STAT10x9.6.05.MeanandVariance

STAT10x9.6.05.MeanandVariance - Announcements Turn in sheet...

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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 100-0a 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 101-106 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 101-106 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 12 144 163 166-3 9 168 166 2 4 167 166 1 1 170 166 4 16 150 166-16 256 Sum = =- 6 1 2 ) ( i i x x = 430 STAT 101-106 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 n-1 ? 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 101-106 Introduction to Statistics 32 IQR changes little, SD changes quite a bit STAT 101-106 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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STAT10x9.6.05.MeanandVariance - Announcements Turn in sheet...

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