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Chapter 3 Numerically Summarizing Data Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section 3.2 Measures of Dispersion Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1. Determine the range of a variable from raw data 2. Determine the standard deviation of a variable from raw data 3. Determine the variance of a variable from raw data 4. Use the Empirical Rule to describe data that are bell shaped 5. Use Chebyshev’s Inequality to describe any data set 3-3 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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? 3-4 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Wait Time at Wendy’s 1.50 2.53 1.88 3.99 0.90 0.79 1.20 2.94 1.90 1.23 1.01 1.46 1.40 1.00 0.92 1.66 0.89 1.33 1.54 1.09 0.94 0.95 1.20 0.99 1.72 0.67 0.90 0.84 0.35 2.00 Wait Time at McDonald’s 3.50 0.00 1.97 0.00 3.08 3-5 0.00 0.26 0.71 0.28 2.75 0.38 0.14 2.22 0.44 0.36 0.43 0.60 4.54 1.38 3.10 1.82 2.33 0.80 0.92 2.19 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 3.04 2.54 0.50 1.17 0.23 (a) The mean wait time in each line is 1.39 minutes. 3-6 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. (b) 3-7 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 • Determine the Range of a Variable from Raw Data 3-8 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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 3-9 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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 3-10 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 • Determine the Standard Deviation of a Variable from Raw Data 3-11 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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). 3-12 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. x1 x2 2 2 L x N 2 N x i 2 N where x1, x2, . . . , xN are the N observations in the population and μ is the population mean. 3-13 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the population standard deviation is x 2 3-14 x 2 i i N N Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Computing a Population Standard Deviation The following data represent the travel times (in minutes) to work for all seven employees of a startup web development company. 23, 36, 23, 18, 5, 26, 43 Compute the population standard deviation of this data. 3-15 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. xi μ xi – μ (xi – μ)2 23 36 23 18 24.85714 24.85714 24.85714 24.85714 -1.85714 11.14286 -1.85714 -6.85714 3.44898 124.1633 3.44898 47.02041 5 26 43 24.85714 24.85714 24.85714 -19.8571 1.142857 18.14286 394.3061 1.306122 329.1633 x i 3-16 x i N 2 2 902.8571 902.8571 11.36 minutes 7 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Using the computational formula, yields the same result. xi (xi )2 23 36 529 1296 23 18 5 26 529 324 25 676 43 1849 Σ xi = 174 Σ (xi)2 = 5228 3-17 x 2 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. x 2 i i N N 174 5228 2 7 7 11.36 minutes 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. x s x 2 i n 1 x1 x x2 x 2 2 L xn x 2 n 1 where x1, x2, . . . , xn are the n observations in the sample and x is the sample mean. 3-18 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the sample standard deviation is x 2 s 3-19 x 2 i i n 1 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. n 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. 3-20 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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. 3-21 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. x xi xi x xi x 2 5 26 22.33333 22.33333 -17.333 3.667 300.432889 13.446889 36 22.33333 13.667 186.786889 x x 500.66667 2 i s 3-22 x i x n 1 2 500.66667 15.82 minutes 2 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Using the computational formula, yields the same result. xi (xi )2 5 26 25 676 36 1296 Σ xi = 67 Σ (xi)2 x 2 x 2 i i n 1 n 67 1997 2 = 1997 3 2 15.82 minutes 3-23 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 3-24 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Comparing Standard Deviations Determine the standard deviation waiting time for Wendy’s and McDonald’s. Which is larger? Why? 3-25 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Wait Time at Wendy’s 1.50 2.53 1.88 3.99 0.90 0.79 1.20 2.94 1.90 1.23 1.01 1.46 1.40 1.00 0.92 1.66 0.89 1.33 1.54 1.09 0.94 0.95 1.20 0.99 1.72 0.67 0.90 0.84 0.35 2.00 Wait Time at McDonald’s 3.50 0.00 1.97 0.00 3.08 3-26 0.00 0.26 0.71 0.28 2.75 0.38 0.14 2.22 0.44 0.36 0.43 0.60 4.54 1.38 3.10 1.82 2.33 0.80 0.92 2.19 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 3.04 2.54 0.50 1.17 0.23 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. 3-27 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 3 • Determine the Variance of a Variable from Raw Data 3-28 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The variance of a variable is the square of the standard deviation. The population variance is σ2 and the sample variance is s2. 3-29 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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. 3-30 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Computing a Population Variance Recall that the population standard deviation (from slide #49) is σ = 11.36 so the population variance is σ2 = 129.05 minutes and that the sample standard deviation (from slide #55) is s = 15.82, so the sample variance is s2 = 250.27 minutes 3-31 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 4 • Use the Empirical Rule to Describe Data that are Bell Shaped 3-32 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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. 3-33 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 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 x used in place of μ and s used in place of σ. 3-34 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 3-35 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Using the Empirical Rule The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor. 41 62 67 60 54 45 3-36 48 75 69 60 54 47 43 77 69 60 55 47 38 58 70 61 56 48 35 82 65 62 56 48 37 39 72 63 56 50 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 44 85 74 64 57 52 44 55 74 64 58 52 44 54 74 64 59 53 (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. 3-37 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. (a) Using a TI-83 plus graphing calculator, we find 57.4 and 11.7 (b) 3-38 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 22.3 34.0 45.7 57.4 69.1 80.8 92.5 (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. 3-39 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 5 • Use Chebyshev’s Inequality to Describe Any Set of Data 3-40 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chebyshev’s Inequality For any data set or distribution, at least 1 1 2 100% of the observations lie within k k standard deviations of the mean, where k is any number greater than 1. That is, at least 1 1 2 100% of the data lie between μ – kσ k and μ + kσ for k > 1. Note: We can also use Chebyshev’s Inequality based on sample data. 3-41 Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Using Chebyshev’s Theorem Using the data from the previous example, use Chebyshev’s Theorem to (a) determine the percentage of patients that have serum HDL within 3 standard deviations of the mean. 1 1 2 100% 88.9% 3 (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% 3-42 Copyright © 2013, 2010 and 2007 Pearson Education, Inc.

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