Measures of Variation Section -5 1 Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank 6.5 6.6 6.7 6.8 7.1 7.3 7.4 Bank of Providence 4. 5.4 5.8 6. 6.7 8.5 9.3 10.0 Mean Median Mode Jefferson Valley Bank 7.15 7.0 Bank of Providence 7.15 7.0 Midrange 7.10 7.10 Dotplots of Waiting Times 3
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, 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
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
Same Means (x = 4) Different Standard Deviations Frequency 7 6 5 4 3 1 s = 0 s = 0.8 s = 1.0 s = 3.0 1 3 4 5 6 7 1 3 4 5 6 7 1 3 4 5 6 7 1 3 4 5 6 7 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 6.5 6.6 6.7 6.8 7.1 7.3 7.4 S = 0.48 minutes 14 Population Standard Deviation σ = Σ (x - µ) N calculators can compute the population standard deviation of data 15
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
Using Your Calculator to find the standard deviation of a data set 6.5 6.6 6.7 6.8 7.1 7.3 7.4 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
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 0-4 5-9 10-14 15-19 0-4 7 1 17 5 8 11 7 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
Estimating the standard deviation using the Range Rule of Thumb 6.5 6.6 6.7 6.8 7.1 7.3 7.4 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 14.4 - (5.9) =.6 maximum x + (s) maximum 14.4 + (5.9) = 6. 7
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