Measures of Variation

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Section 2.4

Section 2.4 Objectives

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Determine the range of a data setDetermine the variance and standard

deviation of a population and of a sampleUse the Empirical Rule and Chebychev’s

Theorem to interpret standard deviationApproximate the sample standard deviation

for grouped data

Range

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RangeThe difference between the maximum and

minimum data entries in the set.The data must be quantitative.Range = (Max. data entry) – (Min. data entry)

Example: Finding the Range

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

Starting salaries (1000s of dollars)41 38 39 45 47 41 44 41 37

42

Solution: Finding the Range

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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

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DeviationThe difference between the data entry, x,

and the mean of the data set.Population data set:

Deviation of x = x – μSample data set:

Deviation of x = x – x

Example: Finding the Deviation

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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

Solution: Finding the Deviation

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Determine the deviation for each data entry.

Salary ($1000s), x Deviation: 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 = 0.5

Σx = 415 Σ(x – μ) = 0

Deviation, Variance, and Standard Deviation

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Population Variance

Population Standard Deviation

22 ( )x

N

Sum of squares, SSx

22 ( )x

N

Finding the Population Variance & Standard Deviation

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In Words In Symbols

1. Find the mean of the population data set.

2. Find 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

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5. Divide by N to get the population variance.

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

22 ( )x

N

2( )x

N

In Words In Symbols

Example: Finding the Population Standard Deviation

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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

42Recall μ = 41.5.

Solution: Finding the Population Standard Deviation

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Determine SSx N = 10 Note that SSx = Σ(x – μ)2

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

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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.

Deviation, Variance, and Standard Deviation

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Sample Variance

Sample Standard Deviation

22 ( )

1

x xs

n

22 ( )

1

x xs s

n

Finding the Sample Variance & Standard Deviation

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In Words In Symbols

1. Find the mean of the sample data set.

2. Find 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

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

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

In Words In Symbols2

2 ( )

1

x xs

n

2( )

1

x xs

n

Sample Standard Deviation Shortcut Formula

n (n - 1)

s =n (x2) - (x)2

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Symbolsfor Standard Deviation

Sample Population

x

xn

s

Sx

xn-1

Book

Some graphicscalculators

Somenon-graphicscalculators

Textbook

Some graphicscalculators

Somenon-graphics

calculators

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Example: Finding the Sample Standard Deviation

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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

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Solution: Finding the Sample Standard Deviation

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Determine SSx

n = 10 Note that

Salary, x Deviation: Squares:

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( ) = 0 SSx = 88.5

x x

x x

2( )x x

2( )xSS x x

Solution: Finding the Sample Standard Deviation

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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

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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 or a computer 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

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Sample Mean

Sample Standard Deviation

Interpreting Standard Deviation

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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.

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

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

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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)

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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

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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 inches, with a sample standard deviation of 2.71 inches. Estimate the percent of the women whose heights are between 64 inches and 69.42 inches.

Solution: Using the Empirical Rule

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3x s x s 2x s 3x sx s x2x s55.87 58.58 61.29 64 66.71 69.42 72.13

34%

13.5%

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

34% + 13.5% = 47.5% of women are between 64 and 69.42 inches tall.

Chebychev’s Theorem

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

2

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k

• k = 2: In any data set, at least2

1 31 or 75%

2 4

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

• k = 3: In any data set, at least2

1 81 or 88.9%

3 9

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

Example: Using Chebychev’s Theorem

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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

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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.

Estimation of Standard DeviationRange Rule of Thumb

x - 2s x x + 2s

Range 4sor

(minimumusual value)

(maximum usual value)

Range

4s =

highest value - lowest value

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Standard Deviation for Grouped Data

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Sample standard deviation for a frequency distribution

When a frequency distribution has classes, estimate the sample mean and 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

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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

Solution: Finding the Standard Deviation for Grouped Data

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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.

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

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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

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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.

Standard Deviation from a Frequency TableShortcut Formula

n (n - 1)S

=

n [(f • x 2)] -[(f • x)]2

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Practice QuestionsQ(2.11)

Compute the sample variance and

sampleStandard deviation.

The number of incidents where policies were needed for a sample of ten schools in Allegheny County is 7, 37, 3, 8, 48, 11, 6, 0, 10, 3. Assume the data represent samples.

Practice QuestionsQ(2.12)Compute the variance and standard

deviation of thegiven grouped data.

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2539-6021475-5390411-4742347-4100283-3465219-2820155-218291-1541327-90fNumber

Practice QuestionsQ(2.13)The mean of a distribution is 20 and the

standarddeviation is 2. Use Chebyshev’s Theorem

to answer thefollowing questions. (1)At least what percentage of the values

will fall between 10 and 30?(2)At least what percentage of the values

will fall between 12 and 28?

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Practice QuestionsQ(2.14)The average U.S yearly per capita consumption of

citrusfruits is 26.8 pounds. Suppose that the distribution

offruits amount consumed is bell-shaped with

standard deviation of 4.2 pounds.

What percentage of Americans would you expect to

consume more than 31 pounds of citrus fruit per year?

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Section 2.4 Summary

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Determined the range of a data setDetermined the variance and standard

deviation of a population and of a sampleUsed the Empirical Rule and Chebychev’s

Theorem to interpret standard deviationApproximated the sample standard

deviation for grouped data