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M ARIO F . T RIOLA

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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

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

Dotplots of Waiting Times

Figure 2-14

Range

lowest

highest

value

value

a measure of variation of the scores about the mean

(average deviation from the mean)

Measures of Variation

Standard Deviation

(x - x)2

S=

n -1

Formula 2-4

calculators can compute the population standard deviation of data

(x - µ)

2

=

N

s

Sx

xn-1

Symbols

for Standard Deviation

Sample

Population

x

xn

Textbook

Book

Some graphics

calculators

Some graphics

calculators

Some

non-graphics

calculators

Some

non-graphics

calculators

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

Variance

standard deviation squared

s

}

2

Sample Variance

Notation

Population Variance

2

(x-x )2

s2 =

n -1

(x-µ)2

2 =

N

Variance Formulas

Sample

Variance

Population

Variance

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.

Estimation of Standard Deviation

Range Rule of Thumb

x + 2s

x - 2s

x

(maximum usual value)

(minimum

usual value)

Range 4s

or

Estimation of Standard Deviation

Range Rule of Thumb

x + 2s

x - 2s

x

(maximum usual value)

(minimum

usual value)

Range 4s

or

Range

4

s

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

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

minimum x - 2(s)

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

maximum x + 2(s)

The Empirical Rule

(applies to bell-shaped distributions)

FIGURE 2-15

x

The Empirical Rule

(applies to bell-shaped distributions)

FIGURE 2-15

68% within

1 standard deviation

34%

34%

x - s

x

x+s

The Empirical Rule

(applies to bell-shaped distributions)

FIGURE 2-15

95% within

2 standard deviations

68% within

1 standard deviation

34%

34%

13.5%

13.5%

x - 2s

x - s

x

x+s

x+2s

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

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