1. Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Chapter 3
Numerically Summarizing Data
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2. Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Section
3.2
Measures of Dispersion
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3. Objectives Determine the range of a variable from raw data
Determine the standard deviation of a variable from raw data
Determine the variance of a variable from raw data
Use the Empirical Rule to describe data that are bell shaped
Use Chebyshev’s Inequality to describe any data set
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4. (a) What was the mean wait time?
To order food at a McDonald’s restaurant, one must choose from multiple lines, while at Wendy’s Restaurant, one enters a single line. The following data represent the wait time (in minutes) in line for a simple random sample of 30 customers at each restaurant during the lunch hour. For each sample, answer the following:
(a) What was the mean wait time?
(b) Draw a histogram of each restaurant’s wait time.
(c ) Which restaurant’s wait time appears more dispersed? Which line would you prefer to wait in? Why?
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5. Wait Time at McDonald’s
Wait Time at Wendy’s
Wait Time at McDonald’s
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6. (a) The mean wait time in each line is 1.39 minutes.
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7. (b)
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8. Objective 1 Determine the Range of a Variable from Raw Data 3-41
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9. The range, R, of a variable is the difference between the largest data value and the smallest data values. That is,
Range = R = Largest Data Value – Smallest Data Value
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10. EXAMPLE Finding the Range of a Set of Data
The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.
23, 36, 23, 18, 5, 26, 43
Find the range.
Range = 43 – 5
= 38 minutes
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11. Objective 2
Determine the Standard Deviation of a Variable from Raw Data
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12. The population standard deviation of a variable is the square root of the sum of squared deviations about the population mean divided by the number of observations in the population, N. That is, it is the square root of the mean of the squared deviations about the population mean.
The population standard deviation is symbolically represented by σ (lowercase Greek sigma).
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13. where x1, x2, . . . , xN are the N observations in the population and μ is the population mean.
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14. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the population standard deviation is
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15. Compute the population standard deviation of this data.
EXAMPLE Computing a Population Standard Deviation
The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.
23, 36, 23, 18, 5, 26, 43
Compute the population standard deviation of this data.
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16. xi μ
xi – μ
(xi – μ)2 23 36 18 5 26 43
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17. Using the computational formula, yields the same result. xi (xi )223
529 36
1296 18
324 5 25 26
676 43
1849
Σ xi = 174
Σ (xi)2 = 5228
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18. The sample standard deviation, s, of a variable is the square root of the sum of squared deviations about the sample mean divided by n – 1, where n is the sample size.
where x1, x2, , xn are the n observations in the sample and is the sample mean.
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19. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the sample standard deviation is
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20. We call n - 1 the degrees of freedom because the first n - 1 observations have freedom to be whatever value they wish, but the nth value has no freedom. It must be whatever value forces the sum of the deviations about the mean to equal zero.
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21. Find the sample standard deviation.
EXAMPLE Computing a Sample Standard Deviation
Here are the results of a random sample taken from the travel times (in minutes) to work for all seven employees of a start-up web development company:
5, 26, 36
Find the sample standard deviation.
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22. xi 5 26
3.667 36
13.667
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23. Using the computational formula, yields the same result. xi (xi )25 25 26
676 36
1296
Σ xi = 67
Σ (xi)2 = 1997
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25. EXAMPLE Comparing Standard Deviations
Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why?
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26. Wait Time at McDonald’s
Wait Time at Wendy’s
Wait Time at McDonald’s
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27. Sample standard deviation for Wendy’s: 0.738 minutes
EXAMPLE Comparing Standard Deviations
Sample standard deviation for Wendy’s:
0.738 minutes
Sample standard deviation for McDonald’s:
1.265 minutes
Recall from earlier that the data is more dispersed for McDonald’s resulting in a larger standard deviation.
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28. Objective 3 Determine the Variance of a Variable from Raw Data 3-61
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29. The variance of a variable is the square of the standard deviation
The variance of a variable is the square of the standard deviation. The population variance is σ2 and the sample variance is s2.
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30. EXAMPLE Computing a Population Variance
The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company.
23, 36, 23, 18, 5, 26, 43
Compute the population and sample variance of this data.
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31. EXAMPLE Computing a Population Variance
Recall that the population standard deviation (from slide #49) is σ = so the population variance is σ2 = minutes
and that the sample standard deviation (from slide #55) is s = 15.82, so the sample variance is s2 = minutes
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32. Objective 4
Use the Empirical Rule to Describe Data That Are Bell Shaped
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33. If a distribution is roughly bell shaped, then
The Empirical Rule
If a distribution is roughly bell shaped, then
• Approximately 68% of the data will lie within 1 standard deviation of the mean. That is, approximately 68% of the data lie between μ – 1σ and μ + 1σ.
• Approximately 95% of the data will lie within 2 standard deviations of the mean. That is, approximately 95% of the data lie between μ – 2σ and μ + 2σ.
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34. If a distribution is roughly bell shaped, then
The Empirical Rule
If a distribution is roughly bell shaped, then
• Approximately 99.7% of the data will lie within 3 standard deviations of the mean. That is, approximately 99.7% of the data lie between μ – 3σ and μ + 3σ.
Note: We can also use the Empirical Rule based on sample data with used in place of μ and s used in place of σ.
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36. EXAMPLE Using the Empirical Rule
The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor.
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37. (a) Compute the population mean and standard deviation.
(b) Draw a histogram to verify the data is bell-shaped.
(c) Determine the percentage of all patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule.
(d) Determine the percentage of all patients that have serum HDL between 34 and 69.1 according to the Empirical Rule.
(e) Determine the actual percentage of patients that have serum HDL between 34 and 69.1.
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38. (a) Using a TI-83 plus graphing calculator, we find
(b)
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39.
(c) According to the Empirical Rule, 99.7% of the all patients that have serum HDL within 3 standard deviations of the mean.
(d) 13.5% + 34% + 34% = 81.5% of all patients will have a serum HDL between 34.0 and 69.1 according to the Empirical Rule.
(e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and 69.1.
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40. Objective 5 Use Chebyshev’s Inequality to Describe Any Set of Data
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41. Chebyshev’s Inequality For any data set or distribution, at least
of the observations lie within k
standard deviations of the mean, where k is any number greater than 1. That is, at least
of the data lie between μ – kσ
and μ + kσ for k > 1.
Note: We can also use Chebyshev’s Inequality based on sample data.
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42. EXAMPLE Using Chebyshev’s Theorem
Using the data from the previous example, use Chebyshev’s Theorem to
determine the percentage of patients that have serum HDL within 3 standard deviations of the mean.
(b) determine the actual percentage of patients that have serum HDL between 34 and 80.8 (within 3 SD of mean).
52/54 ≈ 0.96 ≈ 96%
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