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Unformatted text preview: Announcements Turn in sheet for desired course section today. Assignments made by next class. See times for my and TAs office hours on web. USE THEM! Homework 1 assigned today (available online in materials folder on classes server). DUE NEXT TUESDAY IN CLASS. No exceptions. Remember the antelope! Make sure you have added the course STAT 1000a to your course list! MINITAB sessions underway. Just jump in if you havent been assigned to a time. Suggested readings for each class are listed on the website. STAT 101106 Introduction to Statistics 29 VARIANCE and STANDARD DEVIATION (SD) o Most common and useful measure of SPREAD of a distribution o Relationship: Variance Deviation Standard = o Notation : Sample Variance = s 2 , Standard Deviation = s Idea of variance o How far away are the observations, on average, from the mean? o This calculation involves the DEVIATIONS o A deviation is defined as the difference between an observation and the mean : x x i Aside : i is an index that keeps track of our particular observations. If we collect the heights of 4 people, then our sample size is n = 4 and i is an index that keeps track of the observations : STAT 101106 Introduction to Statistics 30 The S ample Variance s 2 is a S tatistic. This is a number we can calculate from our sample data. Formula for Sample Variance : in words, the average of the squared deviations = = n i i x x n s 1 2 2 ) ( 1 1 Example : Heights of six people in class (cm) : i x (Data) x (Mean ) x x i (Deviations ) 2 ) ( x x i (Squared Deviations) 5 1 = n Sample Variance : 86 5 430 2 = = s Sample Standard Deviation : 3 . 9 86 = = s 178 166 12 144 163 1663 9 168 166 2 4 167 166 1 1 170 166 4 16 150 16616 256 Sum = = 6 1 2 ) ( i i x x = 430 STAT 101106 Introduction to Statistics 31 i = 1 x 1 = 178cm i = 2 x 2 = 156cm i = 3 x 3 = 182cm i = 4 x 2 = 167cm Why squared Deviations? 0. Sum of deviations is just 0. Squaring the deviations converts the negative deviations to positive numbers... 1. Summing squares is a natural operation (think Pythagorus) Why divide by n1 ? If n =1, you shouldnt be calculating a variance! If n is big, it doesnt matter anyway Real Reason it makes the estimate unbiased (more on this later) More on VARIANCE and STANDARD DEVIATION (SD) SD of 3, 3, 3, 3, 3, 3 is zero (no variation) Robustness : IQR is robust; SD is not Example : probabilities of taking this class : 1.0, 0.9, 0.99, 1.0, 0.3, 0.95, 1.0, 0.5, 7.0, 1.0 SD =2.0 , IQR=0.2 Change 7.0 to 0.7 SD =0.25, IQR=0.35 STAT 101106 Introduction to Statistics 32 IQR changes little, SD changes quite a bit STAT 101106 Introduction to Statistics 33 MINITAB notes : To get all of the summary statistics discussed so far (mean, median, IQR, SD), use Stat Basic Statistics Display Descriptive Statistics....
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 Fall '05
 JonathanReuningSchererDonaldGreen
 Variance

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