Refer to Ex 3-18 on page 123-124 Record the info for Brand A in a column. Allow 3 adjacent other...

Refer to Ex 3-18 on page 123-124

Record the info for Brand A in a column. Allow 3 adjacent other columns to be

added.Do the same for Brand B.

Test on Chapter 3

•Friday Sept 27th.•You are expected to provide you own calculator on the test.

3-2 Measures of VariationHow Can We Measure Variability?

RangeVarianceStandard DeviationCoefficient of VariationChebyshev’s TheoremEmpirical Rule (Normal)

3Bluman, Chapter 3

Measures of Variation: Range The range is the difference between the

highest and lowest values in a data set.

R Highest Lowest

4Bluman, Chapter 3

Chapter 3Data Description

Section 3-2Example 3-18/19

Page #123

5Bluman, Chapter 3

Example 3-18/19: Outdoor Paint

Two experimental brands of outdoor paint are tested to see how long each will last before fading. Six cans of each brand constitute a small population. The results (in months) are shown. Find the mean and range of each group.

6Bluman, Chapter 3

Brand A Brand B

10 35

60 45

50 30

30 35

40 40

20 25

Example 3-18: Outdoor Paint

7Bluman, Chapter 3

Brand A Brand B

10 35

60 45

50 30

30 35

40 40

20 25

21035

Brand A: 6

60 10 50

X

N

R

21035

Brand B: 6

45 25 20

X

R

N

The average for both brands is the same, but the rangefor Brand A is much greater than the range for Brand B.

Which brand would you buy?

Measures of Variation: Variance & Standard Deviation The variance is the average of the

squares of the distance each value is from the mean.

The standard deviation is the square root of the variance.

The standard deviation is a measure of how spread out your data are.

8Bluman, Chapter 3

•Uses of the Variance and Standard Deviation

To determine the spread of the data. To determine the consistency of a variable. To determine the number of data values

that fall within a specified interval in a distribution (Chebyshev’s Theorem).

Used in inferential statistics.

9Bluman, Chapter 3

Measures of Variation: Variance & Standard Deviation (Population Theoretical Model) The population variance is

The population standard deviation is

2

2 X

N

2X

N

10Bluman, Chapter 3


Section 3-2Example 3-21

Page #125

11Bluman, Chapter 3

Example 3-21: Outdoor PaintFind the variance and standard deviation for the data set for Brand A paint. 10, 60, 50, 30, 40, 20

12Bluman, Chapter 3

Months, X µ X - µ (X - µ)2

106050304020

353535353535

-252515-55

-15

625625225

2525

225

1750

1750

6

17.1

2

2

1750

6

291.7

X

n

PopulationVariance

PopulationStandard Deviation

Measures of Variation: Variance & Standard Deviation(Sample Theoretical Model) The sample variance is

The sample standard deviation is

2

2

1

X Xs

n

2

1

X Xs

n

13Bluman, Chapter 3

Measures of Variation: Variance & Standard Deviation(Sample Computational Model) Is mathematically equivalent to the

theoretical formula.

Saves time when calculating by hand

Does not use the mean

Is more accurate when the mean has been rounded.

14Bluman, Chapter 3

Measures of Variation: Variance & Standard Deviation(Sample Computational Model) The sample variance is

The sample standard deviation is

15Bluman, Chapter 3

2 2

2

1

X Xn

sn n

2s s



Page #129

16Bluman, Chapter 3

958.94

Example 3-23: European Auto SalesFind the variance and standard deviation for the amount of European auto sales for a sample of 6 years. The data are in millions of dollars.

11.2, 11.9, 12.0, 12.8, 13.4, 14.3

17Bluman, Chapter 3

X X 2

11.211.912.912.813.414.3

125.44141.61166.41163.84179.56204.49

75.6

2 2

2

1

X Xn

sn n

2

2 75.66 958.94

6 5

s

2 1.28

1.13

s

s

2 26 958.94 75.6 / 6 5 s

Finding Variance and Standard deviation of Grouped Data Find the variance and the standard

deviation for the frequency distribution of the data in the next slide. The data represents the number of miles that 20 runners ran during one week.

A B

ClassFrequency

(f)

5.5-10.5 1

10.5-15.5 2

15.5-20.5 3

20.5-25.5 5

25.5-30.5 4

30.5-35.5 3

35.5-40.5 2

A B C D E

ClassFrequency

(f)

Midpoint(Xm)

f•Xm f•Xm2

5.5-10.5 1

10.5-15.5 2

15.5-20.5 3

20.5-25.5 5

25.5-30.5 4

30.5-35.5 3

35.5-40.5 2

1

222

n

nXfXfs mm

7.68

120

20490310,13 2

Example 3-23

Find the sample variance and standard deviation for the amount of European auto sales for a sample of 6 years shown. The data are in millions of dollars.

11.2, 11.9, 12.0, 12.8, 13.4, 14.3

Example 3-24

Find he variance and the standard deviation for the frequency distribution of the data in Example 2-7. the data represent

Uses for standard deviation.

1. Spread of data

2. Consistency

3. Determine the number (or %) of data within an interval.

4. Many other used to be discussed fully second semester!

Measures of Variation: Coefficient of Variation

The coefficient of variation is the standard deviation divided by the mean, expressed as a percentage.

Use CVAR to compare standard deviations when the units are different.

100%s

CVARX

25Bluman, Chapter 3



Page #132

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Example 3-25: Sales of AutomobilesThe mean of the number of sales of cars over a 3-month period is 87, and the standard deviation is 5. The mean of the commissions is $5225, and the standard deviation is $773. Compare the variations of the two.

27Bluman, Chapter 3

Commissions are more variable than sales.

5100% 5.7% Sales

87CVar

773100% 14.8% Commissions

5225CVar

Measures of Variation: Range Rule of Thumb

The Range Rule of Thumb approximates the standard deviation as

when the distribution is unimodal and approximately symmetric.

4

Ranges

28Bluman, Chapter 3

Measures of Variation: Range Rule of Thumb

Use to approximate the lowest value and to approximate the highest value in a data set.

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2X s2X s

Example: 10, 12X Range

123

4s

10 2 3 410 2 3 16

LOWHIGH

Using Symbols

Let 20 and s=3, calculator the following:

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The proportion of values from any data set that fall within k standard deviations of the mean will be at least , • where k >1 • k is not necessarily an integer.

Measures of Variation: Chebyshev’s Theorem

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32Bluman, Chapter 3

# of standard

deviations, k

Minimum Proportion within k standard

deviations

Minimum Percentage within k standard

deviations

2 1-1/4=3/4 75%

3 1-1/9=8/9 88.89%

4 1-1/16=15/16 93.75%

# of standard

deviations, k

Minimum Proportion within k standard

deviations

Minimum Percentage within k standard

deviations

1.5

2.5

3.5


33Bluman, Chapter 3


34Bluman, Chapter 3



Page #135

35Bluman, Chapter 3

Example 3-27: Prices of HomesThe mean price of houses in a certain neighborhood is $50,000, and the standard

deviation is $10,000. Find the price range for which at least 75% of the houses will sell.

Chebyshev’s Theorem states that at least 75% of a data set will fall within 2 standard deviations of the mean.

50,000 – 2(10,000) = 30,000

50,000 + 2(10,000) = 70,000

36Bluman, Chapter 3

At least 75% of all homes sold in the area will have a price range from $30,000 and $75,000.



Page #135

37Bluman, Chapter 3

Example 3-28: Travel AllowancesA survey of local companies found that the mean amount of travel allowance for executives was $0.25 per mile. The standard deviation was 0.02. Using Chebyshev’s theorem, find the minimum percentage of the data values that will fall between $0.20 and $0.30.

38Bluman, Chapter 3

At least 84% of the data values will fall between$0.20 and $0.30.

.30 .25 / .02 2.5.25 .20 / .02 2.5

2.5k

2 21 1/ 1 1/ 2.50.84

k

The Empirical Rule

The empirical rule is only valid for bell-shaped (normal) distributions. The following statements are true.

Approximately 68% of the data values fall within one standard deviation of the mean.

Approximately 95% of the data values fall within two standard deviations of the mean.

Approximately 99.7% of the data values fall within three standard deviations of the mean.

The empirical rule will be revisited later in the chapter on normal probabilities.

Measures of Variation: Empirical Rule (Normal)

41Bluman, Chapter 3

The percentage of values from a data set that fall within k standard deviations of the mean in a normal (bell-shaped) distribution is listed below.

# of standard deviations, k

Proportion within k standard deviations

1 68%

2 95%

3 99.7%

Measures of Variation: Empirical Rule (Normal)

42Bluman, Chapter 3

Homework

Section 3-2 Page 137 1-6 all, 7-17 every other odd, 19, 21 29-41 every other odd

Bluman, Chapter 3 45

Application of Empirical Rule

Given a data set comprised of 5057 measurements that is bell-shaped with a mean of 177. It has a standard deviation of 55. What percentage of the data should lie between 67 and 287?

Bluman, Chapter 3 46

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