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by: Sasha Kunze I


Sasha Kunze I
GPA 3.6


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Class Notes
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This 37 page Class Notes was uploaded by Sasha Kunze I on Thursday October 22, 2015. The Class Notes belongs to SLAV 5 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/227024/slav-5-university-of-california-santa-barbara in Slavic at University of California Santa Barbara.




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Date Created: 10/22/15
PSTAT 5E Statistics with Economics and Business Applications Chapter 3 Descriptive Statistics Numerical Measures Yan Xu Review In Chl 1 Key terms of a Data set Variable Observation 2 Types of data Categorical Quantitative Discrete and Cont 3 Statistics and Stati i 71 Inf rn Pr edure Review In Ch2 How to summarize Data by tables and graphs Tables freq relative freq percent cumulative Graphs for a single variable 1 Categorical data a Pie charts b Bar charts 2 Quantitative data a Dot plot b Histograms c Cumulative 3 Describing data distributions a Shapes b Outliers In this lecture Ch2 StemandLeaf Display Ch3 Descriptive Stats Numerical Measures mean median mode variance etc StemandLeaf Plot A simple graph for quantitative data It uses the actual numerical values of CaCu uaLa puult Exauipiu 1cLW llulllbelo 114 14 U1 I 1U7 Sorted 6 9 69 72 78 7 2 8 97 8 107 112 9 7 10 7 11 Stel s to draw a stem and leaf 1 lot Divide each number into two parts the stem and the leaf List the stems in a column with a vertical line to the right For each number record the leaf in the same row as its matching stem Order the leaves from lowest to highest for each stem Example Stem and leaf The ces 0f18lnandsofshoe 9 7O 7O 7O 75 7O 65 68 6O 74 7O 95 75 7 68 65 4O 65 4 4I 5 5 6 6m 7 000504050 7 0000004551 8 8 9 05 9E Ch3 Describing data With Numerical Measures Numerical measures can be created for both populations and samples A population parameter is a numerical measure computed for a population A sample statistic is a numerical measure computed for a sample Measures of Location Mean 231311 383 183 Percentile I Quartile Some Notations Suppose we have n observations The values we observe x1 XZ X39 XH I Example Suppose we ask five people how many hours of they spend on the internet in a week and get the following numbers 2 9 11 5 6 Then n5x12x29x311x45x56l Sample Mean Where n number of sample observations Population Mean Z N Where N number of the Whole population Example Time spend on internet 2 9 11 5 6 9 5 2x1 291156 5 n Median The median is the value in the middle When the data are arranged in ascendinv order smallest value to largest value 0 Once the data have been ordered the position of the median is n is odd the median is the middle value 5n 1 n is even the median is the average of the middle two values Example Theset 2498653 n7 0 Sort 2 3 4 6 8 9 Position 5 1 57 1 4th Median 4 largest number 0 Theset 249865 n6 Sort 2 8 9 Position 5 1 56 1 53m Median 5 62 we average of the 3rd and 4th numbers Mode The mode is the value that occurs with greatest frequency Example The set 2 4 9 8 8 5 3 The mode is 8 which occurs twice The set 2 2 9 8 8 5 3 There are two modes 8 and 2 bimodal The set 2 4 9 8 5 3 There is no mode each value is unique Example The number of quarts of milk purchased by A nousenol M5 7 ms Re atwe frequency 2 0 Mode hXel39Clse A 43 44 43 J 34 1 Compute the mean and median 2 Which one is better to describe the center of location n6 Mean 223 24253 326 7367 Median 24252 245 The mean is more easily affected by extremely large or small values than the median The median is often used as a measure of center when the distribution is skewed Relationship between mean and median g i g Symmetric Mean Median o o 3 3 o o Skewed right Mean gt Median x 3 m Skewed left Mean lt Median o Measures of Variability Measures used to describe the spread of the distribution from the center I s L Range The range R of a set of n observations is the difference between the largest and smallest values Example Studying hours per day for 5 students 5 12 6 8 14 The range is R 14 5 9 uick and easy but only uses 2 0f the 5 values The Variance The variance is measure of variability that uses all the measurements It measures the average deviation of the measurements about their mean Studying Hours 5 12 6 8 14 Vorion The 0 ulation variance of N measurements is the average of the squared deviations of the values about their population mean m 2 20939 u2 I N Oquot The sample variance of n measurew is e sum of the squared deviations of the values about their sam le mean divided b n 1 S2 2 209 f2 n l Standard Dovis in PopulationSample Standard deviation is the positive square root of the o ulationsam le variance Population standard deviation 0 x 02 Sample standard deviation s xsi2 Exercise Calculate the Sample Variance and Standard deviation for the data 5 12 6 8 I4 xi 3 5 xi 3 02 5 4 16 12 3 9 6 3 9 8 1 1 14 5 25 Sum 45 0 60 2 Z 20939 f2 S n l 15 4 2 SZ VS Some Notes The value of s is ALWAYS positive 0 Meaning of s2 The larger the value of s2 or s the larger the variability of the data set Why divided by n 1 in sample variance instead of n The sample standard deviation sis often used to estimate the population standard deviation 6 Dividing by n 1 gives us a better estimate of o Percentile I A percentile provides information about how the data are spread over the interval from the smallest value to the largest value I One Application 80100 A 5080 B 1050 C 010 D l The pth percentile of a data set is a value such that at least p percent of the observations are less than or equal to this value and at least 100 p percent of the observations are greater than or equal to this value Calculating the pth Percentiles Arrange the data in ascending order Compute index i the position of the pth percentile i p100n If i is not an integer round up The p th percentile is the value in the i th position If i is an integer the p th percentile is the average of the values in positions 139 and 139 1 Example Find 80th percentile for 12 students nal scores 85 65 98 88 65 92 32 43 67 77 90 80 Step 1 Arrange the data in ascending order Step 2 compute indeX i p100n 80100 12 96 Step 3 96 is not an integer round up The position of the 80th percentile is the next integer greater then 96 that is the 10th position Therefore the 80th percentile is 90 Exercise nd the 50th percentile based on the ordered data Ste 2 com ute index i p100n 50100 12 6 Step 3 i6 is an integer The position of the 50th percentile is the average of the values in positions 1 and 139 1 that is the average of values in position 6 and 7 Therefore the 50th percentile is 7 7802 7 85 lt 39 20 7 80 90 is the 80th Score 90 percentile Quartile are some specific percentiles 25th Percentile LOWCY Quartile Q1 50th Percentile E Madian Q2 75th Percentile E Upper Quartile Q3 Quartiles and the IQR The lower quartile Ollis the value which is larger than 25 and less than 75 of the ordered data 0 The upper quartile Qilis the value Which is larger than 75 and less than 25 of the ordered data 0 The range of the middle 50 is de ned as the interquartile range IQR Q3 Q1 Calculating Q1 and Q3 o The lower and upper quartiles 01 and 0 can be calculated as follows 0 The position of Q1 is The position of Q3 is i25100n i75100 n If i is not an integer round up The p th percentile is the value in the i th position If i is an integer the p th percentile is the average of the values in positions 239 and 139 1 Example The prices of 18 brands of walking shoes 40 6O 65 65 65 68 68 7O 7O 7O 7O 7O 7O 74 75 75 9O 95 Position of Q1 2518 45 round up to 5 Position 0139 Q3 7518 135 round up to 14 Q1 UJ Q3 74 IQR Q3Q1 7465 9 Key Concepts Measures of Location 1 Mean a Population mean u b Sample mean of size n 2 Median the value in the middle 3 Mode number with the greatest freq Notes The median may be preferred to the mean ifthe data are hivhl skewed Key Concepts ll Measures of Variability 1 Range R 2 largest smallest 2 Variance 2 Zxl u2 039 a Population of N measurements N b Sample of n measurements 2 S2 20939 X 11 1 Population standard deviation o l 0392 3 Standard deviation Sample standard deviation s Key Concepts Ill Measures of Relative Standing 1 pth percentile p of the observations are smaller and 100 p are larger 2 ip100n sort data first Two situations integer or not integer i is not integer round up to the next integer iinteger average of ith and i1th 3 First Lower quartile 01 position of Q1 2 25100n 4 ThirdUpper quartile Q3 position of Q3 2 75100n 5 Interquartile range IQR Q3 Q1 Ex Sample 15 20 25 25 27 28 30 34 1 Compute mean median and mode Sample Variance and sample standard deviation Compute 20th 65th percentile 4 Compute 1 3 IQR Solution n8 1 Mean sumn2048255 Median2527226 lVlOGGZZD


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