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M ARIO F . T RIOLA

S TATISTICS. E LEMENTARY. Section 2-5 Measures of Variation. M ARIO F . T RIOLA. E IGHTH. E DITION. Jefferson Valley Bank Bank of Providence. Waiting Times of Bank Customers at Different Banks in minutes. 6.5 4.2. 6.6 5.4. 6.7 5.8. 6.8 6.2. 7.1 6.7. 7.3 7.7. 7.4 7.7.

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M ARIO F . T RIOLA

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  1. STATISTICS ELEMENTARY Section 2-5 Measures of Variation MARIO F. TRIOLA EIGHTH EDITION

  2. Jefferson Valley Bank Bank of Providence Waiting Times of Bank Customers at Different Banks in minutes 6.5 4.2 6.6 5.4 6.7 5.8 6.8 6.2 7.1 6.7 7.3 7.7 7.4 7.7 7.7 8.5 7.7 9.3 7.7 10.0

  3. Jefferson Valley Bank Bank of Providence Waiting Times of Bank Customers at Different Banks in minutes 6.5 4.2 6.6 5.4 6.7 5.8 6.8 6.2 7.1 6.7 7.3 7.7 7.4 7.7 7.7 8.5 7.7 9.3 7.7 10.0 Bank of Providence Jefferson Valley Bank Mean Median Mode Midrange 7.15 7.20 7.7 7.10 7.15 7.20 7.7 7.10

  4. Dotplots of Waiting Times Figure 2-14

  5. Range lowest highest value value Measures of Variation

  6. a measure of variation of the scores about the mean (average deviation from the mean) Measures of Variation Standard Deviation

  7. Sample Standard Deviation Formula (x - x)2 S= n -1 Formula 2-4

  8. calculators can compute the population standard deviation of data Population Standard Deviation (x - µ) 2  = N

  9. s Sx xn-1 Symbols for Standard Deviation Sample Population  x xn Textbook Book Some graphics calculators Some graphics calculators Some non-graphics calculators Some non-graphics calculators Articles in professional journals and reports often use SD for standard deviation and VAR for variance.

  10. Measures of Variation Variance standard deviation squared s  } 2 Sample Variance Notation Population Variance 2

  11. (x-x )2 s2 = n -1 (x-µ)2 2 = N Variance Formulas Sample Variance Population Variance

  12. Carry one more decimal place than is present in the original set of values. Round only the final answer, never in the middle of a calculation. Round-off Rulefor measures of variation

  13. Estimation of Standard Deviation Range Rule of Thumb x + 2s x - 2s x (maximum usual value) (minimum usual value) Range  4s or

  14. Estimation of Standard Deviation Range Rule of Thumb x + 2s x - 2s x (maximum usual value) (minimum usual value) Range  4s or Range 4 s 

  15. Estimation of Standard Deviation Range Rule of Thumb x + 2s x - 2s x (maximum usual value) (minimum usual value) Range  4s or Range 4 highest value - lowest value s  = 4

  16. minimum ‘usual’ value  (mean) - 2 (standard deviation) minimum x - 2(s) maximum ‘usual’ value  (mean) + 2 (standard deviation) maximum x + 2(s) Usual Sample Values

  17. The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 x

  18. The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 68% within 1 standard deviation 34% 34% x - s x x+s

  19. The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 95% within 2 standard deviations 68% within 1 standard deviation 34% 34% 13.5% 13.5% x - 2s x - s x x+s x+2s

  20. 0.1% The Empirical Rule (applies to bell-shaped distributions) FIGURE 2-15 99.7% of data are within 3 standard deviations of the mean 95% within 2 standard deviations 68% within 1 standard deviation 34% 34% 2.4% 2.4% 0.1% 13.5% 13.5% x - 3s x - 2s x - s x x+s x+2s x+3s

  21. For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations. Measures of Variation Summary

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