Section 3-2

Measures of Variation

Section 3-2 Objectives

• Determine the range of a data set

• Determine the variance and standard deviation of a population and of a sample

• Use the Empirical Rule and Chebychev’s Theorem to interpret standard deviation

• Approximate the sample standard deviation for grouped data

Range

Range

• The difference between the maximum and minimum data entries in the set.

• The data must be quantitative.

• Range = (Max. data entry) – (Min. data entry)

Solution: Finding the Range

• Ordering the data helps to find the least and greatest salaries.

37 38 39 41 41 41 42 44 45 47

• Range = (Max. salary) – (Min. salary)

= 47 – 37 = 10

The range of starting salaries is 10 or $10,000.

minimum maximum

Deviation, Variance, and Standard Deviation

Deviation

• The difference between the data entry, x, and the mean of the data set.

• Population data set: Deviation of x = x – μ

• Sample data set: Deviation of x x x

Example: Finding the Deviation

A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the deviation of the starting salaries.

Starting salaries (1000s of dollars)

41 38 39 45 47 41 44 41 37 42

Solution:• First determine the mean starting salary.

41541.5

10

x

N

6 of 149© 2012 Pearson Education, Inc. All rights reserved.

Solution: Finding the Deviation

• Determine the deviation for each data entry.

Salary ($1000s), xDeviation ($1000s)

x – μ

41 41 – 41.5 = –0.5

38 38 – 41.5 = –3.5

39 39 – 41.5 = –2.5

45 45 – 41.5 = 3.5

47 47 – 41.5 = 5.5

41 41 – 41.5 = –0.5

44 44 – 41.5 = 2.5

41 41 – 41.5 = –0.5

37 37 – 41.5 = –4.5

42 42 – 41.5 = 0.5Σx = 415 Σ(x – μ) = 0

Finding the Population Variance & Standard Deviation

In Words In Symbols

1. Find the mean of the population data set.

2. Find the deviation of each entry.

3. Square each deviation.

4. Add to get the sum of squares.

x

N

x – μ

(x – μ)2

SSx = Σ(x – μ)2

Finding the Population Variance & Standard Deviation

5. Divide by N to get the population variance.

6. Find the square root of the variance to get the population standard deviation.

22 ( )x

N

2( )x

N

In Words In Symbols

Example: Finding the Population Standard Deviation

A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the population variance and standard deviation of the starting salaries.

Starting salaries (1000s of dollars)

41 38 39 45 47 41 44 41 37 42

Recall μ = 41.5.

Solution: Finding the Population Standard Deviation

• Determine SSx

• N = 10

Salary, x Deviation: x – μ Squares: (x – μ)2

41 41 – 41.5 = –0.5 (–0.5)2 = 0.25

38 38 – 41.5 = –3.5 (–3.5)2 = 12.25

39 39 – 41.5 = –2.5 (–2.5)2 = 6.25

45 45 – 41.5 = 3.5 (3.5)2 = 12.25

47 47 – 41.5 = 5.5 (5.5)2 = 30.25

41 41 – 41.5 = –0.5 (–0.5)2 = 0.25

44 44 – 41.5 = 2.5 (2.5)2 = 6.25

41 41 – 41.5 = –0.5 (–0.5)2 = 0.25

37 37 – 41.5 = –4.5 (–4.5)2 = 20.25

42 42 – 41.5 = 0.5 (0.5)2 = 0.25

Σ(x – μ) = 0 SSx = 88.5

Solution: Finding the Population Standard Deviation

Population Variance

•

Population Standard Deviation

•

22 ( ) 88.5

8.910

x

N

2 8.85 3.0

The population standard deviation is about 3.0, or $3000.

12 of 149

Deviation, Variance, and Standard Deviation

Sample Variance

•

Sample Standard Deviation

•

22 ( )

1

x xs

n

22 ( )

1

x xs s

n

Finding the Sample Variance & Standard Deviation

In Words In Symbols

1. Find the mean of the sample data set.

2. Find the deviation of each entry.

3. Square each deviation.

4. Add to get the sum of squares.

xx

n

2( )xSS x x

2( )x x

x x

Finding the Sample Variance & Standard Deviation

5. Divide by n – 1 to get the sample variance.

6. Find the square root of the variance to get the sample standard deviation.

In Words In Symbols2

2 ( )

1

x xs

n

2( )

1

x xs

n

Example: Finding the Sample Standard Deviation

The starting salaries are for the Chicago branches of a corporation. The corporation has several other branches, and you plan to use the starting salaries of the Chicago branches to estimate the starting salaries for the larger population. Find the sample standard deviation of the starting salaries.

Starting salaries (1000s of dollars)

41 38 39 45 47 41 44 41 37 42

Solution: Finding the Sample Standard Deviation

• Determine SSx

• n = 10

Salary, x Deviation: x – μ Squares: (x – μ)2

41 41 – 41.5 = –0.5 (–0.5)2 = 0.25

38 38 – 41.5 = –3.5 (–3.5)2 = 12.25

39 39 – 41.5 = –2.5 (–2.5)2 = 6.25

45 45 – 41.5 = 3.5 (3.5)2 = 12.25

47 47 – 41.5 = 5.5 (5.5)2 = 30.25

41 41 – 41.5 = –0.5 (–0.5)2 = 0.25

44 44 – 41.5 = 2.5 (2.5)2 = 6.25

41 41 – 41.5 = –0.5 (–0.5)2 = 0.25

37 37 – 41.5 = –4.5 (–4.5)2 = 20.25

42 42 – 41.5 = 0.5 (0.5)2 = 0.25

Σ(x – μ) = 0 SSx = 88.5

Solution: Finding the Sample Standard Deviation

Sample Variance

•

Sample Standard Deviation

•

22 ( ) 88.5

9.81 10 1

x xs

n

2 88.53.1

9s s

The sample standard deviation is about 3.1, or $3100.

Example: Using Technology to Find the Standard Deviation

Sample office rental rates (in dollars per square foot per year) for Miami’s central business district are shown in the table. Use a calculator to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.)

Office Rental Rates

35.00 33.50 37.00

23.75 26.50 31.25

36.50 40.00 32.00

39.25 37.50 34.75

37.75 37.25 36.75

27.00 35.75 26.00

37.00 29.00 40.50

24.50 33.00 38.00

Solution: Using Technology to Find the Standard Deviation

Sample Mean

Sample Standard Deviation

Interpreting Standard Deviation

• Standard deviation is a measure of the typical amount an entry deviates from the mean.

• The more the entries are spread out, the greater the standard deviation.

Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule)

For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics:

• About 68% of the data lie within one standard deviation of the mean.

• About 95% of the data lie within two standard deviations of the mean.

• About 99.7% of the data lie within three standard deviations of the mean.

Interpreting Standard Deviation: Empirical Rule (68 – 95 – 99.7 Rule)

3x s x s 2x s 3x sx s x2x s

68% within 1 standard deviation

34% 34%

99.7% within 3 standard deviations

2.35% 2.35%

95% within 2 standard deviations

13.5% 13.5%

Example: Using the Empirical Rule

In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64.3 inches, with a sample standard deviation of 2.62 inches. Estimate the percent of the women whose heights are between 59.06 inches and 64.3 inches.

Solution: Using the Empirical Rule

• Because the distribution is bell-shaped, you can use the Empirical Rule.

34% + 13.5% = 47.5% of women are between 59.06 and 64.3 inches tall.

Chebychev’s Theorem

• The portion of any data set lying within k standard deviations (k > 1) of the mean is at least:

2

11

k

• k = 2: In any data set, at least 2

1 31 or 75%

2 4

of the data lie within 2 standard deviations of the mean.

• k = 3: In any data set, at least 2

1 81 or 88.9%

3 9

of the data lie within 3 standard deviations of the mean.

Example: Using Chebychev’s Theorem

The age distribution for Florida is shown in the histogram. Apply Chebychev’s Theorem to the data using k = 2. What can you conclude?

Solution: Using Chebychev’s Theorem

k = 2: μ – 2σ = 39.2 – 2(24.8) = – 10.4 (use 0 since age can’t be negative)

μ + 2σ = 39.2 + 2(24.8) = 88.8

At least 75% of the population of Florida is between 0 and 88.8 years old.

Standard Deviation for Grouped Data

Sample standard deviation for a frequency distribution

•

• When a frequency distribution has classes, estimate the sample mean and the sample standard deviation by using the midpoint of each class.

2( )

1

x x fs

n

where n = Σf (the number of entries in the data set)

Example: Finding the Standard Deviation for Grouped Data

You collect a random sample of the number of children per household in a region. Find the sample mean and the sample standard deviation of the data set.

Number of Children in 50 Households

1 3 1 1 1

1 2 2 1 0

1 1 0 0 0

1 5 0 3 6

3 0 3 1 1

1 1 6 0 1

3 6 6 1 2

2 3 0 1 1

4 1 1 2 2

0 3 0 2 4

x f xf

0 10 0(10) = 0

1 19 1(19) = 19

2 7 2(7) = 14

3 7 3(7) =21

4 2 4(2) = 8

5 1 5(1) = 5

6 4 6(4) = 24

Solution: Finding the Standard Deviation for Grouped Data

• First construct a frequency distribution.

• Find the mean of the frequency distribution.

Σf = 50 Σ(xf )= 91

911.8

50

xfx

n

The sample mean is about 1.8 children.

Solution: Finding the Standard Deviation for Grouped Data

• Determine the sum of squares.

x f

0 10 0 – 1.8 = –1.8 (–1.8)2 = 3.24 3.24(10) = 32.40

1 19 1 – 1.8 = –0.8 (–0.8)2 = 0.64 0.64(19) = 12.16

2 7 2 – 1.8 = 0.2 (0.2)2 = 0.04 0.04(7) = 0.28

3 7 3 – 1.8 = 1.2 (1.2)2 = 1.44 1.44(7) = 10.08

4 2 4 – 1.8 = 2.2 (2.2)2 = 4.84 4.84(2) = 9.68

5 1 5 – 1.8 = 3.2 (3.2)2 = 10.24 10.24(1) = 10.24

6 4 6 – 1.8 = 4.2 (4.2)2 = 17.64 17.64(4) = 70.56

x x 2( )x x 2( )x x f

2( ) 145.40x x f

Solution: Finding the Standard Deviation for Grouped Data

• Find the sample standard deviation.

x x 2( )x x 2( )x x f2( ) 145.401.7

1 50 1

x x fs

n

The standard deviation is about 1.7 children.