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

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


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



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


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


Figure 2-14

Dotplots of Waiting Times


Measures of Variation



Range

valuehighest lowest

value


a measure of variation of the scores about the mean

(average deviation from the mean)


Standard Deviation


Sample Standard Deviation Formula

calculators can compute the sample standard deviation of data

Formula 2-4

(x - x)2

n - 1S =


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


x - x

Mean Absolute Deviation Formula

n


Population Standard Deviation

calculators can compute the population standard deviation

of data

2 (x - µ)

N =


Symbolsfor Standard Deviation


Symbolsfor Standard DeviationSample



s



sTextbook



s

Sx

Textbook



s

Sx

Textbook

Some graphicscalculators



s

Sx

xn-1

Textbook




s

Sx

xn-1

Textbook


Somenon-graphics

calculators


Symbolsfor Standard DeviationSample Population

s

Sx

xn-1

Textbook


Somenon-graphics

calculators



x

xn

s

Sx

xn-1

Textbook


Somenon-graphics

calculators



x

xn

s

Sx

xn-1

Book


Somenon-graphicscalculators

Textbook


Somenon-graphics

calculators



x

xn

s

Sx

xn-1

Book


Somenon-graphicscalculators

Textbook


Somenon-graphics

calculators

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



Variance



Variance

standard deviation squared



Variance

standard deviation squared

s

2

2

}

use square keyon calculatorNotation


SampleVariance

PopulationVariance

Variance

(x - x )2

n - 1s2 =

(x - µ)2

N 2 =


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.


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


Estimation of Standard DeviationRange Rule of Thumb

x - 2s x x + 2s

Range 4sor

(minimumusual value)

(maximum usual value)



x - 2s x x + 2s

Range 4sor



Range

4s



x - 2s x x + 2s

Range 4sor



Range

4s =

highest value - lowest value

4


Usual Sample Values


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

minimum x - 2(s)

Usual Sample Values


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


x

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


x - s x x + s

68% within1 standard deviation

34% 34%

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


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


34% 34%

95% within 2 standard deviations

The Empirical Rule(applies to bell-shaped distributions)

13.5% 13.5%

FIGURE 2-15


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


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


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.

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

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