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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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Mario f triola 1346239

STATISTICS

ELEMENTARY

Section 2-5 Measures of Variation

MARIO F. TRIOLA

EIGHTH

EDITION


Mario f triola 1346239

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



Measures of variation

Range

lowest

highest

value

value

Measures of Variation


Measures of variation1

a measure of variation of the scores about the mean

(average deviation from the mean)

Measures of Variation

Standard Deviation


Population standard deviation
Population Standard Deviation

(x - µ)

2

 =

N

Sample Standard Deviation

(x - x)2

S=

n -1


Mario f triola 1346239

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 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.

Round-off Rulefor measures of variation


Mario f triola 1346239

0.1% set of values.

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


Usual sample values

minimum ‘usual’ value set of values. (mean) - 2 (standard deviation)

minimum x - 2(s)

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

maximum x + 2(s)

Usual Sample Values


Mario f triola 1346239

Estimation of Standard Deviation set of values.

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


Measures of variation summary

For typical data sets, it is set of values.unusual for a score to differ from the mean by more than 2 or 3 standard deviations.

Measures of Variation Summary