1Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICSSection 2-5 Measures of Variation

2Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

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

7.7

8.5

7.7

9.3

7.7

10.0

3Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

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

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

Waiting Times of Bank Customers at Different Banks

in minutes

4Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Figure 2-14

Dotplots of Waiting Times

5Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Measures of Variation

6Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Measures of Variation

Range

valuehighest lowest

value

7Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

a measure of variation of the scores about the mean

(average deviation from the mean)

Measures of Variation

Standard Deviation

8Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Sample Standard Deviation Formula

calculators can compute the sample standard deviation of data

Formula 2-4

(x - x)2

n - 1S =

9Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Sample Standard Deviation Shortcut Formula

Formula 2-5

n (n - 1)s =

n (x2) - (x)2

calculators can compute the sample standard deviation of data

10Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - x

Mean Absolute Deviation Formula

n

11Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Population Standard Deviation

calculators can compute the population standard deviation

of data

2 (x - µ)

N =

12Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard Deviation

13Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample

14Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample

s

15Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample

sTextbook

16Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample

s

Sx

Textbook

17Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample

s

Sx

Textbook

Some graphicscalculators

18Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample

s

Sx

xn-1

Textbook

Some graphicscalculators

19Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample

s

Sx

xn-1

Textbook

Some graphicscalculators

Somenon-graphics

calculators

20Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample Population

s

Sx

xn-1

Textbook

Some graphicscalculators

Somenon-graphics

calculators

21Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample Population

x

xn

s

Sx

xn-1

Textbook

Some graphicscalculators

Somenon-graphics

calculators

22Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample Population

x

xn

s

Sx

xn-1

Book

Some graphicscalculators

Somenon-graphicscalculators

Textbook

Some graphicscalculators

Somenon-graphics

calculators

23Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Symbolsfor Standard DeviationSample Population

x

xn

s

Sx

xn-1

Book

Some graphicscalculators

Somenon-graphicscalculators

Textbook

Some graphicscalculators

Somenon-graphics

calculators

Articles in professional journals and reports often use SD for standard deviation and VAR for variance.

24Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Measures of Variation

Variance

25Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Measures of Variation

Variance

standard deviation squared

26Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Measures of Variation

Variance

standard deviation squared

s

2

2

}

use square keyon calculatorNotation

27Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

SampleVariance

PopulationVariance

Variance

(x - x )2

n - 1s2 =

(x - µ)2

N 2 =

28Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Round-off Rulefor 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.

29Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. 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 =

n [(f • x 2)] -[(f • x)]2

30Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Estimation of Standard DeviationRange Rule of Thumb

x - 2s x x + 2s

Range 4sor

(minimumusual value)

(maximum usual value)

31Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Estimation of Standard DeviationRange Rule of Thumb

x - 2s x x + 2s

Range 4sor

(minimumusual value)

(maximum usual value)

Range

4s

32Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Estimation of Standard DeviationRange Rule of Thumb

x - 2s x x + 2s

Range 4sor

(minimumusual value)

(maximum usual value)

Range

4s =

highest value - lowest value

4

33Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

Usual Sample Values

34Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

minimum ‘usual’ value (mean) - 2 (standard deviation)

minimum x - 2(s)

Usual Sample Values

35Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. 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

36Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x

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

37Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - s x x + s

68% within1 standard deviation

34% 34%

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

38Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - 2s x - s x x + 2sx + s

68% within1 standard deviation

34% 34%

95% within 2 standard deviations

The Empirical Rule(applies to bell-shaped distributions)

13.5% 13.5%

FIGURE 2-15

39Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman

x - 3s x - 2s x - s x x + 2s x + 3sx + s

68% within1 standard deviation

34% 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% 0.1%

2.4% 2.4%

13.5% 13.5%

FIGURE 2-15

40Chapter 2. Section 2-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. 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.