section 2.4
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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 - PowerPoint PPT PresentationTRANSCRIPT
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Section 2.4
Measures of Variation
Larson/Farber 4th ed.
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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.
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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.
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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.
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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
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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.
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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.
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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
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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.
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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
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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
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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
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Compare Variance
Population Sample
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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.
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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
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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
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Solution: Finding the Sample Standard Deviation
Larson/Farber 4th ed.
Sample Standard Deviation
The sample standard deviation is about 1.41 or 1410.
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Interpreting Standard Deviation
• Do Problem #26
Larson/Farber 4th ed.
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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.
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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%
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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?
Larson/Farber 4th ed.
2400 + 2(450) = 3300
2400 - 2(450) = 1500
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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.
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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.
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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.
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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)
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Example: Finding the Standard Deviation for Grouped Data
Larson/Farber 4th ed.
Do #40 on page 97
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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.