1 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 2-5 Measures of Variation

2 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank Bank of Providence

3 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Jefferson Valley Bank Bank of Providence Jefferson Valley Bank Bank of Providence Mean Median Mode Midrange Waiting Times of Bank Customers at Different Banks in minutes

4 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Figure 2-14 Dotplots of Waiting Times

5 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation

6 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Range value highest lowest value

7 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman a measure of variation of the scores about the mean (average deviation from the mean) Measures of Variation Standard Deviation

8 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Sample Standard Deviation Formula calculators can compute the sample standard deviation of data Formula 2-4  ( x - x ) 2 n - 1 S =S =

9 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Sample Standard Deviation Shortcut Formula Formula 2-5 n ( n - 1) s = n (  x 2 ) - (  x ) 2 calculators can compute the sample standard deviation of data

10 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman  x - x Mean Absolute Deviation Formula n

11 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Population Standard Deviation calculators can compute the population standard deviation of data 2  ( x - µ ) N  =

12 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation

13 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample

14 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample s

15 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample s Textbook

16 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample sSxsSx Textbook

17 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample sSxsSx Textbook Some graphics calculators

18 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample s S x x  n-1 Textbook Some graphics calculators

19 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample s S x x  n-1 Textbook Some graphics calculators Some non-graphics calculators

20 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample Population s S x x  n-1 Textbook Some graphics calculators Some non-graphics calculators

21 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample Population   x x  n s S x x  n-1 Textbook Some graphics calculators Some non-graphics calculators

22 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample Population   x x  n s S x x  n-1 Book Some graphics calculators Some non-graphics calculators Textbook Some graphics calculators Some non-graphics calculators

23 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Symbols for Standard Deviation Sample Population   x x  n s S x x  n-1 Book Some graphics calculators Some non-graphics calculators Textbook Some graphics calculators Some non-graphics calculators Articles in professional journals and reports often use SD for standard deviation and VAR for variance.

24 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Variance

25 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Variance standard deviation squared

26 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Measures of Variation Variance standard deviation squared s  2 2 } use square key on calculator Notation

27 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Sample Variance Population Variance  ( x - x ) 2 n - 1 s 2 =  (x - µ)2 (x - µ)2 N  2 =

28 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Round-off Rule for measures of variation 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.

29 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Standard Deviation from a Frequency Table Use the class midpoints as the x values Calculators can compute the standard deviation for frequency table Formula 2-6 n ( n - 1) S =S = n [  ( f x 2 )] -[  ( f x )] 2

30 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value)

31 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value) Range 4 s 

32 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value) Range 4 s  = highest value - lowest value 4

33 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman Usual Sample Values

34 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman minimum ‘usual’ value  (mean) - 2 (standard deviation) minimum  x - 2(s) Usual Sample Values

35 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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

36 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15

37 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - s x x + sx + s 68% within 1 standard deviation 34% The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15

38 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - 2s x - s x x + 2s x + sx + s 68% within 1 standard deviation 34% 95% within 2 standard deviations The Empirical Rule (applies to bell-shaped distributions ) 13.5% FIGURE 2-15

39 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman x - 3s x - 2s x - s x x + 2s x + 3s x + sx + s 68% within 1 standard deviation 34% 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 ) 0.1% 2.4% 13.5% FIGURE 2-15

40 Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright Addison Wesley Longman 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.