but also have remained evident in foraging societies Where resources are scare it makes sense for people to remain in nuclear families whereby retaining a certain level of mobility independence Nuclear families are therefore called the basic food collecting unit in addition to being the most dominant mode of family life in many modern day families around the world Extended Families Blood ties are more important than ties of marriage which form the basis of extended families Extended families can be matrilineal or patrilineal The Anthropological Atlas of 1967 noted 46 out of 862 societies Copyright Virtual University of Pakistan 40 Introduction to Cultural Anthropology SOC401 VU as having some form of extended family organization These numbers have no doubt increased over the past few decades given the increasing world population ModernDay Families Modernization and urbanization have seen progressive movement towards nuclear family structures In developing countries this correlation is not necessary The lack of employment security makes extended families serve as social safety nets Migrant families also hold onto traditional family structures even after having gone to live abroad In western societies even nuclear families are not so common given high divorce and separation rates Useful Terms Prevalent commonly found in different places Migrant refugee Correlation association between two entities Scarce in short supply Evident obvious Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 9 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 20 23971 Lil z bropoogy by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Family enwikipediaorg wiki Family Copyright Virtual University of Pakistan 41 Introduction to Cultural Anthropology SOC401 VU Lesson 18 GENDER AND CULTURE Meaning of Gender Gender refers to the way members of the two sexes are perceived evaluated and expected to behave It is not possible to determine the extent to which culture or biology determines differences in behaviors or attitudes between males or females Although biology sets broad limits on gender de nitions there is a wide range of ideas about what it means to be feminine or masculine Margaret Mead demonstrated this gender based variation in her classical study of sex and temperament in New Guinea Gender Roles In some cultures gender roles are rigidly defined in other cultures they can overlap In general terms however there is considerable uniformity in gender roles found throughout the world Men engage in warfare clear land hunt and trap animals build houses fish and work with hard substances Women on the other hand tend crops prepare food collect firewood clean house launder clothes and carry water tasks compatible with child rearing Yet there are many exceptions to the rule For example in parts of Eastern Africa and in other parts of the developing world women carry enormous amounts of firewood on their backs For the foraging Agta of the Philippines hunting is not an exclusively male activity Status of Women The status of women is multidimensional involving such aspects as the division of labor the value placed on women s contributions economic autonomy social and political power legal rights levels of deference and the extent to which women control the everyday events of their lives The status of women varies around the world but it is unfortunate that in most cases it continues to remain below that of men Gender Stratification Gender stratification contrasts the status assigned by different cultures on the basis of gender It is important to release that status is itself a multidimensional notion involving issues of economic social and political empowerment Stratification on the basis of gender is a common phenomenon The relationships between men and women vary in both degree and in extent across different cultures of the world Many cultures in Asia for example are very stratified along gender lines On the other hand foraging societies like the Mbuti Pygmies of Central Africa possess a very egalitarian gender approach all their elders are called tata Gender stratification need not be static However in most critical areas women tend to be subordinate to men in most societies of the world It is difficult to measure the comparative status of men and women in different societies since there are various components of stratification which can vary independently of each other Useful Terms Status social ranking Copyright Virtual University of Pakistan 42 Introduction to Cultural Anthropology SOC401 VU Acquisition gaining Interaction communication Abundance profusion or great quantity Multidimensional many sided Stratification hierarchical division Irrational Without logic Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 11 in Cuz um A lbropology A Applied Pmpecz z39ve 7y Fermer andor Cbapz er 79 23971 Lil z bropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture Which provide useful and interesting information Gender and Cultural Anthropology htt vlibanthrotechcom Cultural Anthro 010 Gender Feminism Copyright Virtual University of Pakistan 43 Introduction to Cultural Anthropology SOC401 VU Lesson 19 GENDER ROLES IN CULTURE continued Gender Ideology Gender ideology is used in most societies to justify the universal male dominance Deeply rooted values about the superiority of men the ritual impurity of women and the preeminence of men s work are used to justify subjugation of women However it has been demonstrated in recent years that women do not perceive themselves in the same ways that they are portrayed in largely male gender ideologies Negative Impact of Biased Gender Ideologies In some societies gender ideologies become so extreme that females suffer serious negative consequences such as female infanticide female nutritional deprivation honor killings and domestic violence These atrocities are due to the negative impact of gender ideologies as well as due to the disempowerment of females which is another simultaneous consequence of these ideologies Women Employment Although the words breadwinner and housewife accurately described the middle class western household around the beginning of the twentieth century the separate gender roles implied by these two terms have become more myth than reality Over the past four decades the number of women in working outside the home has increased dramatically This is true for not only industrialized but also developing countries due to the ongoing phenomenon of globalization which has led more and more women into the workforce Occupational Segregation The economy of most countries is characterized by a high rate of occupational segregation along gender lines Not only are occupations gender segregated but women tend to earn considerably less than men Feminization of Poverty There has been a trend in recent decades toward the feminization of poverty Being disempowered women fall victims to poverty much more easily then men They also have less access to resources with which to fight against poverty Women often are responsible for looking after their children and their poverty results in declining health standards of both women and their children Useful Terms Segregation separation Resources means of production or more generally the nancial means required to do something Ideology an established way of thinking Decade a period of ten years Disempowered without any say or without any authority or power Suggested Readings Copyright Virtual University of Pakistan 44 Introduction to Cultural Anthropology SOC401 VU Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 11 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermer andor Cbapz er 79 23971 Lil z bropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture Which provide useful and interesting information Gender and Cultural Anthropology htt vlibanthrotechcom Cultural Anthro 010 Gender Feminism Copyright Virtual University of Pakistan 45 Introduction to Cultural Anthropology SOC401 VU Lesson 20 STRATIFICATION AND CULTURE Stratification and its Manifestations Individuals in different cultures and societies have varying amounts of access to wealth power and prestige This evident inequality leads to stratification whereby groups or categories of people are ranked hierarchically relative to one another Social Ranking Social ranking is an important feature found to one degree or another in all societies The degree to which societies rank individuals however varies and results in varying amounts of inequality to be found in the world Dimensions of Inequality According to Max Weber stratification takes place on the basis of three reasons People are distinguished from each other on the basis of wealth or economic resources they posses Secondly stratification takes place on the basis of differing levels of power Power is the ability to achieve one s goals and objectives even against the will of others The amount of power often correlates to amount of wealth individuals possess Types of Societies Stratified societies which are associated with the rise of Civilization range from open Class societies which permit high social mobility to more rigid caste societies which allow for little or no social mobility Class societies are associated with achieved status the positions that the individual can Choose or at least have some control over Caste societies on the other hand are based on ascribed statuses into which one is born and cannot change The United States is often cited as a prime example of a class society with maximum mobility Although its national credo includes a belief in the possibility of going from rags to riches most people in the United States remain in the class into which they are born because social environment has an appreciable effect on a person s life Chances The mobility in less developed countries is even more restricted Hindu India is often Cited as the most extreme form of caste society found in the world Social boundaries among castes are strictly maintained by caste endogamy and strongly held notions of ritual purity and pollution Useful Terms Inequality unevenness Purity cleanliness Pollution environmental degradation or physical corruption deterioration Copyright Virtual University of Pakistan 46 Introduction to Cultural Anthropology SOC401 VU Social mobility ability to change one s status Ritual a social routine Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 13 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermer andor Cbapz er 78 23971 Lil z bropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture Which provide useful and interesting information Stratification WWWsocicanterburyacnZ resources glossary socialstshtml Copyright Virtual University of Pakistan 47 Introduction to Cultural Anthropology SOC401 VU Lesson 21 THEORIES OF STRATIFICATION continued Prominent Theories of Stratification Theories of stratification try to explain the existing inequality of wealth in and between different cultures The Functional Theory and the Con ict Theory provide two con icting interpretations of social stratification evident around the world today The Functionalists Functionalists adopt a conservative position and maintain that social inequality exists because it is necessary for the functioning of society Functionalists emphasize the integrative nature of stratification which results in stability and social order They point out that class systems contribute to the overall well being of a society and encourage constructive endeavor Functionalists argue that differential awards are necessary if societies are to recruit the best trained and most highly skilled people for highly valued positions They maintain that highly skilled people need to be given greater rewards to act as an inventive for them to acquire the required skills For example a brain surgeon needs to spend enormous amounts of time and energy to develop his skills and help society and society must in turn reward him more than it does other people who do not have to make a similar investment in obtaining a skill Functionalists cannot account for non functional success of pop icons for example Famous personalities are often given enormous amounts of money to make public appearances due to their popularity rather than their exceptional amount of skill Functionalists ignore the barriers to participation of certain segments of society Con ict Theorists Con ict theorists assume that the natural tendency of all societies is toward change and con ict Con ict theorists believe that stratification exists because the upper classes strive to maintain their superior position at the expense of the lower classes Con ict theorists do not view stratification systems as enviable or desirable Lack of social mobility leads to exploitation crime revitalization reform and even to revolution Con ict theory is in uenced by the wirings of Karl Marx Functionalists versus Con ict Theorists Integrative aspects of stratification are beneficial for society but the exploitation of under classes does cause tensions and con ict Neither theory can alone explain the existing use and dysfunctional aspects of stratification Useful Terms Revitalization recuperation or revival Dysfunctional no longer able to function or have utility in the given circumstances Exploitation taking advantage of someone else sue to their inability to safeguard their own interests Differential Awards different remunerations or rewards Social Inequality a state of being where certain segments of society are more well off than others Suggested Readings Copyright Virtual University of Pakistan 48 Introduction to Cultural Anthropology SOC401 VU Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 13 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermer andor Cbapz er 78 23971 Lil z bropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Stratification WWWsocicanterburyacnZ resources glossary socialstshtml Copyright Virtual University of Pakistan 49 Introduction to Cultural Anthropology SOC401 VU Lesson 22 CULTURE AND CHANGE Cultural Change Although the rate of change varies from culture to culture no cultures remain unchanged Small scale cultures that are less reliant on technology are seen to change more slowly than industrialized cultures and societies However nothing is as constant as change There is no culture or society which can safeguard itself from the processes of change How Cultures Change The two principal ways that cultures change are internally through the processes of invention and innovation and externally through the process of diffusion It is generally recognized that the majority of cultural features things ideas and behavior patterns found in any society got there by diffusion rather than invention Inventions Inventions can be either deliberate or unintentional Although intentional inventors usually receive the most recognition and praise over the long run unintentional inventors have probably had the greatest impact on cultural change Consider for example the common phrase necessity is the mother of all invention which implies that often circumstances are a more compelling factor inducing innovations in society than the declared intention to make something new Because they are not bound by conventional standards many inventors and innovators tend to be marginal people living on the fringes of society Anthropologists examine the backgrounds and psychological factors that in uence innovative personalities Some of them maintain that inventors are often amongst the well off segments of society yet there are other anthropologists who present other arguments concerning innovators Diffusion The following generalizations can be made about the process of diffusion Cultural diffusion is selective in nature selectivigl not all things diffuse from one culture to another at the same rate Diffusion is a two way process 766707on1 both cultures change as a result of diffusion Cultural elements are likely to involve changes in form or function modz mz z39o a diffused cultural item will not remain exactly the same as it is to be found in its original culture Consider for example the case of Chinese food or pizza which are modified according to the taste of different countries The idea of chicken flk d topping is an example of cultural modification Cultural items involving material aspects are more likely candidates for diffusion than those involving non material aspects Diffusion is affected by a number of important variables duration and intensity of contact degree of cultural integration and similarities between donor and recipient cultures Useful Terms Variables values which are subject to change Cultural items these include both material and non material items ranging from clothing to ideas Donor a country or even an individual entity which is at the giving end of a relationship Copyright Virtual University of Pakistan 50 Introduction to Cultural Anthropology SOC401 VU Recipient a country or even an individual entity which is at the receiving taking end of a relationship Conventional standard or acceptable Intentional bez39 g motivated by an intention Intentional innovators for example clearly state that they are trying to deal With a particular problem and Will attempt to identify a solution for it Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 16 in Cuz um A lbrepelegy A Applied Penpeez z39be by Fermrre meder Cbapz er 73 23971 Lil z brepelegy by Ember emd Peegrzbe Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Culture Change An Introduction to the Processes and Consequences of Culture Change htt anthro alomaredu chan e defaulthtm Copyright Virtual University of Pakistan 51 Introduction to Cultural Anthropology SOC401 VU Lesson 23 CULTURE AND CHANGE continued Acculturation Acculturation is a specialized form of cultural diffusion that is a result of sustained contact between two cultures one of which is subordinate to another Whereas diffusion involves a single or complex of traits acculturation involves widespread cultural reorganization over a shorter period of time There are events in history like colonization which have caused acculturation to occur in many parts of the world Some anthropologists have described situations of acculturation in which the non dominant culture has voluntarily chosen the changes Other anthropologists claim that acculturation always involves some measure of coercion and force Cultural Interrelations Because the parts of a culture are interrelated a change in one part of a culture is likely to bring about changes in other parts of the given culture This is the reason why people are often reluctant to accept change since its consequences cannot be exactly predicted nor controlled This insight of cultural anthropology should be kept in mind by applied anthropologists who are involved in planned programs of cultural change Reaction to Change In every culture there are two sets of opposing forces those interested in preserving the status quo and others desiring change The desire for prestige economic gain and more efficient ways of solving a problem are reasons why people embrace change but the threat of loss of these can lead other people to oppose change as well Barriers to Cultural Change Some societies can maintain their cultural boundaries through the exclusive use of language food and clothing Some societies can resist change in their culture because the proposed change is not compatible with their existing value systems Barriers to Cultural Change Societies resist change because it disrupts existing social and economic relationships The functional interrelatedness of cultures serves as a conservative force discouraging change Cultural boundaries include relative values customs language and eating tastes Change Agents Change agents including development workers for example facilitate change in modern times Change agents sometimes fail to understand why some people are resistant to change and should realize cultural relativity and barriers to change Useful Terms Facilitate to make easier or to promote Functional useful or practical aspects Cultural relativity the realization that cultural traits fit in logically within their own cultural environments and that since circumstances around the world differ cultures are also different Copyright Virtual University of Pakistan 52 Introduction to Cultural Anthropology SOC401 VU Status Quo The existing conditions or circumstances There are always those who are interested in maintaining the status quo since they are doing well due to it and others who oppose the status quo since it tends to exploit them or puts them in a disadvantaged position Coercion An act of force rather than that based on the need or desire of a particular individual or society Interrelations interconnections Subordinate in an inferior or subordinated position Dominant in a position of power over others Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 16 in Cuz um A lbropology A Applied Pmpecz z39be by Fermer andor Cbapz er 73 23971 Lil z bropology by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Culture Change1 htt anthro alomaredu chan e defaulthtm Copyright Virtual University of Pakistan 53 Introduction to Cultural Anthropology SOC401 VU Lesson 24 CULTURE AND CHANGE continued The Complex Process of Change Accepting change in one part of a culture is likely to bring about changes to other parts of a culture To understand socio cultural aspects of urbanization it is important to view the rural area the urban areas and the people who move between them as parts of a complex system of change Until some decades ago anthropologists made differentiations between the mechanical solidarity of rural areas and the organic solidarity of cities Recent research notes that there is not a simple flow of migrants from rural areas to urban areas but rather a circulation of people between these areas Urbanization or the process of rural development therefore needs to take into account the fact that there is a constant criss crossing of people ideas and resources from urban to rural areas Rural migrants rely on kinsmen for land purchase dispute resolution or general household management while they go to the cities in search for cash based employment Conversely rural kinsmen may in turn obtain economic support from a urban wage earner or seek his support in finding work or a place to stay in the city for other kinsmen Planned Change Planned programs of change have been introduced into developing countries for decades under the assumption that they benefit the local people Yet a number of studies have shown that although some segments of the local population may benefit many more do not Globalization Globalization is a broad based term which is used to describe the intensification of the flow of money goods and information across the world which is seen to be taking place since the 19805 Globalization has made the study of culture change more complex due to its varied effects on various cultural processes including that of change In some cases globalization is responsible for an accelerated pace of change in world cultures In other situations the forces of globalization may stimulate traditional cultures to redefine themselves Developing countries in the attempt to better deal with the forces of globalization such as trade liberalization have begun to revamp their own economic systems in order to make them more competitive internationally This economic revamping has tremendous cultural impacts as well Globalization has resulted in diffusion of technology but also compounded existing inequalities There are human and environmental costs associated with globalization For example increased productivity has led to pollution and there are many theorists who argue that globalization has also increased the gap between the rich and the poor with those with wealth doing even better and those without it experiencing even worse poverty than before Useful Terms Globalization intensification of the flow of money goods and information across the world Urbanization the process of people moving from rural areas into the cities This phenomenon is taking place in both developed and developing countries and cultural anthropologists are very interested in studying why and how urbanization takes place and the cultural changes it brings Revamping reforming or changing Competitive the process of trying to do better than those engaging in the same activity Copyright Virtual University of Pakistan 54 Introduction to Cultural Anthropology SOC401 VU Environmental Costs the impact of a particular activity on land water or air and on various other species which inhabit the Earth alongside human beings Impacts results or effects Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 16 in Cuz um A lbropology A Applied Pmpm ive 7y Fermer andor Cbapz er 73 23971 24 z brop00gy 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Culture Change2 htt anthro alomaredu chan e defaulthtm Copyright Virtual University of Pakistan 55 Introduction to Cultural Anthropology SOC401 VU Lesson 25 POLITICAL ORGANIZATION Need for Political Organization All societies have political systems that function to manage public affairs maintain social order and resolve con ict Yet the forms of these political systems are diverse sometimes embedded in other social structures Studying Political Organization Political organization involves issues like allocation of political roles levels of political integration concentrations of power and authority mechanisms of social control and resolving con icts Anthropologists recognize four types of political organization based on levels of political integration concentration specialization Political organization is found within bands tribes Chiefdoms and states Nowadays non state forms of political organization have state systems superimposed on them Types of Political Systems Societies based on bands have the least amount of political integration and role specialization Kung in Kalahari Bands Bands are most often found in foraging societies and are associated with low population densities distribution systems based on reciprocity and egalitarian social relations Tribal Organizations Tribal organizations are most commonly found among horticulturists and pastoralists Neur in Sudan With larger and more sedentary populations than are found in band societies tribal organizations do also lack centralized political leadership and are egalitarian Tribally based societies have certain pan tribal mechanisms that integrate clan members to face external threats Clan elders do not hold formal political offices but usually manage affairs of their clans settling disputes representing clan in negotiation with other clans etc Chiefdoms Chiefdoms involve a more formal and permanent political structure than is found in tribal societies Political authority in Chiefdoms rests with individuals who acts alone or with advice of a council Most chiefdom tends to have quite distinct social ranks rely on feasting and tribute as a major way of distributing goods In the late nineteenth and twentieth century many societies had Chiefdoms imposed on them by colonial powers for administrative convenience for eg British impositions in Nigeria Kenya and Australia The pre colonial Hawaiian political system of the 18th century was a typical chiefdom Useful Terms Public Affairs issues concerning the public at large instead of specific individuals only Social Order the state of being where society functions as per the expectations of people and can provide them with a sense of security Sedentary settled Copyright Virtual University of Pakistan 56 Introduction to Cultural Anthropology SOC401 VU Colonial powers at different phases of history different nations have been powerful enough to colonize other nations In the 19th century Britain was a colonial power which was able to colonize many other countries located on the African and the Asian continents Precolonial the period in history when a particular nation had not yet been colonized Allocation distribution Integration tied together or linked in a particular manner Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 12 in Cuz um A lbmpoogy A Applied Pmpecz z39ve 7y Fermer andor Cbapz er 23 23971 Lil z bropoogy 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Political Organization3 htt anthro alomaredu olitical defaulthtm Please use hyperlinks on the website to read the introductory materials and the information provided on bands tribes and chiefdoms Copyright Virtual University of Pakistan 57 Introduction to Cultural Anthropology SOC401 VU Lesson 26 POLITICAL ORGANIZATION continued State Systems State systems have the greatest amount of political integration specialized political roles and maintain authority on the basis of an ideology States are associated with intensive agriculture market economies urbanization and complex forms of social stratification States began to be formed 5500 years ago with the Greek city states and the Roman Empire providing impressive examples of state based political organization States have a monopoly on the use of force and can make and enforce laws collect taxes and recruit labor for military service and public works which differentiates them from other forms of political organization States are now the most prominent form of political organization found around the world today NationStates A nation is a group pf people sharing a common symbolic identity culture history and religion A state is a distinct political structure like bands tribes and chiefdoms Nation state refers to a group of people sharing a common cultural background and unified by a political structure that they consider to be legitimate Few of the world s 200 nation states have homogenous populations to fit the description of a nation state Political Organization Theories Theories explaining the rise of state systems of government have centered on the question of why people surrender some of their autonomy to the power and authority of the state There are theorists who argue that political organization is influenced by self interest and other theorists argue that self interest is not enough to give shape to political systems and that such organization often involves a certain amount of coercion Voluntaristic State Formation Some theorists suggest that those engaging in specialized labor voluntarily gave up their autonomy in exchange for perceived benefits Political integration can mediate between and protect interests of varied groups and provide them an economic superstructure required for specialization Chide 1936 Hydraulic Theory of State Formation Small scale farmers in arid or semi arid areas also voluntarily merged into larger political entities due to the economic advantage of large scale irrigation Karl Wittfogel 1957 Coercive Theory of State Formation Another explanation for state based political organization is that offered by Carneiro hold that states developed as a result of warfare and coercion rather than due to voluntary self interest Useful Terms Coercion use of force Copyright Virtual University of Pakistan 58 Introduction to Cultural Anthropology SOC401 VU Arid dry Smallscale farmers farmers possessing a little amount of land Irrigation the channeling of water from its natural route for the purposes of agriculture Monopoly dominating the production of a particular product Hydraulic water based Homogenous identical to others opposite of heterogeneous Recruit to include or to involve Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 12 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermer andor Cbapz er 23 23971 Lil z bropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Political Organization4 htt anthro alomaredu olitical defaulthtm 4 Please follow the hyperlink on the website to read about state systems Copyright Virtual University of Pakistan 59 Introduction to Cultural Anthropology SOC401 VU Lesson 2quot POLITICAL ORGANIZATION continued Need for Social Control All forms of political organization must provide means for social control Every culture has defined what are considered to be normal proper or expected ways of behaving in society These expected ways of behaving are referred to as social norms Social norms range from etiquette to laws and imply different forms of enforcement and sanctions Breaking some social norms does not result in serious consequences whereas others can result in severe punishment Consider for example the consequence of taking another person s life or of stealing something Social Norms All social norms are sanctioned to varying degrees according to the values held by different cultures Positive social norms reward people for behaving in socially expectable ways ranging from praise or social approval to awards or medals Negative social norms punish people for violating the norms ranging from disapproval to corporal punishment Maintaining Social Control Band and tribal societies Inuit and Kung maintain social control by means of informal mechanisms such as socialization public opinion lineage obligations age organizations and sanctions Societies control behavior by more formal mechanisms such as through laws and law enforcement agencies whose major function is maintaining social order and resolving conflicts Social Control Band and tribal societies Inuit and Kung maintain social control by means of informal mechanisms such as socialization public opinion lineage obligations age organizations and sanctions Societies control behavior by more formal mechanisms whose major function is maintaining social order and resolving con icts Informal Mechanisms Socialization ensures that people are taught what their social norms are Public opinion or social pressure often serves as an effective mechanism to avoid censure and rejection Age organization provides distinct age categories with defined sets of social roles Formal Mechanisms Song Duets amongst the Inuits to settle disputes Social Intermediaries like the Leopard skin Chief of the Neur in southern Sudan settles murder disputes by property settlements Moots are formal airings of disputes involving kinsmen and friends of litigants and the adjudicating bodies are ad hoc Courts and Codified Laws forbid individual use of force and provides legal frameworks established by legislative bodies interpreted by judicial bodies and implemented by administrative systems like law enforcement agencies Copyright Virtual University of Pakistan 60 Introduction to Cultural Anthropology SOC401 VU Useful Terms Administrative systems the system of government officials bureaucrats who are responsible for running public affairs Judicial systems the system of courts which interprets the laws Legislative systems systems which provide the laws for a particular society often legislatures or legislative assemblies are elected by the people of a particular locality ie province or a state Law enforcement agencies agencies which enforce the law like the police fro example Litigants aggrieved parties involved in a legal dispute Ad Hoc arbitrary not following any established procedure Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 12 in Cuz um A lbropology A Applied Pmpecz z39be by Fermer andor Cbapz er 23 23971 Lil z bropology by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Social Control http enwikipediaorg wiki Social control Copyright Virtual University of Pakistan 61 Introduction to Cultural Anthropology SOC401 VU Lesson 28 PSYCHOLOGY AND CULTURE Psychological Development Anthropologists are interested in the psychological differences and similarities between societies and cultures of the world Cultural Anthropologists reject stereotypes based on hasty ethnocentric judgments Anthropological Queries in Psychology The major questions of relevance to cultural anthropologists attempting to understand the linkage between different cultures and what they can reveal about the human personality are 0 Do all human beings develop psychologically in the same way 0 What explains the psychological differences in personality characteristics from one society to another 0 How do people in different societies conceive of personality and psychological development 0 What types of cultural variations may be explained due to cultural factors Emotional Development Early research in anthropology was concerned mainly with supposedly universal stages of emotional development which seems to be affected by cultural differences Magma Mead found Samoan girls were much less rebellious or emotional turmoil than those in western societies In western societies adolescence is a time of turmoil that helps prepare emotionally for independence Psychological Universals The ability to make binary contrasts order phenomenon plan for the future and have an understanding of the world are universal psychological traits All people have a concept of the self they can empathize with others and feel and recognize emotions in others Cognition and Culture Recent research on psychological universals focuses on cognitive or intellectual development For example it considers how different cultures acquire thinking habits such as formal operation notions which enable a person to think of the possible outcomes of a hypothetical situation In looking for universals many researchers have discovered some apparent differences Yet most tests used in anthropological research favors thinking patterns taught in formal schools in Western cultures CrossCultural Variations Instead of focusing on uniqueness anthropologists look at psychological differences found within and between different cultures Researchers focus on child rearing practices to account for observable personality differences Some anthropologists believe that child rearing practices are adaptive and societies produce personalities according to their requirements obedience self reliance etc Copyright Virtual University of Pakistan 62 Introduction to Cultural Anthropology SOC401 VU Useful Terms Universal common in all cultures Formal schools schools organized by the public or private sector but with a standardized curriculum and professional teaching staff Variations differences Rearing bringing up Intellectual concerning the intellect and the process of thinking Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 24 in Am bmpoogy 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Social Psychology5 htt wwwtrini edu Nmkearl soc s html 5 Please visit the hyperlinks on the website to read selectively on topics like the nature versus nurture debate Copyright Virtual University of Pakistan 63 Introduction to Cultural Anthropology SOC401 VU Lesson 29 PSYCHOLOGY AND CULTURE continued Socialization Socialization is the term that psychologists and anthropologists use to describe the development of through the in uence of parents and others of patterns of behavior in children that conform to cultural expectations Direct and Indirect Socialization Socialization takes place both directly and indirectly Indirectly the degree to which parents like children the kinds of work children are asked to do and whether children go to school may at least partially in uence how children develop psychologically Origin of Customs Anthropologists not only seek to understand the link between personality traits and customs but also how customs were themselves developed Some anthropologists believe that societies produce the kind of customs best suited for undertaking activities necessary for the survival of society Personality Types Several anthropologists have tried to describe the in uence of culture on personality In the early 195039s for example David Riesman proposed that there are three common types of personalities around the world I The z mdz39z z39o orz39em edperrom u places a strong emphasis on doing things the same way that they have always been done Individuals with this sort of personality are less likely to try new things and to seek new experiences II Those who have z39mer dz39recz edperm7145mm are guilt oriented That is to say their behavior is strongly controlled by their conscience As a result there is little need for police to make sure that they obey the law These individuals monitor themselves If they break the law they are likely to turn themselves in for punishment III In contrast people with ber directedperrom z ier have ambiguous feelings about right and wrong When they deviate from a societal norm they usually don39t feel guilty However if they are caught in the act or exposed publicly they are likely to feel shame Abnormal Behavior Just as there are cross cultural variations in normal behavior there are also variations in abnormal behavior Abnormality is relative to a degree and a culture s ideas about mental illness and how to deal with it can also vary Applied Perspective Anthropologists are interested in understanding the possible cause of psychological differences and the possible consequences of psychological variation Anthropologists are particularly interested in how psychological characteristics may help explain statistical associations between various aspects of culture Projective Testing People tend to project their feelings ideas and concerns onto ambiguous realities Copyright Virtual University of Pakistan 64 Introduction to Cultural Anthropology SOC401 VU In Thematic Appreciation Tests subjects are shown vague drawings and asked to interpret them by projecting their own personalities An aggressive person may see a weapon in a vague drawing whereas a more industrious person may visualize a more productive tool in the same vague drawing Useful Terms Ambiguous unclear or vague Variation differences Socialization the process of learning behavior Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 24 in Am bmpoogy 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Social Psychology6 htt wwwtrini edu mkearl soc s html 6Please visit the hyperlinks on the website to read selectively on topics like collective behavior Copyright Virtual University of Pakistan 65 Introduction to Cultural Anthropology SOC401 VU Lesson 30 IDEOLOGY AND CULTURE Ideology An ideology is a collection of ideas An ideology can be thought of as a comprehensive vision as a way of looking at things Ideology can also be seen as a set of ideas proposed by the dominant class of a society to all members of this society For example different types of gender ideologies would describe what roles are expected of women and men in a society The ideology of economic liberalization could be seen to particularly promote the interests of the business classes Ideology in E vetyda y Life Every sociegl has an ideology that forms the basis of the public opinion or common sense a basis that usually remains invisible to most people within the society This dominant ideology appears as neutral while all others that differ from the norm are often seen as radical no matter what the actual circumstances may be In uencing Ideology Organizations that strive for power in uence the ideology of a society to provide a favorable environment for them Political organizations governments included and other groups eg lobbyists try to in uence people by broadcasting their opinions which is the reason why so often many people in a society seem to think alike A certain ethic usually forms the basis of an ideology Ideology studied as ideology rather than examples of specific ideologies has been carried out under the name systematic ideology There are many different kinds of ideology political social epistemological ethical The popularity of an ideology is in part due to the in uence of moral entrepreneurs who sometimes act in their own interests A political ideology is the body of ideals principles doctrine myth or symbols of a social movement institution class or large group that references some political and cultural plan It can be a construct of political thought often defining political parties and their policy Hegemony When most people in a society think alike about certain matters or even forget that there are alternatives to the current state of affairs we arrive at the concept of Hegemony about which the philosopher Antonio Gramsci wrote The much smaller scale concept of groupthink also owes something to his work The ideologies of the dominant class of a society are proposed to all members of that society in order to make the ruling class39 interests appear to be the interests of all and thereby achieve hegemony To reach this goal ideology makes use of a special type of discourse the lacunar discourse A number of propositions which are never untrue suggest a number of other propositions which are In this way the essence of the lacunar discourse is what is oz told but is suggested Epistemological ideologies Even when the challenging of existing beliefs is encouraged as in science the dominant paradigm or mindset can prevent certain challenges theories or experiments from being advanced Copyright Virtual University of Pakistan 66 Introduction to Cultural Anthropology SOC401 VU The philosophy of science mostly concerns itself with reducing the impact of these prior ideologies so that science can proceed with its primary task which is according to science to create knowledge There are critics who view science as an ideology itself called scientism Some scientists respond that while the scientific method is itself an ideology as it is a collection of ideas there is nothing particularly wrong or bad about it Other critics point out that while science itself is not a misleading ideology there are some fields of study within science that are misleading Two examples discussed here are in the fields of ecology and economics Useful Terms Discourse discussion or dialogue Proposition proposal or plan Paradigm standard pattern or example Doctrine set of guidelines Comprehensive complete all inclusive Ecology concerning the species found in the natural environment Moral entrepreneurs those who make up new morals according to their cultural needs Suggested Readings Please visit the following web site for this lecture which provide useful and interesting information Ideology7 http enwikipediaorg wiki Ideology 7 Please visit the hyperlinks on the website to read more about topics mentioned in the lecture Copyright Virtual University of Pakistan 67 Introduction to Cultural Anthropology SOC401 VU Lesson 31 IDEOLOGY AND CULTURE Continued Political ideologies In social studies a political ideology is a set of ideas and principles that explain how the society should work and offer the blueprint for a certain social order A political ideology largely concerns itself with how to allocate power and to what ends it should be used For example one of the most in uential and well defined political ideologies of the 20th century was communism based on the original formulations of Karl Marx and Friedrich Engels Communism is a term that can refer to one of several things a certain social system an ideology which supports that system or a political movement that wishes to implement that system As a social system communism is a type of egalitarian society with no state no private property and no social classes In communism all property is owned by the communig as a whole and all people enjoy equal social and economic status Perhaps the best known principle of a communist society is quotFrom each according to his ability to each according to his needquot As an ideology communism is synonymous for Marxism and its various derivatives most notably Marxism Leninism Among other things Marxism claims that human society has gone through various stages of development throughout its history and that capitalism is the current stage we are going through The next stage will be socialism and the one after that will be communism Therefore it should be noted that communists do not seek to establish communism right away they seek to establish socialism rst which is to be followed by communism at some point in the future Other examples of ideologies include anarchism capitalism corporate liberalism fascism monarchism nationalism fascism conservativism and social democracy Economic Ideology Karl Marx proposed a mempem mcz me model of society The bane refers to the means of production of society The mpem mcz we is formed on top of the base and comprises that society39s ideology as well as its legal system political system and religions Marx proposed that the base determines the superstructure It is the ruling class that controls the society39s means of production and thus the superstructure of society including its ideology will be determined according to what is in the ruling class39 best interests On the other hand critics of the Marxist approach feel that it attributes too much importance to economic factors in in uencing society This is far from the only theory of economics to be raised to ideology status some notable economically based ideologies include mercantilism Social Darwinism communism laisseZ faire economics and quotfree tradequot There are also current theories of safe trade and fair trade calling for a revision in terms of trade which can be seen as ideologies These ideologies call for a revision of rules based on which trade between developed and developing countries takes place Interaction between Legal and Economic Ideologies Ideologies often interact with and in uence each other in the real world Consider for example the statement 39All are equal before the law39 which is a theory behind current legal systems suggests that all Copyright Virtual University of Pakistan 68 Introduction to Cultural Anthropology SOC401 VU people may be of equal worth or have equal 39opportunities39 This is not true because the concept of private property over the means of production results in some people being able to own more mm9 more than others and their property brings power and in uence the rich can afford better lawyers among other things and this puts in question the principle of equality before the law Useful Terms Fair trade the notion that all countries should be given a fair price for the products they export through international trade Terms of trade the price which products of different countries fetch in international trade Means of production these include land labor capital investments required to produce something Inevitable unavoidable Synonymous another term carrying the same meaning Suggested Readings Please visit the following web site for this lecture which provide useful and interesting information http enwikipediaorg wiki Ideology Copyright Virtual University of Pakistan 69 Introduction to Cultural Anthropology SOC401 VU Lesson 32 ASSOCIATIONS CULTURES AND SOCIETIES What Are Associations Associations are non kin and non territorial groups found amidst all types of societies and cultures around the world Associations possess some kind of formal organizational structure and their members also have common interests and a sense of purpose which binds the varied types of societies together Cultural anthropologists are interested in examining how different cultures give shape to different types of associations and in turn what functions different types of associations perform within particular cultures Variation in Associations Associations can vary from society to society They vary according to whether or not they are voluntary and whether the qualities of members are universally ascribed variably ascribed or achieved Qualifications for Associations Achieved qualities or skills are those acquired through one s own efforts there may be hurdles in acquiring necessary skills but by and large skills have to be learnt through personal effort as they are not biologically transferable Ascribed qualities are determined at birth because of gender or ethnicity or family background A person does not need to make an effort to acquire ascribed qualities nor can effort do much in changing ascribed status since it is largely determined by forces beyond the control of individuals Universally ascribed qualities are found in all societies Gender is an example of an ascribed quality Variably ascribed qualities are unique and thus vary across cultures like ethnicity social class differences etc NonVoluntary Associations In relatively non stratified societies associations tend to be based on universally ascribed characteristics like gender and age An age set is a common form of non voluntary associations evidenced around the world even today Age Sets An age grade includes a category of people who fall into a culturally distinguished age category An age set on the other hand is a group of people of similar age and the same sex who move through some or all of life s stages together Entry into an age set is usually through an initiation ceremony and transitions to new stages are marked by succession rituals In non commercial societies age sets crosscut kinship ties and form strong supplemental bonds Age sets are prominent amongst the Nadi of Kenya for example Young warriors were given spears and shields in the past and told to bring back wealth to the community now they re given pens and paper by their elders and told to go out and do the same The Karimojong are predominantly cattle herders and number 60000 people living in northeastern Uganda who are organized via age and generation sets including 5 age sets covering 25 years Copyright Virtual University of Pakistan 7 0 Introduction to Cultural Anthropology SOC401 VU The retired generation passes on the mantle of authority to the senior generation and the junior generation recruits members until ready to assume authority and thus the society continues to function in a seamless manner Useful Terms Recruit to admit or to actively enlist Supplemental added on so as to help reinforce existing ties Characteristics identifying features Seamless continuous Organizational having features of an organization like defined roles and responsibilities Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 22 in Am bropoogy by Ember Internet Resources Please visit the following web site for this lecture which provide useful and interesting information Voluntary Associations http www fathomcom feature 122550 Copyright Virtual University of Pakistan 71 Introduction to Cultural Anthropology SOC401 VU Lesson 33 ASSOCIATIONS CULTURES AND SOCIETIES continued Regional and Ethnic Associations Regional and ethnic organizations are voluntary associations whose members possess variably ascribed characteristics Both forms of associations are usually found in societies where technological advance is accelerating bringing with it numerous forms of economic and social complexities as well Regional and Ethnic Associations Despite a variety of types regional and ethnic associations commonly emphasize helping members adapt to new conditions particularly if they are migrants Many rural migrants keep members in touch with home area traditions by the help of regional associations These associations promote improved living conditions for members who have recently migrated to urban areas in several countries where urbanization is taking place at a fast pace Examples of Regional Associations Regional associations Jermms help rural migrants adapt to urban life and Lima Peru The J Wd OJ have been seen to actively lobby the government on community issues assist members with enculturation organize fiestas and act as clearing house for flow of information Chinatowns in major cities of the world have associations performing a similar function for Chinese immigrants Ethnic Associations Ethnic Associations are based on ethic ties Such associations are particularly prominent in urban centers of West Africa Even tribal unions are commonly found in Ghana and Nigeria which superimpose the notion of ethnicity with that of tribal ties Rotating Credit Associations Such associations are based on the principle of mutual aid Each group member contributes regularly to a fund which is handed over to one member on a rotation basis Such associations are common in East South and southeastern Asia in western Africa and the West Indies Default is rare in rotating credit associations due to social pressure and the incentive is reasonable since membership ranges from 10 to 30 contributors Since no collateral is needed trustworthiness is considered essential when letting people become members of such groups Multiethnic Associations Associations with a common purpose of economic or socio political empowerment are often multi ethnic Savings and loan associations in New Guinea often link women from different tribal areas Formation of Associations Age sets arise in societies which have frequent warfare breaking out amidst them or it is found amongst groups with varying populations due to which kinship systems are not sufficient for alliance purposes Copyright Virtual University of Pakistan 72 Introduction to Cultural Anthropology SOC401 VU Urbanization and economic compulsions lack of access to credit also form associations due to the need to cooperate out of self interest Useful Terms Collateral the act of pledging an asset in order to qualify from a loan from a lending institution like a bank Empowerment to empower or reinforce the capacity of individuals Multiethnic different ethnic groups coeXisting within the same society Default being unable to pay back a loan Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 22 in Am bropoogy 7y Ember Internet Resources Please visit the following web site for this lecture which provide useful and interesting information http wwwfathomcomfeature122550 Copyright Virtual University of Pakistan 73 Introduction to Cultural Anthropology SOC401 VU Lesson 34 RACE ETHNICITY AND CULTURE Ethnicity Ethnicity refers to selected cultural and sometimes physical characteristics used to classify people into ethnic groups or categories considered to be signi cantly different from others Commonly recognized American ethnic groups include American Indians Latinos Chinese African Americans European Americans etc In some cases ethnicity involves merely a loose group identity with little or no cultural traditions in common This is the case with many Irish and German Americans In contrast some ethnic groups are coherent subcultures with a shared language and body of tradition Newly arrived immigrant groups often fit this pattern Minority versus Ethnic Group It is important not to confuse the term minority with ethnic group Ethnic groups may be either a minority or a majority in a population Whether a group is a minority or a majority also is not an absolute fact but depends on the perspective For instance in some towns along the southern border of the US people of Mexican ancestry are the overwhelming majority population and control most of the important social and political institutions but are still defined by state and national governments as a minority In small homogenous societies such as those of hunters and gatherers and pastoralists there is essentially only one ethnic group and no minorities Ethnic Categorizations For many people ethnic categorization implies a connection between biological inheritance and culture They believe that biological inheritance determines much of cultural identity If this were true for instance African American cultural traits such as quotblack Englishquot would stem from genetic inheritance This is not true The pioneering 19th century English anthropologist E B Taylor was able to demonstrate conclusively that biological race and culture is not the same thing It is clear that any one can be placed into another culture shortly after birth and can be thoroughly encultured to that culture regardless of their skin color body shape and other presumed racial features Race A race is a biological subspecies or variety of a species consisting of a more or less distinct population with anatomical traits that distinguish it clearly from other races This biologist39s definition does not fit the reality of human genetic variation today We are biologically an extremely homogenous species All humans today are 999 genetically identical and most of the variation that does occur is in the difference between males and females and our unique personal traits This homogeneity is very unusual in the animal kingdom Even our closest relatives the chimpanzees have 2 3 times more genetic variation than people Orangutans have 8 10 times more variation It is now clear that our human quotracesquot are cultural creations not biological realities The concept of human biological races is based on the false assumption that anatomical traits such as skin color and specific facial characteristics cluster together in single distinct groups of people They do not There are no clearly distinct quotblackquot quotwhitequot or other races Copyright Virtual University of Pakistan 74 Introduction to Cultural Anthropology SOC401 VU Similarity in Human Adaptations The popularly held view of human races ignores the fact that anatomical traits supposedly identifying a particular race are often found extensively in other populations as well This is due to the fact that similar natural selection factors in different parts of the world often result in the evolution of similar adaptations For instance intense sunlight in tropical latitudes has selected for darker skin color as a protection from intense ultraviolet radiation As a result the dark brown skin color characteristic of sub Saharan Africa is also found among unrelated populations in the Indian subcontinent Australia New Guinea and elsewhere in the Southwest Paci c Safeguarding Against Cultural Biases We must not let our own cultural biases get in the way of understanding the lives of other people Avoiding cultural biases is a very difficult task given the emotionally charged feelings and deep beliefs that we have concerning race and ethnicity However suspending these attitudinal barriers in order to gain a better understanding of the phenomena is well worth the effort Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 13 in Cuz umAm bmpoogy A Applied Pmpm z39ve 7y Ferm o Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Ethnicity and Race An Introduction to the Nature of Social Group Differentiation and Inequality htt anthro alomaredu ethnici defaulthtm Copyright Virtual University of Pakistan 75 Introduction to Cultural Anthropology SOC401 VU Lesson 35 RACE ETHNICITY AND CULTURE continued The Complex Nature of Human Variations The actual patterns of biological variation among humans are extremely complex and constantly changingThey can also be deceptive All of us could be classified into a number of different quotracesquot depending on what genetic traits are emphasized For example if you divide people up on the basis of stature or blood types the geographic groupings are clearly different from those defined on the basis of skin color Using the B blood type for defining races Australian Aborigines could be lumped together with most Native Americans Some Africans would be in the same race as Europeans while others would be categorized with Asians Historically human quotracesquot have been defined on the basis of a small number of superficial anatomical characteristics that can be readily identified at a distance thereby making discrimination easier However focusing on such deceptive distinguishing traits as skin color body shape and hair texture causes us to magnify differences and ignore similarities between people It is also important to remember that these traits are no more accurate in making distinctions between human groups than any other genetically inherited characteristics All such attempts to scientifically divide humanity into biological races have proven fruitless Relevance of Nurture In the final analysis it is clear that people not nature create our identities Ethnicity and supposed quotracialquot groups are largely cultural and historical constructs They are primarily social rather than biological phenomena This does not mean that they do not exist To the contrary quotracesquot are very real in the world today In order to understand them however we must look into culture and social interaction rather than biological evolution Intergroup Relations How ethnic and racial groups relate to each other can be viewed as a continuum ranging from cooperation to outright exploitation and hostility 0 Pluralism Two or more groups living in harmony while retaining their own heritage and identity 0 Assimilation when one racial or ethnic minority is absorbed into the wider society Paci c Islanders assimilation into Hawaiian society provides a good example of assimilation 0 Legal Protection of Minorities While such legislation cannot ensure that minorities have equal rights they provide a measure of security against blatant forms of prejudice and discrimination 0 Population Transfer physical removal of minority to another location The ethnic Tutsi eeing Rwanda to avoid prosecution by the Hutu government is an example of population transfer 0 Longtermed Subjugation Political social and economic suppression evident in political history The example of the black majority s subjugation in South Africa under apartheid is a recent example from history 0 Genocide Mass annihilation of groups of people in Nazi Germany or in Serbia for example Copyright Virtual University of Pakistan 76 Introduction to Cultural Anthropology SOC401 VU Useful Terms Exploitation take undue advantage of another s weakness Subjugation political or social suppression Prosecution to accuse or take legal action against an individual or a group Minority a group with a lesser population in comparison to another groups within the same are Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 13 in Cuz umAm bmpoogy A Applied Pmpecz z39ve 7y Fermrro Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Ethnicity and Race htt anthro alomaredu ethnici defaulthtm Elbm39cz39gl cmd N Mom2km Am bropoogz39m Pmpecz z39ver htt folkuiono eirthe Ethnicit html Copyright Virtual University of Pakistan 77 Introduction to Cultural Anthropology SOC401 VU Lesson 36 CULTURE AND BELIEFS Systems of Beliefs Although all cultures have belief systems the forms these beliefs take vary widely from society to society It is often difficult to de ne belief systems cross culturally because different societies have different ways of expressing faith Anthropological Perspective on Beliefs The anthropological study of belief systems does not attempt to determine which belief systems are right or wrong Cultural anthropologists concentrate on describing various systems of belief how they function and in uence human behavior across cultures Social Function of Religion Belief systems fulfill social needs They can be powerful dynamic forces in society Beliefs provide a basis for common purpose and values that can help maintain social solidarity By reinforcing group norms they help bring about social homogeneity A uniformity of beliefs also helps bind people together to reinforce group identity Beliefs enhance the overall well being of the society by serving as a mechanism of social control and also reduce the stress and frustrations that often lead to social con ict whereby helping intensify group solidarity In most societies beliefs play an important role in social control by defining what is right and wrong behavior If individuals do the right things in life they may earn moral approval If they do the wrong things they may suffer retribution Psychological Function of Beliefs Belief systems perform certain psychological functions by providing emotional comfort by explaining the unexplainable for eg to confront and explain death A belief system also helps a person cope with stress fears and anxieties about the unknown Beliefs lift the burden of decision making from our shoulders because they tell us what is right and wrong which is of tremendous help in times of stress or crisis Even prayers provide psychological comfort and solace Moreover beliefs help ease the stress during life crises such as birth marriage serious illnesses by providing appropriate guidelines and rituals Politics and Beliefs Belief systems have played an important role in global social change through liberation theology whereby believers for social reform and justice for the poor and religious nationalism whereby religious beliefs are merged with government institutions Useful Terms Liberation freedom Rituals standardized way for performing some vital social function Retribution vengeance or payback Merged combined or put together Copyright Virtual University of Pakistan 78 Introduction to Cultural Anthropology SOC401 VU Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 14 in Cuz umAm bmpoogy A Applied Pmpecz z39ve 7y Fermrro cmd Cbapz er 25 23971 Lil z bropology 7y Ember Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Anthropology of Religion Homepage wwwasuaeduantFacultymurphy419419WWWhtm or htt WWWindianaedu Wanthro reli ionhtm Copyright Virtual University of Pakistan 79 Introduction to Cultural Anthropology SOC401 VU Lesson 3quot LOCAL OR INDIGENOUS KNOWLEDGE What is Local Knowledge Local knowledge consists of factual knowledge skills and capabilities possessed by people belonging to a speci c area Given that local knowledge is usually geared to real life practices it can usually only be understood with reference to the situation in which it is to be applied Local knowledge is local to the extent that it is acquired and applied by people with respect to local objectives situations and problems Local knowledge may on the one hand comprise fixed and structured quotknowledgequot which can be defined or on the other hand may by virtue of its combination with the performance of actions involve a more uid process of quotknowingquot Human beings exist in a continuous flux of experiences and practices so local knowledge must include information concerning social management have forms of learning and teaching and decision making routines Local knowledge and its respective knowledge systems are rooted in local or regional cultures the respective social contexts and their economies Therefore it is important to consider these surrounding circumstances when one is considering the content of local knowledge itself Changing Definitions of Local Knowledge Originally quotindigenousquot was equivalent to quotlocalquot or quotfolkquot or when applied to knowledge quotinformal knowledgequot In the 1960s and 70s the word then took on a populist avor of grass roots politics in the sense of quotindigenousquot as opposed to state or quothighquot culture In view of the marginalization and destruction of the eco Zones inhabited by ethnic groups the term quotindigenous knowledge is being used in a context of quotnon westernquot or quotanti westernquot knowledge Local knowledge also refers to knowledge of the minorities contrasted with knowledge at the level of the nation state There are therefore various types of local knowledge Element of Exclusivity in Local Knowledge There are normally various types of public knowledge Some information is shared by all locals other information remains concealed from the majority Some items of knowledge are known only to women or only to men Within a society only a few specialists possess more in depth knowledge extending beyond laypersons knowledge in a particular field for instance specific medical or cropping expertise Using Local Knowledge Use of local knowledge for development should not be restricted to extracting information The availability of local knowledge to multinationals carries the danger of delegating power to authorities which are external to the local communities and therefore restricts establishment of competent leadership and sustainable social structures in local communities Copyright Virtual University of Pakistan 80 Introduction to Cultural Anthropology SOC401 VU There is an ongoing debate on intellectual property rights equal bene t sharing and the role of local knowledge for development Anthropologists investigate not only the behavior and the material products of people but also their thoughts and feelings In all branches of anthropology focus on the Milk view and local knowledge has increased in the last thirty years Many countries have taken political decisions to empower local institutions union councils districts etc based on the idea of giving more power to local authorities which have a closer contact with those at the grassroots level Decentralization should correspond with building local capacities Therefore local knowledge on local natural and social environments of local forest dwellers farmers is often more detailed than that of formal institutions and can be used to assure sustainable development Useful Terms Antiwestern against western values and or economic or political systems mostly instigated by experiences of exploitation Indigenous rooted in a specific locality native Decentralization delegation of authority to lower levels of administration Suggested Readings based on Internet Resources Students are advised to read the following paper available in PDF format from the following web site for this lecture which provides useful and interesting information Local Knowledge and Local Knowing htt wwwuni trierde uni fb4 ethno know df Copyright Virtual University of Pakistan 81 Introduction to Cultural Anthropology SOC401 VU Lesson 38 LOCAL OR INDIGENOUS KNOWLEDGE continued Scientific Knowledge vs Local Knowledge Is local knowledge ultimately equivalent to knowledge gained through science or is it structured entirely differently This is an age old topic of debate in anthropology the debate concerned with rationality and so called allemaz z39ve moder of lbougbz A corresponding practical question is if local knowledge can be utilized within the framework of scienti cally based measures Or is local knowledge a holistic counter model to science to be used to criticize measures founded on analytical science Most characterizations of local knowledge are defined in complete contrast to scientific knowledge But local and scientific knowledge are neither completely different nor entirely the same they display both commonalities and differences Similarities between Local amp Scientific Knowledge Local knowledge and knowledge derived from science are similar primarily in having an empirical and a methodological basis Both local knowledge and science use observations of the outside world which are in principle accessible and communicable While both forms of knowledge use experiments local knowledge proceeds rather from observations gained through trial and error or so called quotnatural experimentsquot ie inferences drawn from the impacts of natural changes in certain quantities Scientific knowledge on the other hand relies on controlled experiments Distinctions between Local amp Scientific Knowledge Scientific knowledge seeks information which is transferable to any spatial or social situation ie which is not context bound As a result scientists know a great deal about small sections of reality In contrast local knowledge systems seek spatially situation bound or context bound information The validity of items of local knowledge is locally restricted ie they cannot be transferred to other local contexts The potential for generalization and thus also mutual learning is in principle limited with local knowledge Owners of local knowledge are often only inadequately aware of market mechanisms Potential for Anthropological Contribution The inter cultural perspective of anthropologists enables them to re ect on and integrate both ways of knowing and for seeing where to draw the line Local knowledge out of its cultural situation loses its frame of reference and without the necessary skills to decipher it becomes meaningless The Need for Caution While local knowledge increases people s empowerment enhances the visibility of their problems is geared to subsistence and risk minimization leading to more sustainable solutions a cautious approach has to be adopted Practices which are based on local knowledge are not per re ecologically sound necessarily socially just or even democratic Neither is local knowledge equivalent to quotpeople s knowledgequot in the sense that it would always be shared by most or even all members of a group Useful Terms Copyright Virtual University of Pakistan 82 Introduction to Cultural Anthropology SOC401 VU Democratic a system based on sentiments of the majority Risk minimization measures taken to decrease given risks associated with a particular activity Subsistence survival Suggested Readings based on Internet Resources Students are advised to read the following paper available in PDF format from the following web site for this lecture which provides useful and interesting information Indigenous knowledge biodiversity conservation and development httpwwwciesinorgdocsOO4 173004 173html Copyright Virtual University of Pakistan 83 Introduction to Cultural Anthropology SOC401 VU Lesson 39 ANTHROPOLOGY AND DEVELOPMENT What is Development In the popular meaning of the term development is a transition towards directed change towards modernization industrialization and capitalization However major development agencies and multilateral organizations often interpret development in terms of poverty Poverty defined in relation to the absence of basic services and in income terms less than one dollar a day becomes a proxy for the absence of development and a justification for intervention Poverty and development are measured by indicators and targets some global others national which become standard devices for undertaking development But even focusing on poverty does not necessarily imply that poor people are more involved in the development planning process Often the poor cannot represent themselves they are represented It has also been noticed by anthropologists that development is often defined in negative terms not so much as the presence of something as the elimination of an unacceptable state like that of poverty Role of Anthropology in Development Anthropological studies focus on the processes of social transformation positive and negative conventionally associated with development Anthropology helps development initiatives realize the context in which their activities are to be introduced The cultural insights and the kinds of understandings that anthropology offers enables social development professional to envision what kinds of impacts particular interventions may have on particular types of social relations and institutions Comparing Development and Anthropology Development approaches and methods have much common with anthropology but there are also substantial differences What constitutes social development knowledge is determined by the need to meet policy priorities rather than the pursuit of knowledge Social development presents itself as a technical discipline using social analysis as a precondition for social transformation Like anthropological methods development is people focused and uses qualitative techniques But unlike anthropological methods requiring extended fieldwork social development methodologies are designed to fit into short timeframes Who Undertakes Development Development Organizations include multilateral agencies like the World Bank and UN agencies bilateral agencies national and international NGOs Typical partner organizations include national governments national NGOs and the lower tier community based organizations In uence of Development Notions The in uence of development extends far beyond the formal institutions charged with implementing development oriented programs Cultural attitudes informed by development aspirations are entwined in popular cultures of developed and developing countries For eg rural communities in Nepal utilize the category of developed bikas as a means of classifying people according to perceived class position and Copyright Virtual University of Pakistan 84 Introduction to Cultural Anthropology SOC401 VU social networks Wealthy individuals in developed countries provide money for communities perceived as poor via child sponsorship schemes for example Useful Terms Social Development the effort to meet basic needs and to assure access to basic human right Entwined joined or merged together Perceived considered or viewed NGOS Non government organizations Suggested Readings based on Internet Resources Students are advised to read the following paper available in PDF format from the following web site for this lecture which provides useful and interesting information Applying Anthropology in and to Development htt wwwlboroacuk de artments ss a licationsofanthro olo reen a erhtm Copyright Virtual University of Pakistan 85 Introduction to Cultural Anthropology SOC401 VU Lesson 40 ANTHROPOLOGY AND DEVELOPMENT Continued Development and Change From an anthropological point of view culture is an asset even though managing it is difficult since cultures change and do not have sharp borders Examples of development planners39 and development workers39 ignorance of local culture have had devastating repercussions on the local level What Development Anthropologists do Development anthropologists in interpret practices which are difficult for others to access who lack detailed comparative knowledge of social organization gender kinship property resources Anthropological input is often restricted to appraisal and analysis of planned outcome failures Besides international development use of applied anthropology has grown in the West as well Anthropology in the US and in South America is often associated with cultural brokerage between indigenous groups and national governments and between indigenous groups and private companies often those associated with natural resource extraction Changing Notion of Development Development necessitates a kind of social analysis of the situations which the proposed intervention will be designed to address From an anthropological view this essentially requires matching two representations of reality that of development practioners and that of local environments Research on development and culture during the past years has emphasized a culture sensitive approach in development Emphasis on people undertaking their own development instead of imposing development on them it is suggested that research into local culture is one of the most important features for ensuring participatory development Participation means that development should involve all its stakeholders Even the World Bank has recognized the compleX local environments in which development policy was supposed to operate and had failed was due to lack of participation A modified policy discourse spoke the need to include local people civil society and social networks in planning and implementation Contentions in Development If anthropology has conventionally been suspicious of unplanned changes it has been particularly distrustful of directed change and of the international development project which has had directed change as its objective The ambivalent relationship between anthropology and development has its origins in the colonial systems of governance British anthropology strove to be useful to practical men of colonial administration in the 1930 s to access public funds In France anthropological methods were used to improve colonial government This history accounts for the suspicion with which anthropology is still viewed in many countries which have a fairly recent history of colonial domination Copyright Virtual University of Pakistan 86 Introduction to Cultural Anthropology SOC401 VU A New Role for Anthropologists The involvement of anthropology in development did not end with the dawning of the post colonial era The inclusion of the discipline in the institutional structures of international development from the late 1970 s on has created a number of anthropological positions within development agencies Induction of anthropologists in development agencies in the 1980 s and 1990 s coincided with a new people oriented discourse in international development and a renewed focus on social exclusion and marginality Useful Terms Contentions controversies or opposing points of view Conventionally standardized way of doing something Natural resource extraction extraction of resources from the natural environment from the land or the sea for productive purposes Post Colonial the time period commencing after the colonization period is over although the in uence of colonizing countries may still remain after they have physically vacated a colony Ambivalent ambiguous or lacking a clear cut def1nition Suggested Readings using Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information The cultural process of development Some impressions of anthropologists working in development htt wwwvalthelsinkif1 kmi ulkais WPt 1998 W898HTM Copyright Virtual University of Pakistan 87 Introduction to Cultural Anthropology SOC401 VU Lesson 41 ANTHROPOLOGY AND DEVELOPMENT Continued Expectations from an Anthropologist Commonly it is expected that an anthropologist can assist development programmes by bringing in the anthropological perspective Anthropologists are expected to address social rather that technical aspects of development programs It is anticipated that an anthropologist should take care of the soft elements of the project This is a diffuse expectation which can imply many tasks The anthropologist can be expected to report on for example the division of labour in an area or why cultivators prefer a special crop In the latter case the anthropologist collaborates with an agronomist on the given project An anthropologist is expected to give answers to certain questions which should lead to action for eg to drill a well it is necessary to form a water group which will contribute labor and or take the responsibility of maintaining the well after it has become operational Anthropologists entered the field of development when development organizations acknowledged that things often did not work out according to expectations because of cultural factors Anthropologists can help in this regard given their understanding of cultural similarities and differences Anthropology s Contribution to Development Anthropologists have highlighted an appreciation of local knowledge and practices Anthropologists argue that indigenous knowledge practices and social institutions must be considered if local resource management and development plans are to work Interaction between so called experts in the modern sector and people representing local specific knowledge can result in the creation of new knowledge and be a starting point for development activities In an anthropological sense culture is integrated in society and social development and is thus heterogeneous dynamic and holistic Anthropologists have shown that people are not an undifferentiated mass A first step of development workers is to get the whole picture of norms and values and maybe their ideals in a specific area The second step is to look for the variations in the heterogeneity of what first looks like a homogeneous mass of people Hierarchies are found everywhere It is of utmost importance to recognize hierarchies in the process of planned change The manner in which certain groups are left outside the decision making process also deserves attention Requirements amp Rewards of Anthropological Input Research into culture and development requires time It involves considering the interaction and interchange of different kind of knowledge and learning between development agents the so called experts and people representing local knowledge all this also requires much effort and resources Much work done by the anthropologist is anticipatory in nature Anthropological experience helps anticipate potential both negative and positive changes A well done cultural analysis of development initiatives also helps to anticipate conflicts which can be addressed before they become serious problems Useful Terms Hierarchies segmented responsibilities accompanied by differences in rewards and prestige Copyright Virtual University of Pakistan 88 Introduction to Cultural Anthropology SOC401 VU Undifferentiated lacking differentiation similar Integrated tied or connected to each other Operational functional or workable Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Addressing livelihoods in Afghanistan http WWWareuorgpk publications livelihoods Addressing20Livelihoods pdf Copyright Virtual University of Pakistan 89 Introduction to Cultural Anthropology SOC401 VU Lesson 42 CULTURAL ANTHROPOLOGY AND ART What is Art Art can be de ned as the process and products of applying certain skills to any activity that transforms matter sound or motion into a form that is deemed aesthetically meaningful to people in a society Yet there is no universal definition of art Art re ects the human urge to express oneself and to take pleasure from aesthetics The creative process of art is enjoyable produces an emotional response and conveys a message Verbal art includes myths and folktales Myths tend to involve supernatural beings whereas folktales are more secular in nature Like other art forms verbal arts are connected to other aspects of a culture Art and Anthropology Art plays a useful social function and is prominent in ceremonies and customs of most cultures The forms of artistic expression of relevance to cultural anthropologists include graphic and plastic arts such as painting carving and weaving music dance and verbal art such as myth and folklore Examples of Art Painting sculpture and ceramics are common forms of western art Religiously inspired art forms are also impressive including architecture Smaller societies also have distinct art forms the Nubian body decorations Eskimo body tattooing and Navajo sand paintings are examples of art Relevance of Art Art contributes to the well being of individuals and society For individuals art provides emotional gratification to the artist and the beholder From the social perspective art strengthens and reinforces social bonds and cultural themes acts as a mechanism of social control and is a symbol of high status particularly in complex societies Differences in Art Forms Major differences in art forms are found between different cultures of the world In small scale societies of foragers pastoralists or shifting cultivators with nomadic or semi nomadic residence patterns the art in these societies either involves performing arts song dance or story telling or is highly portable jewelry tattooing Judging the Quality of Art In modern societies what constitutes good art is largely determined by the professional art establishment experts critics academics In societies lacking professional art establishments artistic standards are less elaborate and more diffuse and democratic relying on public reaction Complex societies with specialization and sophisticated institutions invest in elaborate buildings larger than life canvases kept in museums Copyright Virtual University of Pakistan 90 Introduction to Cultural Anthropology SOC401 VU Art and Politics It is possible to see symbols of political power expressed via art In Polynesia leadership based on centralized chiefdoms results in chiefs using permanent tattoos to reflect their hereditary high status In Melanesia on the other hand power is more uid and the big men indicate their authority using temporary body paints Useful Terms Tattoo form of body art which illustrates onto the skin using permanent ink Canvas the cloth on which paintings are done Diffuse spread out Art establishment art experts critics and academics Museums and other art institutions are also part of this establishment Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 15 in Cuz umAm bropoogy A Applied Pmpecz z39be by Fermrro cmd Cbapz er 26 23971 Lil z bropology by Ember Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Art and Anthropology wwwanthroarcheartorg or wwwartandanthropologycom or wwwaugieedu deptz art Copyright Virtual University of Pakistan 91 Introduction to Cultural Anthropology SOC401 VU Lesson 43 CULTURAL ANTHROPOLOGY AND ART continued Functionalist Perspectives Concerning Art Manilowski tended to emphasize how various cultural elements function for the psychological well being of the individual Radcliff Brown stressed how a cultural functional element of art functions to contribute to the well being or continuity of society Psychological Benefits of Art For the artist artistic impressions enable expression of emotional energy in a concrete and visible manner The creative tension released via artistic expression brings personal gratification Works of art evoke emotional responses from their viewers which can be positive or negative but do help relieve stress Art and Social Integration Art functions to sustain longevity of the society in which it is found Art is connected to other parts of the social system and used to evoke positive feelings for its rulers Even in ancient Aztec and Egyptian civilizations the ziggurats and pyramids served to provide a visual reinforcement of the awesome power of the rulers Art forms like music also help reinforce social bonds and cultural themes Martial music on the other hand helps rally people against a common enemy Story telling also passes on social values from one generation onto the next whereby helping social integration Art and Social Control A popular perception concerning artists is that they are non conformist visionary and aloof Art often reinforces existing socio cultural systems It also instills important cultural values and in uences people to behave in socially appropriate ways Art can buttress inequalities of existing stratification systems In highly stratified societies state governments use art for maintaining the status quo and to solicit obedience and respect Art as a Status Symbol Acquiring art objects provides a convincing way to display one s wealth and power Possessing art objects implies high prestige due to its uniqueness Art in ancient Egypt was the personal property of the pharos Art galleries often exhibit personal collections obtained from high ranking members of society Art as a Form of Protest Art functions as a vehicle for protest resistance and even revolution Various artists have attempted to raise the consciousness of their countrymen through their poems painting and plays and helped instigate socio political changes Useful Terms Consciousness the feelings sentiments and thoughts of a person or of a given people Copyright Virtual University of Pakistan 92 Introduction to Cultural Anthropology SOC401 VU Acquiring obtaining Status quo the existing system Ziggurats ancient places of worship in the South American continent renowned for their architectural design Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 15 in Cuz umAm bropoogy A Applied Pmpecz z39ve 7y Fermrro cmd Cbapz er 26 23971 Lil z bropology 7y Ember Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Art and Anthropology wwwanthroarcheartorg or wwwartandanthropologycom or wwwaugieedu dept art Copyright Virtual University of Pakistan 93 Introduction to Cultural Anthropology SOC401 VU Lesson 44 ETHICS IN ANTHROPOLOGY Ethical Condemnation Since the 1960s cultural anthropology has been the target of critical attacks both from within and without the discipline The condemnation of anthropology and anthropologists by postmodernism literary theory and post colonialism among others has been directed at its status as a science and its participation in the oppression of minorities and justification of colonialism Critics assert that anthropology has been used solely to objectify oppressed peoples and that it cannot be considered a science Anthropologists are blamed for asserting domination over his or her subject due to negative and inaccurate representations formed by the critics Anthropology is charged with ignoring history in studying non Western societies and so anthropologists have been blamed for treating cultures as isolated from neighbors and the world at large Anthropologists can also reinforce biases and stereotypes by using flawed methodology in their works Orientalism By studying the orient the scholar separates him or herself from the culture they study and recreate it as another world Said believes that Asians are confined by the Oriental label that has been constructed by the European scholar It is natural for the human race to divide itself into quotusquot and quotthemquot It is this division that leads to hostility The separation that arises due to scholarly study only strengthens this hostility Response of Anthropologists In order to continue the study of culture anthropology developed the term relativism which stated that all cultures were equal but not necessarily alike Cultural anthropology could not however accept relativism because issues of morality became controversial The study of anthropology became obsessed with data analysis in order to avoid moral judgment Classic anthropologists feared domination of the discipline by psychology and sociology therefore anthropology had to be redefined in order to shift the focus of the discipline back to the study of culture Past research existed only on exotic cultures and the theories developed from that research were used to try to define modern or first world culture Several problems arose from this movement Few people were interested in studies in cities or familiar places the exotic areas broke the rule that all cultures are equal and therefore these areas drew the attention of anthropologists Another problem was that all previous studies were done on societies with no recorded history and therefore no changes in patterns or traditions were observed Defending Anthropological Integrity Leading and influential anthropologists generally believed in uniformity in the actions and nature of humankind not in the idea of self and the Other They wanted to study all forms of culture at home and abroad to discover similarities There are several examples of anthropologists who recognized the importance of borrowing diffusion and regional and global interactions in shaping society Anthropology should base their criticisms on a careful scrutiny of facts Copyright Virtual University of Pakistan 94 Introduction to Cultural Anthropology SOC401 VU Using Criticism Constructively Questions and ideas put forth by anthropology s critics must be used to help avoid misperceptions and poorly founded opinions from passing on as common knowledge to the new generation of anthropologists Reexamination of the prevalent attitudes in anthropology can move away the notion of anthropologists as authoritarian figures to humanistic scientific scholars interested in comparing and contrasting cultures Assuring Anthropological Integrity Objectivity and functional analysis combined with today s knowledge of psychology that is the key to comprehensiveness and objectivity in anthropology Useful Terms Objectivity unbiased observation of facts Authoritarian monopolized exertion of power Prevalent existing or in current use Scrutiny study or careful observation Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 28 in Am bropoogy by Ember Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Ethics in Anthropology htt www ublicanthro 010 or ournals En a in Ideas Rt ano Peterslhtm Copyright Virtual University of Pakistan 95 Introduction to Cultural Anthropology SOC401 VU Lesson 45 RELEVANCE OF CULTURAL ANTHROPOLOGY Change and the Future of Anthropology Change is occurring at such an accelerated pace that it is difficult to keep up with all the changes in the world today The recent revolution in transportation and telecommunications and the resulting increase in communications and travel are diffusing cultures at a much greater rate today than ever before Some argue that cultural anthropology will loose importance in the future since it is only a matter of time when all cultures will be homogenized Yet few cultural anthropologists are studying pristine cultures as the discipline is adapting to the realities of this changing world Concern for survival of indigenous cultures and the study of complex societies is now the new focus area for many cultural anthropologists There is also greater emphasis on using anthropological perspectives to deal with developmental problems There is little evidence to suggest that the world is becoming a cultural melting pot so despite cultural changes there is enough diversity in the world to keep cultural anthropologists occupied for a long time to come Ensuring Cultural Survival Cultural patterns and in some cases people themselves have been eradicated as a direct result of progress and economic development The indigenous population of Tasmania in 19th century by white settlers for sheep herding is a tragic example of cultural extinction The 1884 Berlin Conference was a civilized way of dividing spoils of Africa but not safeguarding rights of indigenous people and numerous conflicts on the African continent are based on this insensitive division and lumping together of different ethnic groups The Brazilian Amazon shelters the largest population of the world s still indigenous people But by building roads through the Amazonian frontier the Brazilian government has introduced diseases such as in uenza and measles amongst the indigenous communities Contemporary Anthropologists Anthropological research has great relevance for the public at large Consider for example the role archaeology played in society during the nineteenth century Books on the subject were widely read Darwin s work for example significantly changed beliefs on human history and development of the modern world Throughout this era of advancements academic archaeology was on the rise This movement finally phased out the participation of amateurs in the field creating a more elitist and inaccessible discipline While professionalization has certainly had numerous benefits including developments in quotmethod theory and culture historical knowledgequot its negative aspects are causing a significant deterioration of popular interest in archaeology A movement towards popularization through accessible writing must take place in order to involve the public and rekindle active interest in archaeology and indeed in other branches of anthropology Accessibility glorifies the field of anthropology rather than denigrates it Nowadays rather than writing holistic ethnographies cultural anthropologists bring to the study of cities and complex societies a more nuanced sensitivity towards understanding and dealing with the issue of ethnic diversity Copyright Virtual University of Pakistan 96 Introduction to Cultural Anthropology SOC401 VU Anthropologists practicing quotaction anthropologyquot collaborate with other disciplines concerning the development of culture and how it relates to current pertinent issues Useful Terms Holistic ethnographies overarching description concerning all aspects of life of a given community Ethnic diversity different ethnic groups or the differences within or between them Pertinent relevant or important Nuanced having various aspects Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 17 in Cultural Anthropology A Applied Pmpecz z39ve by Ferrarro Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Intellectuals and the Responsibilities of Public Life An Interview with Chomsky htt www ublicanthro 010 or ournals En a in Ideas choms htm Copyright Virtual University of Pakistan 97

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