Measures of Variation. Section 2-5. Dotplots of Waiting Times. Waiting Times of Bank Customers at Different Banks in minutes. Bank of Providence

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1 Measures of Variation Section -5 1 Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence Mean Median Mode Jefferson Valley Bank Bank of Providence Midrange Dotplots of Waiting Times 3

2 Measures of Variation Range highest value lowest value 4 Measures of Variation Standard Deviation A measure of variation of the scores about the mean Average deviation from the mean Average distance scores are from the mean 5 Rough Definition Average distance the scores are from the mean Σ (x - x) n 6

3 , 3, 3, 5, 7 x =4 (x - x) = -, -1, -1, 1, 3 Σ (x - x) = 0/5 = 0 n 7 Mean Absolute Deviation Formula Σ x - x n 8, 3, 3, 5, 7 x =4 x - x =, 1, 1, 1, 3 Σ x - x = 8/5 = 1.6 n 9

4 Standard Deviation Formula for a Sample S = Σ (x - x) n - 1 Formula -4 calculators can compute the sample standard deviation of data 10 Sample Standard Deviation Shortcut Formula s = n (Σx ) - (Σx) n (n - 1) Formula -5 calculators can compute the sample standard deviation of data 11 Important Properties of Standard Deviation A measure of variation of all values from the mean Usually positive; is zero (0) when all data are the same Value can increase dramatically with outliers Units are the same as the units of the original data 1

5 Same Means (x = 4) Different Standard Deviations Frequency s = 0 s = 0.8 s = 1.0 s = When data is more Standard varied, the Deviations standard deviation gets larger. 13 Using Your Calculator to find the standard deviation of a data set S = 0.48 minutes 14 Population Standard Deviation σ = Σ (x - µ) N calculators can compute the population standard deviation of data 15

6 Measures of Variation Variance standard deviation squared 16 Measures of Variation Variance standard deviation squared Notation } s σ use square key on calculator 17 s = Variance Σ (x - x ) n - 1 Sample Variance σ = Σ (x - µ) N Population Variance 18

7 Using Your Calculator to find the standard deviation of a data set S = 0.48 minutes S = 0.3 minutes 19 Notation Textbook Some graphics calculators Some non-graphics calculators Sample s Sx xσ n-1 Population σ σ x xσ n Book Some graphics calculators Some non-graphics calculators Articles in professional journals and reports often use SD for standard deviation and VAR for variance. 0 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, never in the middle of a calculation. 1

8 Standard Deviation from a Frequency Distribution Formula -6 S = n [Σ(f x )] -[Σ(f x)] n (n - 1) Use the class midpoints as the x values. Calculators can compute the standard deviation for a frequency distribution. Standard Deviation and Variance of a Frequency Distribution Quiz Scores Midpoints Frequency S = 5.9 S = 34.6 (using S not rounded) σ = 5.8 σ = 33.5 (using σ not rounded) 3 Estimation of Standard Deviation Range Rule of Thumb x - s x x + s (minimum usual value) Range 4s (maximum usual value) Range s = 4 highest value - lowest value 4 4

9 Estimating the standard deviation using the Range Rule of Thumb S Range / 4 = ( 6.5) / 4 = 1. / 4 = 0.3 min (estimate) S = 0.48 minutes (actual) 5 Rough Estimates of Usual Sample Values minimum usual value (mean) - (standard deviation) minimum x - (s) maximum usual value (mean) + (standard deviation) maximum x + (s) 6 Rough Estimates of Usual Sample Values Quiz Scores minimum x - (s) minimum (5.9) =.6 maximum x + (s) maximum (5.9) = 6. 7

10 FIGURE -13 The Empirical Rule (applies to bell-shaped distributions) 99.7% of data are within 3 standard deviations of the mean 95% within standard deviations 68% within 1 standard deviation 34% 34%.4%.4% 0.1% 0.1% 13.5% 13.5% x - 3s x - s x - 1s x x + 1s x + s x + 3s 8 Measures of Variation Summary For typical data sets, it is unusual for a score to differ from the mean by more than or 3 standard deviations. 9