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740 CHAPTER 15 y Descriptive Statistics 15.3 Measures of Dispersion Objectives 1. Compute the range of a data set. 2. Understand how the standard deviation measures the spread of a distribution. 3. Use the coefficient of variation to compare the standard deviations of different distributions. Let’s imagine that you are a cardiac surgeon faced with deciding which of two heart pacemaker batteries you should choose. Because you must operate to replace a failing battery, it is important that you choose the one that will last the longest. Assume that battery A lasts for a mean time of 45,000 hours (slightly more than 5 years) and battery B has a mean time of 46,000 hours. On the surface, it would seem that you should choose battery B. However, suppose that we also tell you that in testing the batteries, we found that all of A’s times were within 500 hours of the mean, but B’s times varied widely. In fact, some of B’s times were as much as 2,500 hours below the mean. So B’s times could be as low as 46,000 - 2,500 = 43,500 hours, whereas A’s times were never below 44,500. Based on this information, it appears that battery A is the better choice. The point of our story is that although the mean and median tell you something about a distribution, they do not tell the whole story. For example, the following two distributions both have a mean and median of 25, but Y’s values are much more spread out than X’s: X: 24, 25, 25, 25, 25, 26 Y: 1, 2, 3, 47, 48, 49 From these examples, it is clear that we need to develop some method for measuring the spread of a distribution. KEY POINT The range is a crude measure of the spread of a data set. The Range of a Data Set It is clear from the discussion of pacemaker batteries that a numerical measure of the dispersion, or spread, of a data set can provide useful information. One simple way to describe the spread of a set of data is to subtract the smallest data value from the largest. D E F I N I T I O N The range of a data set is the difference between the largest and smallest data values in the set. EXAMPLE 1 Comparing Heights Find the range of the heights of the people listed in the accompanying table. Person Height Height in Inches Leonid Stadynk (World’s Tallest Person) 8 feet, 6 inches 102 inches LaDainian Tomlinson 5 feet, 10 inches 70 inches Madge Bester (World’s Shortest Person) 2 feet, 2 inches 26 inches LeBron James 6 feet, 8 inches 80 inches SOLUTION: The range of these data is range = largest data value - smallest data value = 102 - 26 = 76 inches = 6 feet, 4 inches. ] Copyright © 2010 Pearson Education, Inc. 15.3 y Measures of Dispersion 741 HISTORICAL HIGHLIGHT ¶ ¶ ¶ The Origins of Statistics Ring a ring of roses, A pocket full of posies, Asha Asha, We all fall down.* It may have been King Henry VII’s fear of the dreaded Black Plague that prompted him to begin publishing weekly Bills of Mortality in 1532. John Graunt, a merchant, noticed patterns in these reports regarding deaths due to accident, suicide, and disease and concluded that social phenomena do not occur randomly. He published his observations in a paper titled, “Natural and Political Observations . . . Made upon the Bills of Mortality.” King Charles II, impressed by Graunt’s work, nominated him to the Royal Society of London even though he had few academic credentials. Edmund Halley, the noted English astronomer, continued the work of Graunt in a paper titled “An Estimate of the Degrees of the Mortality of Mankind, Drawn from Curious Tables of the Births and Funerals at the City of Breslaw; with an Attempt to Ascertain the Price of Annuities upon Lives.” By studying the mathematics of life expectancy, Halley helped lay the foundations for actuarial science, which insurance companies use today to determine their premiums. The range is generally not useful to measure the spread of a distribution because it can be influenced by a single outlier. For example, the range of the distribution 2, 2, 2, 2, 3, 4, 100 is 100 - 2 = 98. Standard Deviation KEY POINT The standard deviation is a reliable measure of dispersion. Data Value, x Deviation from Mean, x ⴚ xq 16 -1 14 -3 12 -5 21 4 22 5 Total 0 TABLE 15.11 Deviations from the mean for data values in distribution A. A better way to find the spread of data is to use the standard deviation, a measure based on calculating the distance of each data value from the mean. D E F I N I T I O N If x is a data value in a set whose mean is xq, then x - xq is called x ’s deviation from the mean. To illustrate the deviation from the mean, consider distribution A: 16, 14, 12, 21, 22, whose mean is 17. We list the deviation of each data value from the mean in Table 15.11. It might seem reasonable to measure the spread in A by averaging the deviations from the mean in Table 15.11. However, this will not work. As you see in Figure 15.10, for scores above the mean that have positive deviations from the mean, there must be scores below the mean that have negative deviations from the mean. When we add the positive and negative deviations, they cancel each other, giving a total of zero. To avoid this cancellation, we square each of these deviations, as shown in Table 15.12. (–5) 12 (–3) 14 (–1) 16 12 14 16 (+4) (+5) 21 22 18 20 22 Mean = 17 FIGURE 15.10 Values having positive and negative deviations from the mean must balance about the mean. *Some see this rhyme as describing the rosy rash, the pockets full of medicinal herbs, the sneezing and eventual death associated with the Black Plague. If so, it owes its origin, like statistics, to that horrible time in European history. Copyright © 2010 Pearson Education, Inc. 742 CHAPTER 15 y Descriptive Statistics Data Value, x Deviation from Mean, x ⴚ xq Deviation Squared, ( x ⴚ xq ) 2 16 -1 1 14 -3 9 12 -5 25 21 4 16 22 5 25 TABLE 15.12 The squares of deviations from the mean for distribution A. If we now average* these squared deviations, we get 76 1 9 25 16 25 19. 4 51 n1 Clearly, this number is too large to represent the spread of scores from the mean. To compensate for the fact that we squared the deviations in doing our calculations, we take the square root of this number to get 119 L 4.36. This number is a more reasonable measurement of the way scores vary from the mean. The quantity that we just calculated is called the standard deviation. D E F I N I T I O N We denote the standard deviation of a sample of n data values by s,† which is defined as follows: s= a (x - xq ) A n-1 2 We compute the standard deviation in four steps. C O M P U T I N G T H E S T A N D A R D D E V I A T I O N To compute the standard deviation for a sample consisting of n data values, do the following: 1. Compute the mean of the data set; call it xq. 2. Find ( x - xq )2 for each score x in the data set. 3. Add the squares found in step 2 and divide this sum by n - 1; that is, find 2 a1x - xq2 , n-1 which is called the variance. 4. Compute the square root of the number found in step 3. EXAMPLE 2 Calculating the Standard Deviation Dunder Mifflin has hired six interns. After 4 months, their work records show the following number of work days missed for each worker: 0, 2, 1, 4, 2, 3 Find the standard deviation of this data set. *When doing this calculation for a sample (as opposed to a population), statisticians divide by n - 1 instead of n. There are technical reasons for doing this that are beyond the scope of this text. In doing this calculation for a population, we would divide by n rather than n - 1. †We represent the standard deviation of a population by s (instead of s), which we calculate using the formula s= A 2 a(x - m) . n Copyright © 2010 Pearson Education, Inc. 15.3 y Measures of Dispersion 743 0 + 2 + 1 + 4 + 2 + 3 12 = = 2. We calculate the 6 6 squares of the deviations of the data values from the mean in the following table. SOLUTION: The mean of this data set is Number of Days Missed Deviation from Mean Square of Deviation from the Mean 0 -2 4 2 0 0 1 -1 1 4 2 4 2 0 0 3 1 1 ©(x - 2)2 = 10 There are six scores in the distribution, so the standard deviation is Quiz Yourself s= 9 Find the standard deviation of the following sample of data values: 3, 4, 5, 6, 4, 2, 0, 8, 4 2 10 a(x - 2) = L 1.41. A 6-1 A5 n-1 This relatively small standard deviation shows that the data values are not spread too far from the mean of 2. Now try Exercises 5 to 14. ] 9 In Example 3, we calculate the standard deviation of a distribution using a frequency table. EXAMPLE 3 Using a Frequency Table to Compute the Standard Deviation Anya is considering investing in WebSoft, a software company that specializes in Internet applications. She intends to compute the standard deviation of its recent price changes to measure how steady its stock price has been recently. The following are the closing prices for the stock for the past 20 trading sessions: 37, 39, 39, 40, 40, 38, 38, 39, 40, 41, 41, 39, 41, 42, 42, 44, 39, 40, 40, 41 What is the standard deviation for this data set? S O LU T I O N : In Table 15.13, we see that the sum of the 20 closing prices is 800, so the mean is 800 20 = 40. We have added columns to the table for the deviations from the mean, the squares of these deviations, and so on. Closing Price, x Frequency, f Product, x·f Deviation, (x ⴚ 40) Deviation Squared, (x ⴚ 40)2 Product, (x ⴚ 40)2 · f 37 1 37 -3 9 9 38 2 76 -2 4 8 39 5 195 1 5 40 5 200 -1 0 0 0 41 4 164 1 1 4 42 2 84 2 4 8 44 1 44 4 16 16 © f = 20 ©(x · f ) = 800 ©(x - 40)2 · f = 50 TABLE 15.13 Computations necessary to find the standard deviation of WebSoft closing prices. Copyright © 2010 Pearson Education, Inc. 744 CHAPTER 15 y Descriptive Statistics When calculating the standard deviation, we must multiply the squared deviation of each price from the mean by its frequency. We list these products in the column labeled “Product, (x - 40)2 · f,” and show the sum of these products at the bottom of the column. We can now calculate the standard deviation. Remember that ©f is the number of data values that we have been representing by n. The standard deviation is therefore s= 2# 50 a(x - 40) f = L 1.62. A A 19 n-1 This relatively small standard deviation indicates that the closing prices for the WebSoft stock have not been varying very much lately. If Anya is a cautious investor, she will find the stability of WebSoft’s prices appealing. Now try Exercises 15 to 20. ] We summarize the method we use for finding the standard deviation in the following formula. F O R M U L A F O R C O M P U T I N G T H E S A M P L E S TA N DA R D D E V I AT I O N F O R A F R E Q U E N C Y D I S T R I B U T I O N We calculate the standard deviation, s, of a sample that is given as a frequency distribution as follows: s= 2# a(x - qx ) f A n-1 where xq is the mean of the distribution, f is the frequency of data value x, and n = ©f, the number of data values in the distribution. 10 Quiz Yourself 10 The investor in Example 3 also gathered the closing stock prices for WebRanger, another Internet software company. We list these stock prices in Table 15.14. The mean closing price is 42. Find the standard deviation for this distribution. Closing Price, x 36 37 38 39 40 41 42 43 44 45 2 0 0 3 1 3 0 1 5 5 Frequency, f TABLE 15.14 WebRanger closing stock prices for 20 days. Comparing the standard deviation of 1.62 for WebSoft with the standard deviation of 3.01 for WebRanger in Quiz Yourself 10, we see that WebRanger’s stock is much more volatile than WebSoft’s stock. We have said that the standard deviation describes the spread of a distribution. In Figure 15.11, all three distributions have a mean and median of 5; however, as the spread of the distribution increases, so does the standard deviation. 5 4 3 2 1 0 3 4 5 6 7 s 0.82 3 4 5 6 7 s 1.29 2 3 4 5 6 s 2.38 FIGURE 15.11 As the spread of the distribution increases, so does the standard deviation. Copyright © 2010 Pearson Education, Inc. 7 8 15.3 y Measures of Dispersion 745 HIGHLIGHT ¶ ¶ ¶ Using a Graphing Calculator to Find the Standard Deviation* Because calculating the standard deviation can become tedious, we often use a calculator to save work, as we show in the accompanying TI calculator screen. We have stored the data from Example 3 in list L1, and then used the calculator to find the standard deviation, as we see in the screen below. With such powerful technology available, it is very important that you think critically to set up problems correctly rather than just focusing on doing computations. mean sample standard deviation population standard deviation KEY POINT We use the coefficient of variation to compare the standard deviations of different data sets. The Coefficient of Variation In Example 3, we used the standard deviation with one set of data. If you use the standard deviation to compare two sets of data, the data must be similar and have comparable means. The following situation shows why. Suppose we are comparing the body weights of two groups of people and find the standard deviation to be 3 pounds for the first group and 10 pounds for the second group. Can we state that there is more uniformity in the first group than the second? Before answering this question, here is more information. The first group consists of preschool children, whereas the second is a group of National Football League linemen. Clearly, the standard deviation of 3 is more significant, relatively speaking, for a group of preschoolers with a mean weight of 30 pounds than the standard deviation of 10 for a group of football players with a mean weight of 300 pounds. In order to use the standard deviation effectively to compare different sets of data, we must first make the numbers comparable. We do this by finding the coefficient of variation. D E F I N I T I O N For a set of data with mean xq and standard deviation s, we define the coefficient of variation, denoted by CV, as CV = s# 100%. qx Note that the coefficient of variation compares the standard deviation to the mean and is expressed as a percentage. The coefficient of variation for the group of preschool children is CV = 3 # 100% = 10%. 30 In contrast, the coefficient of variation for the group of NFL football linemen is CV = 10 # 100% = 3.3%. 300 Because the coefficient of variation is larger for the group of preschoolers, we conclude that there is more variation, relatively speaking, in their weights than in the weights of the football players. *Ask your instructor for tutorials on using technology in statistics. Copyright © 2010 Pearson Education, Inc. 746 CHAPTER 15 y Descriptive Statistics EXAMPLE 4 Using the Coefficient of Variation to Compare Data Use the coefficient of variation to determine whether the women’s 100-meter race (328 feet) or the men’s marathon (26 miles) has had more consistent times over the five Olympics listed in Table 15.15. Women’s 100 Meters Men’s Marathon 2004 10.93 sec 2 h, 10 m, 55 sec (7,855 sec) 2000 10.75 sec 2 h, 10 m, 11 sec (7,811 sec) 1996 10.94 sec 2 h, 12 m, 36 sec (7,956 sec) 1992 10.82 sec 2 h, 13 m, 23 sec (8,003 sec) 1988 10.54 sec 2 h, 10 m, 32 sec (7,832 sec) TABLE 15.15 Olympic times in women’s 100-meter race and men’s marathon. S O LU T I O N : Using a calculator, we found that the mean time for the 100-meter race is 10.796 and the standard deviation is approximately 0.163. For the marathon, the mean is 7,891.4 and the standard deviation is approximately 83.58. Thus, the coefficient of variation for the women’s 100-meter race is 0.163 # 100% L 1.51%. 10.796 For the men’s marathon, the coefficient of variation is 83.5 # 100% L 1.06%. 7,891.4 Surprisingly, using the coefficient of variation as a measure, there is less variation in the times for the marathon than for the 100-meter race! ] Exercises 15.3 Looking Back* These exercises follow the general outline of the topics presented in this section and will give you a good overview of the material that you have just studied. Find the range, mean, and standard deviation for the following data sets. 5. 19, 18, 20, 19, 21, 20, 22, 18, 18, 17 6. 89, 72, 100, 87, 65, 98, 77, 92 1. What problem did we point out in Example 1 that can occur when we use the range to measure the spread of a distribution? 7. 5, 7, 9, 4, 6, 8, 7, 10 2. What do you get if you average the deviations from the mean without squaring them? 10. 21, 3, 5, 11, 7, 1, 9, 7 3. What is the purpose of the coefficient of variation? 8. 8, 4, 7, 6, 5, 5, 4, 9 9. 2, 12, 3, 11, 14, 5, 8, 9 11. 18, 3, 8, 7, 7, 9, 13, 7 12. 22, 18, 15, 21, 21, 15, 19, 13 4. What was our point in the technology Highlight? 13. 3, 3, 3, 3, 3, 3, 3 Sharpening Your Skills 14. 4, 6, 4, 6, 4, 6, 4, 6 If you want, use a calculator or computer software to solve the following exercises. Some of your answers may differ slightly from ours due to differences in the way we round off our calculations. 15. The following frequency table shows the number of fires per month in a city. Complete the table entries to find the mean and standard deviation for this distribution. *Before doing these exercises, you may find it useful to review the note How to Succeed at Mathematics on page xix. Copyright © 2010 Pearson Education, Inc. 15.3 y Exercises Number, x Frequency, f 2 1 3 1 4 0 5 2 6 3 7 4 8 4 9 3 10 Deviation2, ( x ⴚ xq )2 Deviation, ( x ⴚ xq ) Product, x·f Product, ( x ⴚ xq )2 # f 2 ©f = 747 ©(x - qx )2 # f = ©x · f = Table for Exercise 15 16. The following frequency table summarizes the number of job offers made to graduates of a network administrator certification program. Complete the table entries to find the mean and standard deviation for this distribution. Number, x Frequency, f 4 2 5 0 6 3 7 5 8 4 9 5 10 Product, x·f Deviation2, ( x ⴚ xq )2 Deviation, ( x ⴚ xq ) 1 ©f = ©(x - qx )2 # f = ©x · f = 19. Find the mean and standard deviation for the following frequency distributions. 17. Product, ( x ⴚ xq )2 # f 20. x f x f 3 1 12 3 x f 4 1 13 0 2 8 3 5 0 14 5 3 4 9 2 6 2 15 0 4 2 10 1 7 5 16 1 5 0 11 2 8 3 17 2 6 4 12 3 9 3 18 3 x f 2 18. Use the following graphs to solve Exercises 21 and 22. 8 7 6 5 4 3 2 1 0 4 5 (a) 6 2 3 4 5 (b) 6 7 8 2 3 4 5 (c) Copyright © 2010 Pearson Education, Inc. 6 7 8 3 4 5 (d) 6 7 CHAPTER 15 y Descriptive Statistics 748 21. a. Which data set has the smallest standard deviation? b. The largest? 22. Rank the data sets in order from smallest standard deviation to largest. You can do this just by looking at the graphs and without doing any calculations. 26. Summarizing vital statistics. The following is the 2008 roster of the Houston Comets in the WNBA. Find the mean and the population standard deviation of these data. Player Weight (lbs) Matee Ajavon 160 Applying What You’ve Learned Latasha Byears 206 23. Summarizing test score data. The following are the scores of 20 people who took a paramedics licensing test. Find the mean and standard deviation for these data. Tamecka Dixon 148 Sequoia Holmes 155 Shannon Johnson 152 Crystal Kelly 190 Sancho Lyttle 175 Mwadi Mabika 165 Hamchétou Maïga-Ba 160 Ashley Shields 155 Michelle Snow 158 Tina Thompson 178 Marcedes Walker 253 Erica White 135 Mistie Williams 184 Score Frequency 72 73 78 84 86 93 7 1 3 6 2 1 24. Summarizing age data. The following are the ages of 16 World War II veterans who are attending a reunion commemorating the Normandy invasion. Find the mean and the standard deviation for these data. Source: www.wnba.com 27. Summarizing customer data. The manager at the local Starbucks counted the customers once each hour over a busy weekend. We summarize the results she obtained in the bar graph below. Find the mean and standard deviation of these data. 14 Frequency 12 Age Frequency 82 83 84 86 93 95 4 2 4 2 3 1 10 8 6 4 2 0 0 25. Summarizing tax data. The following table lists the state income tax for a person earning $50,000 per year for several states. Find the mean and sample standard deviation for these data. (Source: The World Almanac, 2008) 1 2 3 4 5 6 7 8 9 Number of Customers 10 11 12 28. Find the mean and sample standard deviation for the distribution given by the following bar graph. 5 Income Tax (%) Frequency 4 State 3 Arizona 3.36 Colorado 4.63 Hawaii 8.25 1 Kansas 6.45 0 Massachusetts 5.3 New York 6.85 Pennsylvania 3.07 Virginia 5.75 2 1 2 3 4 Value 5 6 29. Family incomes. The following table gives the annual incomes for eight families, in thousands of dollars. Find the number of standard deviations family H’s income is from the mean. Copyright © 2010 Pearson Education, Inc. 15.3 y Exercises Family A B C D E F G H Annual Income (in $thousands) 47 48 50 49 51 47 49 51 Table for Exercise 29 30. Family incomes. The following table gives the annual incomes for eight families, in thousands of dollars. Find the number of standard deviations family A’s income is from the mean. with a standard deviation of 1.2, did WebRanger have more or less volatility than the DJIA during that week? 37. Price stability. In the following table, we list the monthly price (in dollars) of a pound of coffee and a gallon of unleaded gasoline for 2007. Find the population standard deviation of each set of data. (Source: Bureau of Labor Statistics) 2007 Coffee Prices $/lb Gasoline Prices $/gal Jan. 3.288 2.274 Feb. 3.456 2.285 Mar. 3.475 2.592 In Exercises 31 and 32, you are given a distribution of averages in a literature class. The professor in this class assigns grades as follows:* Apr. 3.437 2.860 May 3.308 3.130 A: averages that are at least 1.5 standard deviations above the mean Jun. 3.407 3.052 B: averages that are between 0.5 standard deviation above the mean, and 1.5 standard deviations above the mean Jul. 3.529 2.961 Aug. 3.497 2.782 C: averages that are between 0.5 standard deviation below the mean, and 0.5 standard deviations above the mean Sep. 3.537 2.789 D: averages that are between 1.5 standard deviations below the mean, and 0.5 standard deviation below the mean Oct. 3.577 2.793 Nov. 3.607 3.069 Dec. 3.685 3.020 Family A B C D E F G H Annual Income (in $thousands) 49 51 52 51 51 50 52 52 749 F: averages that are more than 1.5 standard deviations below the mean Source: Bureau of Labor Statistics, U.S. Department of Energy 31. Assigning grades. The averages in the class are 80, 76, 81, 84, 79, 80, 90, 75, 75, and 80. What grade does the person earning the 76 get? 38. Comparing price stability. Consider the table given in Exercise 37. Use the coefficient of variation to determine whether coffee prices or gasoline prices were more stable in 2007. 32. Assigning grades. The grades in the class are 72, 71, 73, 70, 71, 79, 65, 73, 74, and 72. What grade does the person earning the 74 get? 39. Comparing family incomes. Consider the tables given in Exercises 29 and 30, respectively. Suppose that the eight families are the same in both tables and the incomes are for two different years that are 2 years apart. Which of the families had the greatest improvement in annual income over the 2 year period with respect to the other families? Explain how you arrived at your conclusion. In Exercises 33–36, we present information on the performance of various stocks during a given month. Stocks with greater coefficients of variation are considered more volatile. 33. Comparing stocks. Suppose for a given month that the mean daily closing price for Apple Computer common stock was 123.76 and the standard deviation was 12.3. For Dell stock, the mean daily closing price was 78.6 with a standard deviation of 7.2. Which stock was more volatile? 34. Comparing stocks. Suppose for a given month that the mean daily closing price for Netflix was 37.4 and the standard deviation was 4.1. For Blockbuster, the mean daily closing price was 18.6 with a standard deviation of 3.2. Which stock was more volatile? 35. Comparing stocks. The Dow Jones Industrial Average (DJIA) measures the stock prices of a large group of stocks. If, for a given week, the DJIA had a mean daily closing price of 11,261.12 with a standard deviation of 72.17 and WebMaster stock had a mean closing price of 37.6 with a standard deviation of 1.7, did WebMaster have more or less volatility than the DJIA during that week? 36. Comparing stocks. If, for a given week, the DJIA had a mean daily closing price of 10,834.8 with a standard deviation of 144.5 and WebRanger stock had a mean closing price of 123.6 40. Comparing family incomes. Suppose that the mean family income in the United States was $48,000 with a standard deviation of $1,000 in year X. Also, the mean family income was $51,000 with a standard deviation of $2,250 three years later, in year X + 3. If a family earned $50,000 during year X and $54,000 three years later, then in which of these 2 years did the family have a better annual income in relation to the rest of the population? Explain how you arrived at your answer. Communicating Mathematics 41. What do we do differently when we calculate the standard deviation of a sample versus a population? Try to answer each of the following as true or false without doing any computations. Explain your answers in words or give appropriate examples or counterexamples to support your answers. 42. The standard deviation of the set of numbers -2, 2, -2, 2, -2, 2, -2, 2 is zero. 43. If the standard deviation of a set of data is zero, then all the numbers in the data set are the same. *If an average falls on the border of two grades, the professor will assign the higher grade. Copyright © 2010 Pearson Education, Inc. 750 CHAPTER 15 y Descriptive Statistics 44. The more numbers there are in a distribution, the larger the standard deviation. Using Technology to Investigate Mathematics d. What conclusion can you make about changes in the mean and the standard deviation when the same number is added to or subtracted from each score in a data set? e. Use the conclusion reached in part (d) to simplify the calculation of the mean and standard deviation for the distribution 598, 597, 599, 596, 600, 601, 602, 603. 45. Search the Internet for “statistical calculators.” Duplicate some of the examples in Sections 15.2 and 15.3. Report on your findings. 49. a. Pick any five numbers and compute the mean and standard deviation for this data set. 46. See your instructor for tutorials on using a graphing calculator to do statistical calculations. Use a graphing calculator to redo some of the examples and exercises in this section. b. Multiply each number in the data set you created in part (a) by 4, and compute the mean and standard deviation for this new data set. 47. Data on the Internet. Do a search on the Internet to locate data in an area that is of interest to you. For example, you can go to the New York Times Web site and find the number of weeks that the top 10 fiction books have been on the bestseller list. Or, you can locate statistics from the Web site of your favorite sports team. Other good sources of data are the Bureau of Labor Statistics and the U.S. Department of Transportation Web sites. After you have located a site, choose a data set and find its mean and standard deviation. c. Multiply each number in the data set you created in part (a) by 9, and compute the mean and standard deviation for this new data set. For Extra Credit 48. a. Pick a data set consisting of any five numbers. Compute the mean and the standard deviation of this data set. (Consider the data set to be a sample.) b. Add 20 to each number in the data set you created in part (a), and compute the mean and standard deviation for this new data set. c. Subtract 5 from each number in the data set you created in part (a), and compute the mean and standard deviation for this new data set. d. What conclusion can you make about changes in the mean and standard deviation when each score in a data set is multiplied by the same number? e. The mean is 4 and the standard deviation is 2.2 for the data set 3, 4, 7, 1, 5. Consider your conclusion from part (d), and compute the mean and standard deviation for the data set 15, 20, 35, 5, 25. 50. Consider the two distributions given by the bar graphs in Exercises 27 and 28. Can you simply look at the graphs to decide which of the two distributions has the greater standard deviation? If so, describe what you look for in the graphs on which to base your decision. 51. Comparative cost of living. According to an Internet service that compares the cost of living in one location with the cost of living in another, a salary of $100,000 in Manhattan is comparable to $38,000 in Allentown, Pennsylvania. What statistical methods do you think this service uses in determining this information? Objectives Copyright © 2010 Pearson Education, Inc.