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9781133105060_App_C1.qxp 12/27/11 1:31 PM Page C1 Appendix C.1 C ■ C1 Representing Data and Linear Modeling Further Concepts in Statistics C.1 Representing Data and Linear Modeling ■ Use stem-and-leaf plots to organize and compare sets of data. ■ Use histograms and frequency distributions to organize and represent data. ■ Use scatter plots to represent and analyze data. ■ Fit lines to data. Stem-and-Leaf Plots Statistics is the branch of mathematics that studies techniques for collecting, organizing, and interpreting data. In this section, you will study several ways to organize and interpret data. One type of plot that can be used to organize sets of numbers by hand is a stemand-leaf plot. A set of test scores and the corresponding stem-and-leaf plot are shown below. Test Scores 93, 70, 76, 58, 86, 93, 82, 78, 83, 86, 64, 78, 76, 66, 83, 83, 96, 74, 69, 76, 64, 74, 79, 76, 88, 76, 81, 82, 74, 70 Stems 5 6 7 8 9 Leaves 8 Key: 5 8 ⫽ 58 4469 0044466666889 122333668 336 ⱍ Note from the key in the stem-and-leaf plot that the leaves represent the units digits of the numbers and the stems represent the tens digits. Stem-and-leaf plots can also be used to compare two sets of data, as shown in the following example. Example 1 Comparing Two Sets of Data Use a stem-and-leaf plot to compare the test scores given above with the test scores below. Which set of test scores is better? 90, 81, 70, 62, 64, 73, 81, 92, 73, 81, 92, 93, 83, 75, 76, 83, 94, 96, 86, 77, 77, 86, 96, 86, 77, 86, 87, 87, 79, 88 SOLUTION Begin by ordering the second set of scores. 62, 64, 70, 73, 73, 75, 76, 77, 77, 77, 79, 81, 81, 81, 83, 83, 86, 86, 86, 86, 87, 87, 88, 90, 92, 92, 93, 94, 96, 96 9781133105060_App_C1.qxp C2 12/27/11 Appendix C ■ 1:31 PM Page C2 Further Concepts in Statistics Now that the data have been ordered, you can construct a double stem-and-leaf plot by letting the leaves to the right of the stems represent the units digits for the first group of test scores and letting the leaves to the left of the stems represent the units digits for the second group of test scores. Leaves (2nd Group) 42 977765330 877666633111 6643220 Stems 5 6 7 8 9 Leaves (1st Group) 8 4469 0044466666889 122333668 336 By comparing the two sets of leaves, you can see that the second group of test scores is better than the first group. Example 2 Using a Stem-and-Leaf Plot The table below shows the percent of the population of each state and the District of Columbia that was at least 65 years old in July 2009. Use a stem-and-leaf plot to organize the data. (Source: U.S. Census Bureau) AL 13.8 CO 10.6 GA 10.3 IA 14.8 MD 12.2 MO 13.7 NJ 13.5 OH 13.9 SC 13.7 VT 14.5 WY 12.3 SOLUTION AK CT HI KS MA MT NM OK SD VA 7.6 13.9 14.5 13.0 13.6 14.6 13.0 13.5 14.5 12.2 AZ DE ID KY MI NE NY OR TN WA 13.1 14.3 12.1 13.2 13.4 13.4 13.4 13.5 13.4 12.1 AR DC IL LA MN NV NC PA TX WV 14.3 11.7 12.4 12.3 12.7 11.6 12.7 15.4 10.3 15.8 CA FL IN ME MS NH ND RI UT WI 11.2 17.2 12.9 15.6 12.8 13.5 14.7 14.3 9.0 13.5 Begin by ordering the numbers, as shown below. 7.6, 9.0, 10.3, 10.3, 10.6, 11.2, 11.6, 11.7, 12.1, 12.1, 12.2, 12.2, 12.3, 12.3, 12.4, 12.7, 12.7, 12.8, 12.9, 13.0, 13.0, 13.1, 13.2, 13.4, 13.4, 13.4, 13.4, 13.5, 13.5, 13.5, 13.5, 13.5, 13.6, 13.7, 13.7, 13.8, 13.9, 13.9, 14.3, 14.3, 14.3, 14.5, 14.5, 14.5, 14.6, 14.7, 14.8, 15.4, 15.6, 15.8, 17.2 Next, construct a stem-and-leaf plot using the leaves to represent the digits to the right of the decimal points. 12/27/11 1:31 PM Page C3 Appendix C.1 Stems 7 8 9 10 11 12 13 14 15 16 17 Leaves 6 ■ Representing Data and Linear Modeling ⱍ Key: 7 6 ⫽ 7.6 C3 Alaska has the lowest percent. 0 336 267 11223347789 0012444455555677899 333555678 468 2 Florida has the highest percent. Histograms and Frequency Distributions When you want to organize large sets of data, such as those given in Example 2, it is useful to group the data into intervals and plot the frequency of the data in each interval. For instance, the frequency distribution and histogram shown in Figure C.1 represent the data given in Example 2. Frequency Distribution Interval Tally 关6, 8兲 | 关8, 10兲 | 关10, 12兲 |||| | 关12, 14兲 |||| |||| |||| |||| |||| |||| 关14, 16兲 |||| |||| || 关16, 18兲 | Histogram 30 Number of states (including District of Columbia) 9781133105060_App_C1.qxp 25 20 15 10 5 6 8 10 12 14 16 18 20 Percent of population 65 or older FIGURE C.1 A histogram has a portion of a real number line as its horizontal axis. A bar graph is similar to a histogram, except that the rectangles (bars) can be either horizontal or vertical and the labels of the bars are not necessarily numbers. Another difference between a bar graph and a histogram is that the bars in a bar graph are usually separated by spaces, whereas the bars in a histogram are not separated by spaces. 9781133105060_App_C1.qxp C4 12/27/11 Appendix C ■ 1:31 PM Page C4 Further Concepts in Statistics Example 3 Constructing a Bar Graph The data below give the average monthly precipitation (in inches) in Houston, Texas. Construct a bar graph for these data. What can you conclude? (Source: National Climatic Data Center) January April July October 3.7 3.6 3.2 4.5 February May August November 3.0 5.2 3.8 4.2 March June September December 3.4 5.4 4.3 3.7 SOLUTION To create a bar graph, begin by drawing a vertical axis to represent the precipitation and a horizontal axis to represent the months. The bar graph is shown in Figure C.2. From the graph, you can see that Houston receives a fairly consistent amount of rain throughout the year—the driest month tends to be February and the wettest month tends to be June. Average monthly precipitation (in inches) 6 5 4 3 2 1 J F M A M J J A S O N D Month FIGURE C.2 Scatter Plots Cable revenue (in billions of dollars) R 100 90 80 70 60 50 40 t 2 3 4 5 6 7 8 9 Year (2 ↔ 2002) FIGURE C.3 Many real-life situations involve finding relationships between two variables, such as the year and the total revenue of the cable television industry. In a typical situation, data are collected and written as a set of ordered pairs. The graph of such a set is called a scatter plot. From the scatter plot in Figure C.3, it appears that the points describe a relationship that is nearly linear. (The relationship is not exactly linear because the total revenue did not increase by precisely the same amount each year.) A mathematical equation that approximates the relationship between t and R is called a mathematical model. When developing a mathematical model, you strive for two (often conflicting) goals—accuracy and simplicity. For the data in Figure C.3, a linear model of the form R ⫽ at ⫹ b appears to be best. It is simple and relatively accurate. Consider a collection of ordered pairs of the form 共x, y兲. If y tends to increase as x increases, the collection is said to have a positive correlation. If y tends to decrease as x increases, the collection is said to have a negative correlation. Figure C.4, on the next page, shows three examples: one with a positive correlation, one with a negative correlation, and one with no (discernible) correlation. 9781133105060_App_C1.qxp 12/27/11 1:31 PM Page C5 Appendix C.1 y y y x Positive Correlation FIGURE C.4 C5 Representing Data and Linear Modeling ■ x Negative Correlation x No Correlation Fitting a Line to Data Finding a linear model that represents the relationship described by a scatter plot is called fitting a line to data. You can do this graphically by simply sketching the line that appears to fit the points, finding two points on the line, and then finding the equation of the line that passes through the two points. Example 4 Fitting a Line to Data Find a linear model that relates the year with the total revenue R (in billions of dollars) for the U.S. cable television industry for the years 2002 through 2009. (Source: SNL Kagan) Year 2002 2003 2004 2005 2006 2007 2008 2009 Revenue, R 48.0 53.2 58.6 64.9 71.9 78.9 85.2 89.5 Let t represent the year, with t ⫽ 2 corresponding to 2002. After plotting the data from the table, draw the line that you think best represents the data, as shown in Figure C.5. Two points that lie on this line are 共3, 53.2兲 and 共8, 85.2兲. Using the point-slope form, you can find the equation of the line to be SOLUTION The model in Example 4 is based on the two data points chosen. If different points are chosen, the model may change somewhat. For instance, if you choose 共2, 48.0兲 and 共6, 71.9兲, the new model is R ⫽ 5.975t ⫹ 36.05. 共t ⫺ 3兲 ⫹ 53.2 ⫽ 6.4t ⫹ 34. 冢85.28 ⫺⫺ 53.2 3 冣 Linear model R Cable revenue (in billions of dollars) STUDY TIP R⫽ 100 90 80 70 60 50 40 t 2 3 4 5 6 7 8 9 Year (2 ↔ 2002) FIGURE C.5 9781133105060_App_C1.qxp C6 12/27/11 Appendix C ■ 1:31 PM Page C6 Further Concepts in Statistics Once you have found a model, you can measure how well the model fits the data by comparing the actual values with the values given by the model, as shown in the table below. t 2 3 4 5 6 7 8 9 Actual R 48.0 53.2 58.6 64.9 71.9 78.9 85.2 89.5 Model R 46.8 53.2 59.6 66.0 72.4 78.8 85.2 91.6 The sum of the squares of the differences between the actual values and the model values is the sum of the squared differences. The model that has the least sum is called the least squares regression line for the data. For the model in Example 4, the sum of the squared differences is 8.32. The least squares regression line for the data is R ⫽ 6.17t ⫹ 34.8. Best-fitting linear model Its sum of squared differences is approximately 4.69. Least Squares Regression Line The least squares regression line, for the points 共x1, y1兲, 共x2, y2兲, 共x3, y3兲, . . . , is given by y ⫽ ax ⫹ b. The slope a and y-intercept b are given by n n a⫽ 兺xy i i i⫽1 n n 兺 xi 2 ⫺ i⫽1 Example 5 n n 兺 x 兺y ⫺ i i⫽1 n i i⫽1 2 and b ⫽ 冢兺 冣 xi 1 n 冢兺 n i⫽1 兺 x 冣. n yi ⫺ a i i⫽1 i⫽1 Finding the Least Squares Regression Line Find the least squares regression line for the points 共⫺3, 0兲, 共⫺1, 1兲, 共0, 2兲, and 共2, 3兲. SOLUTION Begin by constructing a table like the one shown below. x y xy x2 ⫺3 0 0 9 ⫺1 1 ⫺1 1 0 2 0 0 2 n 3 n 6 n 4 n 兺 x ⫽ ⫺2 兺 y ⫽ 6 兺 x y ⫽ 5 兺 x i i⫽1 i i⫽1 2 i i i i⫽1 i⫽1 ⫽ 14 9781133105060_App_C1.qxp 12/27/11 1:31 PM Page C7 Appendix C.1 Representing Data and Linear Modeling ■ C7 Applying the formulas for the least squares regression line with n ⫽ 4 produces n n n i⫽1 n a⫽ n 兺x y ⫺ 兺x 兺y i i i⫽1 n i 兺x ⫺ 冢兺x 冣 2 i n i⫽1 2 i ⫽ 4共5兲 ⫺ 共⫺2兲共6兲 32 8 ⫽ ⫽ 4共14兲 ⫺ 共⫺2兲2 52 13 i i⫽1 i⫽1 and b⫽ 1 n 冢兺 n i⫽1 兺 x 冣 ⫽ 4冤6 ⫺ 13 共⫺2兲冥 ⫽ 52 ⫽ 26. n yi ⫺ a 1 8 94 47 i i⫽1 8 So, the least squares regression line is y ⫽ 13 x ⫹ 47 26 , shown in Figure C.6. y 5 47 8 y = x + 26 13 4 3 2 1 −3 −2 −1 −1 x 1 2 −2 FIGURE C.6 Many graphing utilities have built-in least squares regression programs. If your calculator has such a program, try using it to duplicate the results shown in the following example. Example 6 Finding the Least Squares Regression Line The ordered pairs 共w, h兲 shown below represent the shoe sizes w and the heights h (in inches) of 25 men. Use the regression feature of a graphing utility to find the least squares regression line for these data. 共10.0, 70.5兲, 共10.5, 71.0兲, 共9.5, 69.0兲, 共11.0, 72.0兲, 共12.0, 74.0兲, 共8.5, 67.0兲, 共9.0, 68.5兲, 共13.0, 76.0兲, 共10.5, 71.5兲, 共10.5, 70.5兲, 共10.0, 71.0兲, 共9.5, 70.0兲, 共10.0, 71.0兲, 共10.5, 71.0兲, 共11.0, 71.5兲, 共12.0, 73.5兲, 共12.5, 75.0兲, 共11.0, 72.0兲, 共9.0, 68.0兲, 共10.0, 70.0兲, 共13.0, 75.5兲, 共10.5, 72.0兲, 共10.5, 71.0兲, 共11.0, 73.0兲, 共8.5, 67.5兲 FIGURE C.7 90 SOLUTION Enter the data into a graphing utility. Then, use the regression feature of the graphing utility to obtain the model shown in Figure C.7. So, the least squares regression line for the data is 8 14 50 FIGURE C.8 h ⫽ 1.84w ⫹ 51.9. In Figure C.8, this line is plotted with the data. Notice that the plot does not have 25 points because some of the ordered pairs graph as the same point. 9781133105060_App_C1.qxp C8 12/27/11 Appendix C ■ 1:31 PM Page C8 Further Concepts in Statistics When you use a graphing utility or a computer program to find the least squares regression line for a set of data, the output may include an r-value. For instance, the r-value from Example 6 was r ⬇ 0.981. This number is called the correlation coefficient of the data and gives a measure of how well the model fits the data. Correlation coefficients vary between ⫺1 and 1. Basically, the closer r is to 1, the better the points can be described by a line. Three examples are shown in Figure C.9. ⱍⱍ 18 0 0 18 9 r = 0.981 0 0 18 9 r = −0.866 0 0 9 r = 0.190 FIGURE C.9 Exercises C.1 Exam Scores In Exercises 1 and 2, use the data below which represent the scores on two 100-point exams for a math class of 30 students. See Examples 1 and 2. Exam #1: 77, 100, 77, 70, 83, 89, 87, 85, 81, 84, 81, 78, 89, 78, 88, 85, 90, 92, 75, 81, 85, 100, 98, 81, 78, 75, 85, 89, 82, 75 Exam #2: 76, 78, 73, 59, 70, 81, 71, 66, 66, 73, 68, 67, 63, 67, 77, 84, 87, 71, 78, 78, 90, 80, 77, 70, 80, 64, 74, 68, 68, 68 1. Use a stem-and-leaf plot to organize the scores for Exam #1. 2. Construct a double stem-and-leaf plot to compare the scores for Exam #1 and Exam #2. Which set of test scores is better? 3. Cancer Incidence The table shows the estimated numbers of new cancer cases (in thousands) in the 50 states in 2010. Use a stem-and-leaf plot to organize the data. (Source: American Cancer Society, Inc.) AL CO HI KS MA MT NM OK SD VA 24 21 7 14 36 6 9 19 4 36 AK CT ID KY MI NE NY OR TN WA 3 21 7 24 56 9 103 21 33 35 AZ DE IL LA MN NV NC PA TX WV 30 5 64 21 25 12 45 75 101 11 AR FL IN ME MS NH ND RI UT WI 15 107 33 9 14 8 3 6 10 30 CA GA IA MD MO NJ OH SC VT WY 157 40 17 28 31 48 64 23 4 3 4. Snowfall The data give the seasonal snowfalls (in inches) for Lincoln, Nebraska for the seasons 1970–1971 through 2009–2010 (the amounts are listed in order by year). Use a frequency distribution and a histogram to organize the data. (Source: National Weather Service) 49.0, 21.6, 29.2, 33.6, 42.1, 21.1, 21.8, 31.0, 34.4, 23.3, 13.0, 32.3, 38.0, 47.5, 21.5, 18.9, 15.7, 13.0, 19.1, 18.7, 25.8, 23.8, 32.1, 21.3, 21.7, 30.7, 29.0, 44.6, 24.4, 11.7, 37.9, 29.5, 31.7, 35.9, 16.3, 19.5, 31.0, 20.4, 19.2, 41.6 5. Bus Fares The data below give the base prices of bus fare in selected U.S. cities. Construct a bar graph for these data. (Source: American Public Transportation Association) Seattle Houston New York Los Angeles $2.25 $1.25 $2.25 $1.50 Atlanta Dallas Denver Chicago $2.00 $1.75 $2.25 $2.00 6. Melanoma Incidence The data below give the places of origin and the estimated numbers of new melanoma cases in 2010. Construct a bar graph for these data. (Source: American Cancer Society, Inc.) California Michigan Texas 8030 2240 3570 Florida New York Washington 4980 4050 1930 9781133105060_App_C1.qxp 12/27/11 1:31 PM Page C9 Appendix C.1 Crop Yield In Exercises 7–10, use the data in the table, where x is the number of units of fertilizer applied to sample plots and y is the yield (in bushels) of a crop. x 0 1 2 3 4 y 58 60 59 61 63 5 6 7 8 66 65 67 70 7. Sketch a scatter plot of the data. 8. Determine whether the points are positively correlated, are negatively correlated, or have no discernible correlation. 9. Sketch a linear model that you think best represents the data. Find an equation of the line you sketched. Use the line to predict the yield when 10 units of fertilizer are used. 10. Can the model found in Exercise 9 be used to predict yields for arbitrarily large values of x? Explain. Speed of Sound In Exercises 11–14, use the data in the table, where h is altitude (in thousands of feet) and v is the speed of sound (in feet per second). h 0 5 10 15 20 25 30 v 1116 1097 1077 1057 1037 1016 995 35 973 11. Sketch a scatter plot of the data. 12. Determine whether the points are positively correlated, are negatively correlated, or have no discernible correlation. 13. Sketch a linear model that you think best represents the data. Find an equation of the line you sketched. Use the line to estimate the speed of sound at an altitude of 27,000 feet. 14. The speed of sound at an altitude of 70,000 feet is approximately 968 feet per second. What does this suggest about the validity of using the model in Exercise 13 to extrapolate beyond the data given in the table? Fitting Lines to Data In Exercises 15 and 16, (a) sketch a scatter plot of the points, (b) find an equation of the linear model you think best represents the data and find the sum of the squared differences, and (c) use the formulas in this section to find the least squares regression line for the data and the sum of the squared differences. See Examples 4 and 5. 15. 共⫺1, 0兲, 共0, 1兲, 共1, 3兲, 共2, 3兲 16. 共0, 4兲, 共1, 3兲, 共2, 2兲, 共4, 1兲 Finding the Least Squares Regression Line In Exercises 17–20, sketch a scatter plot of the points, use the formulas in this section to find the least squares regression line for the data, and sketch the graph of the line. See Example 5. ■ 17. 18. 19. 20. C9 Representing Data and Linear Modeling 共⫺2, 0兲, 共⫺1, 1兲, 共0, 1兲, 共2, 2兲 共⫺3, 1兲, 共⫺1, 2兲, 共0, 2兲, 共1, 3兲, 共3, 5兲 共1, 5兲, 共2, 8兲, 共3, 13兲, 共4, 16兲, 共5, 22兲, 共6, 26兲 共1, 10兲, 共2, 8兲, 共3, 8兲, 共4, 6兲, 共5, 5兲, 共6, 3兲 Finding the Least Squares Regression Line In Exercises 21–24, use the regression feature of a graphing utility to find the least squares regression line for the data. Graph the data and the regression line in the same viewing window. See Example 6. 21. 共0, 23兲, 共1, 20兲, 共2, 19兲, 共3, 17兲, 共4, 15兲, 共5, 11兲, 共6, 10兲 22. 共4, 52.8兲, 共5, 54.7兲, 共6, 55.7兲, 共7, 57.8兲, 共8, 60.2兲, 共9, 63.1兲, 共10, 66.5兲 23. 共⫺10, 5.1兲, 共⫺5, 9.8兲, 共0, 17.5兲, 共2, 25.4兲, 共4, 32.8兲, 共6, 38.7兲, 共8, 44.2兲, 共10, 50.5兲 24. 共⫺10, 213.5兲, 共⫺5, 174.9兲, 共0, 141.7兲, 共5, 119.7兲, 共8, 102.4兲, 共10, 87.6兲 25. Advertising The management of a department store ran an experiment to determine if a relationship existed between sales S (in thousands of dollars) and the amount spent on advertising x (in thousands of dollars). The following data were collected. x 1 2 3 4 5 6 7 8 S 405 423 455 466 492 510 525 559 (a) Use the regression feature of a graphing utility to find the least squares regression line for the data. Use the model to estimate sales when $4500 is spent on advertising. (b) Make a scatter plot of the data and sketch the graph of the regression line. (c) Use a graphing utility or computer to determine the correlation coefficient. 26. Horses The table shows the heights (in hands) and corresponding lengths (in inches) of horses in a stable. Height 17 16 16.2 15.3 15.1 16.3 Length 77 73 74 71 69 75 (a) Use the regression feature of a graphing utility to find the least squares regression line for the data. Let h represent the height and l represent the length. Use the model to predict the length of a horse that is 15.5 hands tall. (b) Make a scatter plot of the data and sketch the graph of the regression line. (c) Use a graphing utility or computer to determine the correlation coefficient. 9781133105060_App_C2.qxp C10 12/27/11 Appendix C ■ 1:32 PM Page C10 Further Concepts in Statistics C.2 Measures of Central Tendency and Dispersion ■ Find and interpret the mean, median, and mode of a set of data. ■ Determine the measure of central tendency that best represents a set of data. ■ Find the standard deviation of a set of data. Mean, Median, and Mode In many real-life situations, it is helpful to describe data by a single number that is most representative of the entire collection of numbers. Such a number is called a measure of central tendency. Here are three of the most commonly used measures of central tendency. 1. The mean, or average, of n numbers is the sum of the numbers divided by n. 2. The median of n numbers is the middle number when the numbers are written in order. If n is even, the median is the average of the two middle numbers. 3. The mode of n numbers is the number that occurs most frequently. If two numbers tie for most frequent occurrence, the collection has two modes and is called bimodal. Example 1 Finding Measures of Central Tendency You are interviewing for a job. The interviewer tells you that the average income of the company’s 25 employees is $60,849. The actual annual incomes of the 25 employees are shown. What are the mean, median, and mode of the incomes? Was the interviewer telling you the truth? $17,305, $25,676, $12,500, $34,983, $32,654, SOLUTION $478,320, $28,906, $33,855, $36,540, $98,213, $45,678, $12,500, $37,450, $250,921, $48,980, $18,980, $24,540, $20,432, $36,853, $94,024, $17,408, $33,450, $28,956, $16,430, $35,671 The mean of the incomes is 17,305 ⫹ 478,320 ⫹ 45,678 ⫹ 18,980 ⫹ . . . ⫹ 35,671 25 1,521,225 ⫽ ⫽ $60,849. 25 Mean ⫽ To find the median, order the incomes. $12,500, $18,980, $28,956, $35,671, $48,980, $12,500, $20,432, $32,654, $36,540, $94,024, $16,430, $24,540, $33,450, $36,853, $98,213, $17,305, $25,676, $33,855, $37,450, $250,921, $17,408, $28,906, $34,983, $45,678, $478,320 9781133105060_App_C2.qxp 12/27/11 1:32 PM Page C11 Appendix C.2 ■ Measures of Central Tendency and Dispersion C11 From this list, you can see the median income is $33,450. You can also see that $12,500 is the only income that occurs more than once. So, the mode is $12,500. Technically, the interviewer was telling the truth because the average is (generally) defined to be the mean. However, of the three measures of central tendency Mean: $60,849 Median: $33,450 Mode: $12,500 it seems clear that the median is most representative. The mean is inflated by the two highest salaries. Choosing a Measure of Central Tendency Which of the three measures of central tendency is the most representative? The answer is that it depends on the distribution of the data and the way in which you plan to use the data. For instance, in Example 1, the mean salary of $60,849 does not seem very representative to a potential employee. To a city income tax collector who wants to estimate 1% of the total income of the 25 employees, however, the mean is precisely the right measure. Example 2 Choosing a Measure of Central Tendency Which measure of central tendency is the most representative for each situation based on the data shown in the frequency distribution? a. The data represent a student’s scores on 20 labs assignments worth 10 points each. The professor will use the data to determine the lab grade for the student. Score 5 6 7 8 9 10 Frequency 3 0 3 2 1 11 b. The data represent the ages of the students in a class. The professor will use the data to determine the typical age of a student in the class. Age 18 19 20 21 22 82 Frequency 7 5 5 6 4 1 SOLUTION a. For these data, the mean is 8.55, the median is 10, and the mode is 10. Of these, the mean is the most representative measure because it takes into account all of the student’s lab assignment scores. b. For these data, the mean is about 22.04, the median is 20, and the mode is 18. The mean is greater than most of the data, because it is affected by the extreme value of 82. The mode corresponds to the youngest age in the class. The median appears to be the most representative measure, because it is central to most of the data. ■ Page C12 Further Concepts in Statistics Variance and Standard Deviation Very different sets of numbers can have the same mean. You will now study two measures of dispersion, which give you an idea of how much the numbers in the set differ from the mean of the set. These two measures are called the variance of the set and the standard deviation of the set. Definitions of Variance and Standard Deviation Consider a set of numbers 再x1, x2, . . . , xn冎 with a mean of x. The variance of the set is v⫽ 共x1 ⫺ x兲2 ⫹ 共x2 ⫺ x兲2 ⫹ . . . ⫹ 共xn ⫺ x兲2 n and the standard deviation of the set is ⫽ 冪v ( is the lowercase Greek letter sigma). The standard deviation of a set is a measure of how much a typical number in the set differs from the mean. The greater the standard deviation, the more the numbers in the set vary from the mean. For instance, each of the following sets has a mean of 5. 再5, 5, 5, 5冎, 再4, 4, 6, 6冎, and 再3, 3, 7, 7冎 The standard deviations of the sets are 0, 1, and 2. 冪共5 ⫺ 5兲 共4 ⫺ 5兲 ⫽冪 共3 ⫺ 5兲 ⫽冪 1 ⫽ 2 3 Example 3 2 ⫹ 共5 ⫺ 5兲2 ⫹ 共5 ⫺ 5兲2 ⫹ 共5 ⫺ 5兲2 ⫽0 4 2 ⫹ 共4 ⫺ 5兲2 ⫹ 共6 ⫺ 5兲2 ⫹ 共6 ⫺ 5兲2 ⫽1 4 2 ⫹ 共3 ⫺ 5兲2 ⫹ 共7 ⫺ 5兲2 ⫹ 共7 ⫺ 5兲2 ⫽2 4 Estimations of Standard Deviation Consider the three sets of data represented by the bar graphs in Figure C.10. Which set has the smallest standard deviation? Which has the largest? Set A 5 4 3 2 1 Set B 5 4 3 2 1 Set C Frequency Appendix C 1:32 PM Frequency C12 12/27/11 Frequency 9781133105060_App_C2.qxp 5 4 3 2 1 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Number Number Number FIGURE C.10 9781133105060_App_C2.qxp 12/27/11 1:32 PM Page C13 Appendix C.2 ■ Measures of Central Tendency and Dispersion C13 Of the three sets, the numbers in set A are grouped most closely to the center and the numbers in set C are the most dispersed. So, set A has the smallest standard deviation and set C has the largest standard deviation. SOLUTION Example 4 Finding Standard Deviation Find the standard deviation of each set shown in Example 3. Because of the symmetry of each bar graph, you can conclude that each has a mean of x ⫽ 4. The standard deviation of set A is SOLUTION 冪 (⫺3兲2 ⫹ 2共⫺2兲2 ⫹ 3共⫺1兲2 ⫹ 5共0兲2 ⫹ 3共1兲2 ⫹ 2共2兲2 ⫹ 共3兲2 17 ⬇ 1.53. ⫽ The standard deviation of set B is ⫽ 冪 2共⫺3兲2 ⫹ 2共⫺2兲2 ⫹ 2共⫺1兲2 ⫹ 2共0兲2 ⫹ 2共1兲2 ⫹ 2共2兲2 ⫹ 2共3兲2 14 ⫽ 2. The standard deviation of set C is 冪 5共⫺3兲2 ⫹ 4共⫺2兲2 ⫹ 3共⫺1兲2 ⫹ 2共0兲2 ⫹ 3共1兲2 ⫹ 4共2兲2 ⫹ 5共3兲2 26 ⬇ 2.22. ⫽ These values confirm the results of Example 3. That is, set A has the smallest standard deviation and set C has the largest. The following alternative formula provides a more efficient way to compute the standard deviation. Alternative Formula for Standard Deviation The standard deviation of 再x1, x2, . . . , xn冎 is ⫽ 冪x 2 1 ⫹ x22 ⫹ . . . ⫹ x2n ⫺ x 2. n Because of lengthy computations, this formula is difficult to verify. Conceptually, however, the process is straightforward. It consists of showing that the expressions 共x1 ⫺ x兲2 ⫹ 共x2 ⫺ x兲2 ⫹ . . . ⫹ 共xn ⫺ x兲2 冪 n and 冪 x21 ⫹ x22 ⫹ . . . ⫹ x2n ⫺ x2 n are equivalent. Try verifying this equivalence for the set 再x1, x2, x3冎 with x ⫽ 共x1 ⫹ x2 ⫹ x3兲兾3. 9781133105060_App_C2.qxp C14 12/27/11 Appendix C ■ 1:32 PM Page C14 Further Concepts in Statistics Example 5 Using the Alternative Formula Use the alternative formula for standard deviation to find the standard deviation of the set of numbers. 5, 6, 6, 7, 7, 8, 8, 8, 9, 10 SOLUTION Begin by finding the mean of the set, which is 7.4. So, the standard deviation is 冪 冪 52 ⫹ 2共62兲 ⫹ 2共72兲 ⫹ 3共82兲 ⫹ 92 ⫹ 102 ⫺ 共7.4兲2 10 568 ⫽ ⫺ 54.76 10 ⫽ 冪2.04 ⬇ 1.43. ⫽ You can use the statistical features of a graphing utility to check this result. A well-known theorem in statistics, called Chebychev’s Theorem, states that at least 1⫺ 1 k2 of the numbers in a distribution must lie within k standard deviations of the mean. So, 75% of the numbers in a collection must lie within two standard deviations of the mean, and at least 88.9% of the numbers must lie within three standard deviations of the mean. For most distributions, these percentages are low. For instance, in all three distributions shown in Example 3, 100% of the numbers lie within two standard deviations of the mean. Example 6 Describing a Distribution The table shows the numbers of nurses (per 100,000 people) in each state. Find the mean and standard deviation of the data. What percent of the numbers lie within two standard deviations of the mean? (Source: Bureau of Labor Statistics) AL 899 CO 799 HI 680 KS 894 MA 1218 MT 773 NM 599 OK 734 SD 1244 VA 770 AK CT ID KY MI NE NY OR TN WA 777 1010 710 958 866 1062 867 792 987 792 AZ 581 DE 1034 IL 847 LA 890 MN 1065 NV 610 NC 911 PA 1027 TX 676 WV 932 AR FL IN ME MS NH ND RI UT WI 802 793 884 1065 930 992 988 1078 632 919 CA 657 GA 669 IA 1008 MD 897 MO 1009 NJ 873 OH 997 SC 819 VT 950 WY 807 9781133105060_App_C2.qxp 12/27/11 1:32 PM Page C15 Appendix C.2 ■ C15 Measures of Central Tendency and Dispersion Begin by entering the numbers into a graphing utility that has a standard deviation program. After running the program, you should obtain SOLUTION x ⬇ 875.3 and ⫽ 152.3. The interval that contains all numbers that lie within two standard deviations of the mean is 关875.3 ⫺ 2共152.3兲, 875.3 ⫹ 2共152.3兲兴 or 关570.7, 1179.9兴. From the table, you can see that all but two of the numbers (96%) lie in this interval— all but the numbers that correspond to the number of nurses (per 100,000 people) in Massachusetts and South Dakota. Exercises C.2 Finding Measures of Central Tendency In Exercises 1– 6, find the mean, median, and mode of the set of measurements. See Example 1. 1. 2. 3. 4. 5. 6. 5, 12, 7, 14, 8, 9, 7 30, 37, 32, 39, 33, 34, 32 5, 12, 7, 24, 8, 9, 7 20, 37, 32, 39, 33, 34, 32 5, 12, 7, 14, 9, 7 30, 37, 32, 39, 34, 32 7. Reasoning Compare your answers for Exercises 1 and 3 with those for Exercises 2 and 4. Which of the measures of central tendency is sensitive to extreme measurements? Explain your reasoning. 8. Reasoning (a) Add 6 to each measurement in Exercise 1 and calculate the mean, median, and mode of the revised measurements. How are the measures of central tendency changed? (b) If a constant k is added to each measurement in a set of data, how will the measures of central tendency change? 9. Cost of Electricity A person’s monthly electricity bills are shown for one year. What are the mean and median of the collection of bills? January $67.92 February $59.84 March $52.00 April $52.50 May $57.99 June $65.35 July $81.76 August $74.98 September $87.82 October $83.18 November $65.35 December $57.00 10. Car Rental The numbers of miles of travel for a rental car are shown for six consecutive days. What are the mean, median, and mode of these data? Monday 410 Tuesday 260 Wednesday 320 Thursday 320 Friday 460 Saturday 150 11. Six-Child Families A study was done on families having six children. The table shows the numbers of families in the study with the indicated number of girls. Determine the mean, median, and mode of this set of data. Number of girls 0 1 2 3 4 5 6 Frequency 1 24 45 54 50 19 7 12. Baseball A fan examined the records of a baseball player’s performance during his last 50 games. The table shows the numbers of games in which the player had 0, 1, 2, 3, and 4 hits. Number of hits 0 1 2 3 4 Frequency 14 26 7 2 1 (a) Determine the average number of hits per game. (b) The player had 200 at-bats in the 50 games. Determine the player’s batting average for these games. 13. Think About It Construct a collection of numbers that has the following properties. If this is not possible, explain why. Mean ⫽ 6, median ⫽ 4, mode ⫽ 4 9781133105060_App_C2.qxp C16 12/27/11 Appendix C ■ 1:32 PM Page C16 Further Concepts in Statistics 14. Think About It Construct a collection of numbers that has the following properties. If this is not possible, explain why. 34. Think About It Consider the four sets of data represented by the histograms. Order the sets from the smallest to the largest standard deviation. Mean ⫽ 6, median ⫽ 6, mode ⫽ 4 Finding Standard Deviation In Exercises 17–24, find the mean x, variance v, and standard deviation of the set. See Example 4. 17. 18. 19. 20. 21. 22. 23. 24. 4, 10, 8, 2 3, 15, 6, 9, 2 0, 1, 1, 2, 2, 2, 3, 3, 4 2, 2, 2, 2, 2, 2 1, 2, 3, 4, 5, 6, 7 1, 1, 1, 5, 5, 5 49, 62, 40, 29, 32, 70 1.5, 0.4, 2.1, 0.7, 0.8 Using the Alternate Formula In Exercises 25–30, use the alternative formula to find the standard deviation of the data set. See Example 5. 25. 26. 27. 28. 29. 30. 2, 4, 6, 6, 13, 5 10, 25, 50, 26, 15, 33, 29, 4 246, 336, 473, 167, 219, 359 6.0, 9.1, 4.4, 8.7, 10.4 8.1, 6.9, 3.7, 4.2, 6.1 9.0, 7.5, 3.3, 7.4, 6.0 31. Reasoning Without calculating the standard deviation, explain why the set 再4, 4, 20, 20冎 has a standard deviation of 8. 32. Reasoning When the standard deviation of a set of numbers is 0, what does this imply about the set? 33. Test Scores An instructor adds five points to each student’s exam score. Will this change the mean or standard deviation of the exam scores? Explain. Frequency 8 6 4 2 1 2 3 6 4 2 4 1 2 3 Number Number Set C Set D 8 4 8 Frequency Which measure of central tendency best describes these test scores? 16. Shoe Sales A salesman sold eight pairs of men’s dress shoes. The sizes of the eight pairs were 1012, 8, 12, 1012, 10, 912, 11, and 10 12. Which measure (or measures) of central tendency best describes the typical shoe size for these data? Set B 8 Frequency 99, 64, 80, 77, 59, 72, 87, 79, 92, 88, 90, 42, 20, 89, 42, 100, 98, 84, 78, 91 Set A Frequency 15. Test Scores A philosophy professor records the following scores for a 100-point exam. 6 4 2 1 2 3 4 6 4 2 1 Number 2 3 4 Number 35. Test Scores The scores of a mathematics exam given to 600 science and engineering students at a college had a mean and standard deviation of 235 and 28, respectively. Use Chebychev’s Theorem to determine the intervals containing at least 34 and at least 89 of the scores. 36. Price of Gold The data represent the average prices of gold (in dollars per troy ounce) for the years 1985 through 2009. Use a graphing utility or computer to find the mean, variance, and standard deviation of the data. What percent of the data lie within two standard deviations of the mean? (Source: U.S. Bureau of Mines and U.S. Geological Survey) 318, 385, 386, 280, 446, 368, 363, 389, 272, 606, 448, 345, 332, 311, 699, 438, 361, 295, 365, 768, 383, 385, 280, 411, 950 37. Price of Silver The data represent the average prices of silver (in dollars per troy ounce) for the years 1990 through 2009. Use a graphing utility or computer to find the mean, variance, and standard deviation of the data. What percent of the data lie within one standard deviation of the mean? (Source: U.S. Bureau of Mines and U.S. Geological Survey) 4.82, 5.15, 5.00, 7.34, 4.04, 5.19, 4.39, 11.61, 3.94, 4.89, 4.62, 13.43, 4.30, 5.54, 4.91, 15.02, 5.29, 5.25, 6.69, 13.37