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Section 3-3 Measures of Variation Created by Tom Wegleitner, Centreville, Virginia Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 2 Key Concept Because this section introduces the concept of variation, which is something so important in statistics, this is one of the most important sections in the entire book. 1 Place a high priority on how to interpret values of standard deviation. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 3 See p. 93 Figure 3-3. • The mean of all three samples is x = 6.0 min. • But, the samples are very different in the amounts that the waiting times vary. • Bank #1 All waiting times are 6 min – they don’t vary at all. • Bank #2 and #3 The wait time for customers in multiple lines vary more than those in the single line. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 4 Definition The range of a set of data is the difference between the maximum value and the minimum value. Range = (maximum value) – (minimum value) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 5 Definition The standard deviation of a set of sample values is a measure of variation of values about the mean. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. 2 Slide 6 Sample Standard Deviation Formula s= Σ (x - x)2 n-1 Sample Standard Deviation (Shortcut Formula) s= nΣ(x2) - (Σx)2 n (n - 1) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 7 • See handout for Example problem to calculate standard deviation. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 8 Standard Deviation Important Properties The standard deviation is a measure of variation of all values from the mean. The value of the standard deviation s is usually positive. 3 The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others). The units of the standard deviation s are the same as the units of the original data values. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 9 Population Standard Deviation This formula is similar to the previous formula, but instead, the population mean and population size are used. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 10 Definition The variance of a set of values is a measure of variation equal to the square of the standard deviation. Sample variance: Square of the sample standard deviation s Population variance: Square of the population standard deviation σ Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 11 Variance - Notation standard deviation squared } Notation s2 σ 2 4 Sample variance Population variance Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 12 Round-off Rule for Measures of Variation Carry one more decimal place than is present in the original set of data. Round only the final answer, not values in the middle of a calculation. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Slide 13