section 2.4

Section 2.4

Measures of Variation

Larson/Farber 4th ed.

Section 2.4 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)


Example: Finding the Range

A sample of annual salaries (in thousands of dollars) for private school teachers. Find the range of the salaries.

21.8 18.4 20.3 17.6 19.7 18.3 19.4 20.8


Solution: Finding the Range

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

17.6 18.3 18.4 19.4 19.7 20.3 20.8 21.8

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

= 21.8 – 17.6 = 4.2

The range of starting salaries is 4.2 or $4,200.


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


21.8 18.4 20.3 17.6 19.7 18.3 19.4 20.8


Solution:• First determine the mean annual salary.

Solution: Finding the Deviation


• Determine the deviation for each data entry.

Salary, x Deviation: x – μ

19.54

17.6 17.6 - 19.54 = -1.94

18.3 18.3 - 19.54 = -1.24

18.4 18.4 - 19.54 = -1.14

19.4 19.4 - 19.54 = -0.14

19.7 19.7 - 19.54 = 0.16

20.3 20.3 - 19.54 = 0.76

20.8 20.8 - 19.54 = 1.26

21.8 21.8 - 19.54 = 2.26

Σx = 156.3 0.00

Σ(x – μ) = 0

Finding the Sample Variance & Standard Deviation

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.

Finding the Sample Variance & Standard Deviation


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

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

In Words In Symbols

Finding the Population Variance & Standard Deviation

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 – 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 to get the population standard deviation.

In Words In Symbols

Compare Variance

Population Sample

Example: Finding the Standard Deviation


21.8 18.4 20.3 17.6 19.7 18.3 19.4 20.8


Solution: Finding the Standard Deviation


• Determine SSx

• n = 8Salary, x Deviation: x – μ

19.54

1 17.6 17.6 - 19.54 = -1.94 3.75

2 18.3 18.3 - 19.54 = -1.24 1.53

3 18.4 18.4 - 19.54 = -1.14 1.29

4 19.4 19.4 - 19.54 = -0.14 0.02

5 19.7 19.7 - 19.54 = 0.16 0.03

6 20.3 20.3 - 19.54 = 0.76 0.58

7 20.8 20.8 - 19.54 = 1.26 1.59

8 21.8 21.8 - 19.54 = 2.26 5.12

Σx = 156.3 13.92

Solution: Finding the Sample Variance


Sample Variance

The sample variance is 1.99 or roughly 2 or 1,990.

Population Variance

Solution: Finding the Sample Standard Deviation


Sample Standard Deviation

The sample standard deviation is about 1.41 or 1410.

Interpreting Standard Deviation

• Do Problem #26


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)


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

The mean value of land and buildings per acre from a sample of farms is $2400, with a standard deviation of $450. Between what values do about 95% of the data lie? What percent of the values are between $2400 and $3300?


2400 + 2(450) = 3300

2400 - 2(450) = 1500

Solution: Using the Empirical Rule


$1050 $1500 $1950 $2400 $2850 $3300 $3750

34%

13.5%

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

34% + 13.5% = 47.5% of land values are between $2400 and $3300.

Chebychev’s Theorem

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


• k = 2: In any data set, at least

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

• k = 3: In any data set, at least

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

Example: Using Chebychev’s Theorem

The mean time in a women’s 400-meter dash is 57.07 seconds, with a standard deviation of 1.05. Using Chebychev’s Theorem for k = 2, 4, 6.


57.07 - 2(1.05) = 54.97

57.07 + 2(1.05) = 59.17

75% of the women came in between 54.97 and 59.17 seconds.

Standard Deviation for Grouped Data

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.


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

Example: Finding the Standard Deviation for Grouped Data


Do #40 on page 97

Section 2.4 Summary

• Determined the range of a data set• Determined the variance and standard deviation of a

population and of a sample• Used the Empirical Rule and Chebychev’s Theorem

to interpret standard deviation• Approximated the sample standard deviation for

grouped data• Homework 2.4 EOO


section 2.4

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