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by: skenan

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# STAT 1051, Balaji, Week 2 STAT 1051

Marketplace > George Washington University > STAT 1051 > STAT 1051 Balaji Week 2
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Week 2 notes with in-class examples.
COURSE
Introduction to Business and Economic Statistics
PROF.
Dr. Srinivasan Balaji
TYPE
Class Notes
PAGES
8
WORDS
CONCEPTS
stat, Statistics, intro to statistics, Balaji
KARMA
25 ?

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This 8 page Class Notes was uploaded by skenan on Sunday September 11, 2016. The Class Notes belongs to STAT 1051 at George Washington University taught by Dr. Srinivasan Balaji in Fall 2016. Since its upload, it has received 68 views.

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Date Created: 09/11/16
CHAPTER 2: DESCRIPTIVE STATISTICS Qualitative Data:  measurements that cannot be measured on a natural numerical scale.  Observations can be classified into one of a group of categories.  Ex; political affiliation (Democrat), race, gender, etc. Summarizing Qualitative data:  Tabular Summary: o Frequency Distribution  o Relative Frequency Distribution  Graphical summary: o Bar Chart o Pie Chart Frequency distribution: is a tabular summary of data showing different classes and the  frequency (or number) of items in each of several non­overlapping classes. It shows how the  observations in the sample are distributed. The objective is to provide insights about the data  that cannot be quickly obtained by looking only at the original data. Relative Frequency: of a class is the proportion of the total number of observations belonging to the class. frequency  RF:  totalnumber of values Bar Chart: A graphical method for depicting qualitative data (from a frequency, relative  frequency distribution.)   On the horizontal axis the levels of the qualitative variable are specified.   Vertical axis represents frequency or relative frequency.   Bar of fixed width are drawn for each class, with height beign proportional to the  frequency.   The bars are separated. Pie Chart: graphical method used for presenting relative frequency distributions for  qualitative data.   A circle represents the entire dataset. It is portioned so that each sector represents a  class. Areas of the sectors represent the relative frequency of class it is representing.   Since there are 360° in a circle, a class with a relative frequency of .25 would  consume (.25)*(360)= 90° of a circle.  Example; using the class below, find the relative frequency; create a bar chart and a pie chart. Average Below Good Good Below  Good Good Averag Poor Good e Excellen Average Good Average Good t Excellen Average Good Good Poor t Class Frequency Relative Frequency Pie Chart   Poor 2 2/20 = 0.1 (0.1)*(360) = 36° Below Ave. 2 2/20 = 0.1 (0.1)*(360) = 36° Average 5 5/20 = 0.25 (0.25)*(360) = 90° Good 9 9/20 = 0.45 (0.45)*(360) = 162° Excellent 2 2/20 = 0.1 (0.1)*(360) = 36° =20 Quantitative Data:  Measurements that can be measured on a natural and meaningful  numerical scale. Ex; SAT score of students, the current unemployment rate for 50 states, number of calls made over last week by 10 cell phone users. Distribution: how the observations are spread over the range of the data.  The distribution could be skewed or symmetric.   Different types of distribution; o Skewed to the right OR positively skewed (Ex; income distribution) o Symmetric and mound­shaped o Negatively skewed or skewed to the left Summarizing Quantitative Data: Graphical/Tabular Methods:   Stem and Leaf Plot  Frequency/Relative Frequency Distribution   Histogram Dot Plot:   6    5.5    9    6.5    8    6    7    6.5    6    7    6.5    8    5    6.5    Stem  Leaf 0 3 Stem and Leaf Plot: every observation is divided into two parts.  2 4  Leaf: the right most digit(s) of an observation. 3 5 6   Stem: remaining part (to the left of leaf.) 8 11 2 Ex; a dataset contains observations{3.5,11.2,8,0.3,2.4,3.} Split Stem Plot: used if the normal stem and leaf plot has condensed the data too much.  Stem Leaf Leaf Unit: 0.1 (the decimal place value) 0 23 Where the leaf unit is not shown, it is assumed  0 689 1 134 Leaves < 5 to equal 1. Leaves ≥ 5 1 578 Frequency Distribution: data is summarized in a tabular form.  Range of the dataset is partitioned into a number of classes of equal width.   Frequency distribution table is constructed by counting number of observations  (called frequency) in each class and presenting the classes and their frequencies.  Guidelines: Guidelines for selecting number of classes:  Use between 5 and 20 classes.  Larger datasets require a larger number of classes.  Smaller data sets usually require fewer classes. Guidelines for deciding class­width:  Use classes of equal width.  Adjacent  Class width ≥  LargestObs.−SmallestObs. Bumer of Classes Histogram: a common graphical presentation of frequency distribution.  The variable is placed on the horizontal axis and the relative) / frequency is placed on  the vertical axis.   A rectangle is drawn above each class interval with its height proportional to the  frequency of that class.   Rectangles are adjacent.   There is NO GAP between bars.  Ex; construct a histogram for the following data of scored of students in a statistics class. 85 76 78 89 92 74 83 85 72 94 99 82 75 78 63 91 86 51 58 68 92 73 75 84 Total # of values: 24   # of class: √24 ≈ 5 Class width ≥  largest−smallest  =  99−51  = 9.6 ¿of class 5 Class Tally Marks Frequency Relative Frequency 50 ≤ x < 60 II 2 2/24 60 ≤ x < 70 II 2 2/24 70 ≤ x < 80 ?  III 8 8/24 80 ≤ x < 90 ?  II 7 7/24 90 ≤ x <  ? 5 5/24 100 = 24 Bars overlap in the histogram. Graphical Summary Using SPSS: SPSS can be used to get the histogram as well. Look at the beanie dataset. Consider a typical dataset. There are four variables. Name, Age, Value, and Status. (Retired or not) We want to construct histogram for the   Age variable   Value variable Less than 25 obs/ 5­6 classes           25­50 obs/ 6­10 classes                    More than 50 obs/ 10­20 classes    Descriptive Statistics: Summation Notation: Observations in a dataset are denoted by {x ,x1,x2,x3,4...xn}; n=sample  size  x1 s the first observation,2  is the second and so on.  We use ∑ x io denote x +1 +2 +3 +.4..+x n   In particular, n  x  x  x ......  x i 1 2 n                                  Example: construct the following s.n. for the dataset below.  7 11 3 4 5 6 13 n=7  ∑ x = 7+11+3+4+5+6+13= 49  ∑(x­2)= (7­2)+(11­2)+(3­2)+(4­2)+(5­2)+(6­2)+(13­2)= 35 2 2 2 2 2 2 2 2  ∑x = 7 +11 +3 +4 +5 +6 +13 = 415  (∑x) = (49) = 2401 ∑ x  = 49/7 =7 n Population: Collection of all the units that we are interested in studying. Sample: a subset of the units of the population.  Numerical Summary: summarizing the data using numerical descriptive measures.  Both population and sample data can be summarized.  Two quantities to measure: 1. Center: measure the central tendency a. Mean: the average of a group of numbers.  i. Population mean: (µ), computes the mean of a population data.  X   ...     X 1 X 2 X 3 X N ii. N N iii. Sample mean: computes the mean of a sample data. (xx) X   ...  X    x 1 x 2 x n iv. n n v. Mean and Distribution: Mean is the point where the histogram is  balanced. For positively skewed distribution extreme observations will  pull it up.   b. Median: median is the middle observation. i. Median partitions the histogram into two equal halves. c. Mode: the most frequent observation. For continuos variables, mode is the  point where the histogram has the peak.  2. Variability: spread of the data Example:  1 3 5 6 8 8 9 11 12 n  xi 13568891112 63 x i1    7 Mean:  n 9 9 Median: 8 Mode: 8 Comparing 3m’s:  For negatively (left) skewed distributions: mean < median < mode               For positively skewed distribution: mean > median > mode               For symmetric (not skewed) distributions: Mean = Median = Mode Measures of Spread: different ways of computing the spread/ variability of a dataset.  1. Range: Maximum­minimum.  a. Two very different datasets could have the same range.  2. Variance: the squared distance between a typical observation and the mean of the  data.  a. Population variance (σ ) : measures the spread in population. 2 2  X    N 2 b. Sample Variance (s ): measures the spread in the sample. xx: sample mean n 2 2  i1xix  ,                             3. Standard deviation (SD): square root of variance. Gives the “average” distance  between a typical observation and the mean of the dataset. a. Popular Standard Deviation:  2   X     N b. Sample Standard Deviation: n 2 s   i1 ix  n1 4. Inter quartile range (IQR) Example: Find the mean, median, and the standard deviation for the following datasets. a) 3 7 11 2 5 4 3 b) 4 ­2 5 8 12 5 7 4 9 8 (a) Ordered data: 2 3 3 4 5 7 11 Mean :  xx= +3+3+4+5+7+11 3= =5 7 7 Median: 4 Mode: 3 Range: 11­2=9 Standard Deviation: s = ∑ x−x¯ 2 = n−1 2 2 2 2 2 2 2 2−5 )+(3−5 )+ 3−5 )+(4−5 ) +(5−5 )+ 7−5 ) +(11−5 ) = 7−1 9+4+4+1+0+4+36 58 = =9.67 6 6 Sample Standard Deviation: s=√s =√9.67=3.1 (b) Ordered data: ­2 4 4 5 5 7 8 8 9 12 −2+4+4+5+5+7+8+8+9+12 60 Mean:  xx=  10 = 10 =6 Median: 6 Mode: 4, 5, 8 Range: 12+2=14 Standard Deviation: s = ∑ x−x¯ 2 = n−1 2 2 2 2 2 2 2 2 2 2 −2−6 ) +(4−6 )+ 4−6 )+(5−6 ) +(5−6 )+ 7−6 ) +(8−6 )+ 8−6 )+(9−6 ) +12−6 ) = 10−1 64+4+4+1+1+1+4+4+9+36 128 = =14.22 9 2 9 Sample Standard Deviation: s=√s =√14.22=3.77

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