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 3-18/19Page #123

5Bluman, Chapter 3

Example 3-18/19: Outdoor PaintTwo 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

210 35Brand A: 6

60 10 50

XN

R

210 35Brand B: 6

45 25 20

X

RN

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

22 X

N

2XN

10Bluman, Chapter 3

Chapter 3Data Description

Section 3-2Example 3-21Page #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

6256252252525

2251750

17506

17.1

22

17506

291.7

Xn

PopulationVariance

PopulationStandard Deviation

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

The sample standard deviation is

2

2

1X X

sn

2

1X X

sn

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 Xns

n n

2s s

Chapter 3Data Description

Section 3-2Example 3-23Page #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 Xns

n n

22 75.66 958.94

6 5

s2 1.28

1.13ss

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•Xm

2

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

7.68120

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 data2. Consistency3. Determine the number (or %) of data

within an interval.4. Many other used to be discussed fully

second semester!

Measures of Variation: Coefficient of VariationThe 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%sCVARX

25Bluman, Chapter 3

Chapter 3Data Description

Section 3-2Example 3-25Page #132

26Bluman, Chapter 3

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.

5 100% 5.7% Sales87

CVar

773 100% 14.8% Commissions5225

CVar

Measures of Variation: Range Rule of ThumbThe Range Rule of Thumb approximates the standard deviation as

when the distribution is unimodal and approximately symmetric.

4Ranges

28Bluman, Chapter 3

Measures of Variation: Range Rule of ThumbUse to approximate the lowest value and to approximate the highest value in a data set.

29Bluman, Chapter 3

2X s2X s

Example: 10, 12X Range

12 34

s

10 2 3 410 2 3 16

LOWHIGH

Using SymbolsLet 20 and s=3, calculator the following:

30Bluman, Chapter 3

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

31Bluman, Chapter 3

Measures of Variation: Chebyshev’s Theorem

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

1.52.53.5

Measures of Variation: Chebyshev’s Theorem

33Bluman, Chapter 3

Measures of Variation: Chebyshev’s Theorem

34Bluman, Chapter 3

Chapter 3Data Description

Section 3-2Example 3-27Page #135

35Bluman, Chapter 3

Example 3-27: Prices of HomesThe mean price of houses in a certain neighborhood is $50,000, and the standarddeviation 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) = 30,00050,000 + 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.

Chapter 3Data Description

Section 3-2Example 3-28Page #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