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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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Presentation Transcript

STATISTICS

ELEMENTARY

Section 2-5 Measures of Variation

MARIO F. TRIOLA

EIGHTH

EDITION

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

Range

lowest

highest

value

value

Measures of Variation
a measure of variation of the scores about the mean

(average deviation from the mean)

Measures of Variation

Standard Deviation

Population Standard Deviation

(x - µ)

2

 =

N

Sample Standard Deviation

(x - x)2

S=

n -1

s

Sx

Symbols

for Standard Deviation

Sample

Population

x

Textbook

Textbook

TI-83

calculators

TI-83

calculators

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

Round only the final answer, never in the middle of a calculation.

Round-off Rulefor measures of variation

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

minimum x - 2(s)

maximum ‘usual’ value  (mean) + 2 (standard deviation)

maximum x + 2(s)

Usual Sample Values

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