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
Larson/Farber 4th ed.
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)
Larson/Farber 4th ed.
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
Larson/Farber 4th ed.
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
minimum
maximum
• Range = (Max. salary) – (Min. salary)
= 21.8 – 17.6 = 4.2
The range of starting salaries is 4.2 or $4,200.
Larson/Farber 4th ed.
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
Larson/Farber 4th ed.
Example: Finding the Deviation
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:
• First determine the mean annual salary.
Larson/Farber 4th ed.
Solution: Finding the Deviation
Salary, x
• Determine the
deviation for each
data entry.
19.54
Σ(x – μ) = 0
Σx =
Larson/Farber 4th ed.
Deviation: x – μ
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
156.3
0.00
Finding the Sample Variance & Standard
Deviation
In Words
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.
Larson/Farber 4th ed.
In Symbols
Finding the Sample Variance & Standard
Deviation
In Words
5. Divide by n – 1 to get the
sample variance.
6. Find the square root to get
the sample standard
deviation.
Larson/Farber 4th ed.
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.
x–μ
3. Square each deviation.
4. Add to get the sum of
squares.
Larson/Farber 4th ed.
(x – μ)2
SSx = Σ(x – μ)2
Finding the Population Variance &
Standard Deviation
In Words
5. Divide by N to get the
population variance.
6. Find the square root to get
the population standard
deviation.
Larson/Farber 4th ed.
In Symbols
Compare Variance
Population
Sample
Example: Finding the Standard Deviation
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
Larson/Farber 4th ed.
Solution: Finding the Standard Deviation
• Determine SSx
• n=8
Larson/Farber 4th ed.
Salary, 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
Population Variance
The sample variance is 1.99 or roughly 2 or 1,990.
Larson/Farber 4th ed.
Solution: Finding the Sample Standard
Deviation
Sample Standard Deviation
The sample standard deviation is about 1.41 or 1410.
Larson/Farber 4th ed.
Interpreting Standard Deviation
• Do Problem #26
Larson/Farber 4th ed.
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.
Larson/Farber 4th ed.
Interpreting Standard Deviation:
Empirical Rule (68 – 95 – 99.7 Rule)
99.7% within 3 standard deviations
95% within 2 standard deviations
68% within 1
standard deviation
34%
34%
2.35%
2.35%
13.5%
Larson/Farber 4th ed.
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
Larson/Farber 4th ed.
Solution: Using the Empirical Rule
• Because the distribution is bell-shaped, you can use the
Empirical Rule.
34%
13.5%
$1050
$1500
$1950
$2400
$2850
$3300
34% + 13.5% = 47.5% of land values are between
$2400 and $3300.
Larson/Farber 4th ed.
$3750
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.
Larson/Farber 4th ed.
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.
Larson/Farber 4th ed.
Standard Deviation for Grouped Data
Sample standard deviation for a frequency
distribution
where n= Σf (the number of
entries in the data set)
•
• When a frequency distribution has classes, estimate the
sample mean and standard deviation by using the
midpoint of each class.
Larson/Farber 4th ed.
Example: Finding the Standard Deviation
for Grouped Data
Do #40 on page 97
Larson/Farber 4th ed.
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
Larson/Farber 4th ed.