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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
slide1
STATISTICS

ELEMENTARY

Section 2-5 Measures of Variation

MARIO F. TRIOLA

EIGHTH

EDITION

slide2
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

slide3
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
slide6
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

slide9
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

slide11
(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
slide13
Estimation of Standard Deviation

Range Rule of Thumb

x + 2s

x - 2s

x

(maximum usual value)

(minimum

usual value)

Range  4s

or

slide14
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 

slide15
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
slide17
The Empirical Rule

(applies to bell-shaped distributions)

FIGURE 2-15

x

slide18
The Empirical Rule

(applies to bell-shaped distributions)

FIGURE 2-15

68% within

1 standard deviation

34%

34%

x - s

x

x+s

slide19
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

slide20
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