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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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M ARIO F . T RIOLA

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M ario f t riola

STATISTICS

ELEMENTARY

Section 2-5 Measures of Variation

MARIO F. TRIOLA

EIGHTH

EDITION


M ario f t riola

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


M ario f t riola

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


M ario f t riola

Dotplots of Waiting Times

Figure 2-14


Measures of variation

Range

lowest

highest

value

value

Measures of Variation


M ario f t riola

a measure of variation of the scores about the mean

(average deviation from the mean)

Measures of Variation

Standard Deviation


Sample standard deviation formula

Sample Standard Deviation Formula

(x - x)2

S=

n -1

Formula 2-4


Population standard deviation

calculators can compute the population standard deviation of data

Population Standard Deviation

(x - µ)

2

 =

N


M ario f t riola

s

Sx

xn-1

Symbols

for Standard Deviation

Sample

Population

x

xn

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.


Measures of variation1

Measures of Variation

Variance

standard deviation squared

s



}

2

Sample Variance

Notation

Population Variance

2


M ario f t riola

(x-x )2

s2 =

n -1

(x-µ)2

2 =

N

Variance Formulas

Sample

Variance

Population

Variance


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


M ario f t riola

Estimation of Standard Deviation

Range Rule of Thumb

x + 2s

x - 2s

x

(maximum usual value)

(minimum

usual value)

Range  4s

or


M ario f t riola

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 


M ario f t riola

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


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


M ario f t riola

The Empirical Rule

(applies to bell-shaped distributions)

FIGURE 2-15

x


M ario f t riola

The Empirical Rule

(applies to bell-shaped distributions)

FIGURE 2-15

68% within

1 standard deviation

34%

34%

x - s

x

x+s


M ario f t riola

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


M ario f t riola

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


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.

Measures of Variation Summary


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