Download - 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
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
• 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.
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
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
Larson/Farber 4th ed.
Solution:• First determine the mean annual salary.
Solution: Finding the Deviation
Larson/Farber 4th ed.
• 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
Larson/Farber 4th ed.
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
Larson/Farber 4th ed.
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
Larson/Farber 4th ed.
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
Larson/Farber 4th ed.
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
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
Larson/Farber 4th ed.
• 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
Larson/Farber 4th ed.
Sample Variance
The sample variance is 1.99 or roughly 2 or 1,990.
Population Variance
Solution: Finding the Sample Standard Deviation
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Sample Standard Deviation
The sample standard deviation is about 1.41 or 1410.
Interpreting Standard Deviation
• Do Problem #26
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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:
Larson/Farber 4th ed.
• 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)
Larson/Farber 4th ed.
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?
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2400 + 2(450) = 3300
2400 - 2(450) = 1500
Solution: Using the Empirical Rule
Larson/Farber 4th ed.
$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:
Larson/Farber 4th ed.
• 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.
Larson/Farber 4th ed.
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.
Larson/Farber 4th ed.
where n= Σf (the number of entries in the data set)
Example: Finding the Standard Deviation for Grouped Data
Larson/Farber 4th ed.
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
Larson/Farber 4th ed.