1. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Measures of Variance Section 2-5 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

2. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2 Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence 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 8.5 7.7 9.3 7.7 10.0

3. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3 Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence 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 8.5 7.7 9.3 7.7 10.0 Jefferson Valley Bank 7.15 7.20 7.7 7.10 Bank of Providence 7.15 7.20 7.7 7.10 Mean Median Mode Midrange

4. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Measure of Variation

5. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 Measure of Variation Range score highest lowest score

6. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Measure of Variation Standard Deviation

7. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 a measure of variation of the scores about the mean (the average deviation from the mean is zero) x – x n  = 0 Deviation from the mean: x – x

8. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Sample Standard Deviation Formula

9. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9 Sample Standard Deviation Formula Formula 2 -4 calculators can calculate sample standard deviation of data  (x – x) 2 n – 1 S =S =

10. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 1234567123456712345671234567 1 2 3 4 5 6 7 s = 0 s = 0.8s = 1.0 s = 3.0 Standard deviation gets larger as spread of data increases. Same Means (x = 4) Different Standard Deviations FIGURE Frequency

11. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 Mean Deviation Formula (absolute deviation)

12. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 12 Mean Deviation Formula (absolute deviation)  x – x n

13. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 13 Population Standard Deviation

14. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 Population Standard Deviation  ( x – µ ) N 2  =

15. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 Population Standard Deviation calculators can calculate the population standard deviation of data  ( x – µ ) N 2  =

16. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 Examples

17. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 Measure of Variation Variance

18. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18 Measure of Variation Variance standard deviation squared

19. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 19 Measure of Variation Variance standard deviation squared ss 2 2 } use square key on calculator Notation

20. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 20  (x – x)2 (x – x)2 n – 1 s 2 =  (x – µ)2 (x – µ)2 N  2 = Sample Variance Population Variance

21. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 21 Round-off Rule for measures of variation Carry one more decimal place than was present in the original data

22. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 22 Standard Deviation Shortcut Formula

23. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 23 Standard Deviation Shortcut Formula Formula 2 - 6 n ( n – 1) s = n (  x 2 ) – (  x) 2

24. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 24  f (x – x) 2 Standard Deviation for Group Data (Frequency Table) Formula 2-7 n ( n – 1) S =S = n [  ( f x 2 )] – [  ( f x ) ] 2 where x = class mark calculators can calculate the standard deviation of grouped data shortcut n – 1 S =S =

25. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 25 Examples

26. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 26 Range Rule of Thumb x – 2 s x x + 2 s Range  4 s (minimum) (maximum)

27. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 27 Range Rule of Thumb x – 2 s x x + 2 s Range  4 s or s  Range 4 (minimum) (maximum)

28. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 28 x The Empirical Rule (applies to bell shaped distributions ) FIGURE

29. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 29 x – s x x + sx + s 68% within 1 standard deviation 0.340 The Empirical Rule (applies to bell shaped distributions ) FIGURE

30. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 30 x – 2s x – s x x + 2s x + sx + s 68% within 1 standard deviation 0.340 95% within 2 standard deviations The Empirical Rule (applies to bell shaped distributions ) 0.135 FIGURE

31. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 31 x – 3s x – 2s x – s x x + 2s x + 3s x + sx + s 68% within 1 standard deviation 0.340 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean The Empirical Rule (applies to bell shaped distributions ) 0.001 0.024 0.135 FIGURE

32. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 32 Chebyshev’s Theorem  applies to distributions of any shape  the proportion (or fraction) of any set of data lying within k standard deviations of the mean is always at least 1 – 1/k 2, where k is any positive number greater than 1.

33. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 33 Measures of Variation Summary For typical data sets, it is unusual for a score to differ from the mean by more than 2 or 3 standard deviations.