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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 TA’s 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 haven’t 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 shouldn’t be calculating a variance! • If n is big, it doesn’t 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
 Normal Distribution, Standard Deviation, Variance

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