?Median When data are summarized in a frequency distribution, the median can be found by

Average The defunct website IncomeTaxList.com listed the “average” annual income for Florida as $35,031. What is the role of the term average in statistics? Should another term be used in place of average?

What’s Wrong? USA Today published a list consisting of the state tax on each gallon of gas. If we add the 50 state tax amounts and then divide by 50, we get 27.3 cents. Is the value of 27.3 cents the mean amount of state sales tax paid by all U.S. drivers? Why or why not?

Measures of Center In what sense are the mean, median, mode, and midrange measures of “center”?

Resistant Measures Here are four of the Verizon data speeds (Mbps) from Figure 3-1: 13.5, 10.2, 21.1, 15.1. Find the mean and median of these four values. Then find the mean and median after including a fifth value of 142, which is an outlier. (One of the Verizon data speeds is 14.2 Mbps, but 142 is used here as an error resulting from an entry with a missing decimal point.) Compare the two sets of results. How much was the mean affected by the inclusion of the outlier? How much is the median affected by the inclusion of the outlier?

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Football Player Numbers Listed below are the jersey numbers of 11 players randomly selected from the roster of the Seattle Seahawks when they won Super Bowl XLVIII. What do the results tell us? 89 91 55 7 20 99 25 81 19 82 60

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Football Player Weights Listed below are the weights in pounds of 11 players randomly selected from the roster of the Seattle Seahawks when they won Super Bowl XLVIII (the same players from the preceding exercise). Are the results likely to be representative of all National Football League (NFL) players? 189 254 235 225 190 305 195 202 190 252 305

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Celebrity Net Worth Listed below are the highest amounts of net worth (in millions of dollars) of celebrities. The celebrities are Tom Cruise, Will Smith, Robert De Niro, Drew Carey, George Clooney, John Travolta, Samuel L. Jackson, Larry King, Demi Moore, and Bruce Willis. What do the results tell us about the population of all celebrities? Based on the nature of the amounts, what can be inferred about their precision? 250 200 185 165 160 160 150 150 150 150

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. What Happens in Vegas . . . Listed below are prices in dollars for one night at different hotels located on Las Vegas Boulevard (the “Strip”). If you decide to stay at one of these hotels, what statistic is most relevant, other than the measures of center? Apart from price, identify one other important factor that would affect your choice. 212 77 121 104 153 264 195 244

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Hurricanes Listed below are the numbers of Atlantic hurricanes that occurred in each year. The data are listed in order by year, starting with the year 2000. What important feature of the data is not revealed by any of the measures of center? 8 9 8 7 9 15 5 6 8 4 12 7 8 2

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Peas in a Pod Biologists conducted experiments to determine whether a deficiency of carbon dioxide in the soil affects the phenotypes of peas. Listed below are the phenotype codes, where 1 = smooth-yellow, 2 = smooth-green, 3 = wrinkled-yellow, and 4 = wrinkled-green. Can the measures of center be obtained for these values? Do the results make sense? 2 1 1 1 1 1 1 4 1 2 2 1 2 3 3 2 3 1 3 1 3 1 3 2 2

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. TV Prices Listed below are selling prices (dollars) of TVs that are 60 inches or larger and rated as a “best buy” by Consumer Reports magazine. Are the resulting statistics representative of the population of all TVs that are 60 inches and larger? If you decide to buy one of these TVs, what statistic is most relevant, other than the measures of central tendency? 1800 1500 1200 1500 1400 1600 1500 950 1600 1150 1500 1750

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Cell Phone Radiation Listed below are the measured radiation absorption rates (in W>kg) corresponding to these cell phones: iPhone 5S, BlackBerry Z30, Sanyo Vero, Optimus V, Droid Razr, Nokia N97, Samsung Vibrant, Sony Z750a, Kyocera Kona, LG G2, and Virgin Mobile Supreme. The data are from the Federal Communications Commission (FCC). The media often report about the dangers of cell phone radiation as a cause of cancer. The FCC has a standard that a cell phone absorption rate must be 1.6 W>kg or less. If you are planning to purchase a cellphone, are any of the measures of center the most important statistic? Is there another statistic that is most relevant? If so, which one? 1.18 1.41 1.49 1.04 1.45 0.74 0.89 1.42 1.45 0.51 1.38

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Caffeine in Soft Drinks Listed below are measured amounts of caffeine (mg per 12 oz of drink) obtained in one can from each of 20 brands (7-UP, A&W Root Beer, Cherry Coke, . . . , Tab). Are the statistics representative of the population of all cans of the same 20 brands consumed by Americans? 0 0 34 34 34 45 41 51 55 36 47 41 0 0 53 54 38 0 41 47

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Firefighter Fatalities Listed below are the numbers of heroic firefighters who lost their lives in the United States each year while fighting forest fires. The numbers are listed in order by year, starting with the year 2000. What important feature of the data is not revealed by any of the measures of center? 20 18 23 30 20 12 24 9 25 15 8 11 15 34

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Foot Lengths Listed below are foot lengths in inches of randomly selected Army women measured in the 1988 Anthropometric Survey (ANSUR). Are the statistics representative of the current population of all Army women? 10.4 9.3 9.1 9.3 10.0 9.4 8.6 9.8 9.9 9.1 9.1

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Most Expensive Colleges Listed below are the annual costs (dollars) of tuition and fees at the 10 most expensive colleges in the United States for a recent year (based on data from U.S. News & World Report). The colleges listed in order are Columbia, Vassar, Trinity, George Washington, Carnegie Mellon, Wesleyan, Tulane, Bucknell, Oberlin, and Union. What does this “top 10” list tell us about those costs for the population of all U.S. college tuitions? 49,138 47,890 47,510 47,343 46,962 46,944 46,930 46,902 46,870 46,785

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Diamond Ring Listed below in dollars are the amounts it costs for marriage proposal packages at the different Major League Baseball stadiums. Five of the teams don’t allow proposals. Are there any outliers? 39 50 50 50 55 55 75 85 100 115 175 175 200 209 250 250 350 400 450 500 500 500 500 1500 2500

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Sales of LP Vinyl Record Albums Listed below are annual U.S. sales of vinyl record albums (millions of units). The numbers of albums sold are listed in chronological order, and the last entry represents the most recent year. Do the measures of center give us any information about a changing trend over time? 0.3 0.6 0.8 1.1 1.1 1.4 1.4 1.5 1.2 1.3 1.4 1.2 0.9 0.9 1 1.9 2.5 2.8 3.9 4.6 6.1

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. California Smokers In the California Health Interview Survey, randomly selected adults are interviewed. One of the questions asks how many cigarettes are smoked per day, and results are listed below for 50 randomly selected respondents. How well do the results reflect the smoking behavior of California adults? 9 10 10 20 40 50 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Critical Thinking. For Exercises 5–20, watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, (d) midrange, and then answer the given question. Speed Dating In a study of speed dating conducted at Columbia University, female subjects were asked to rate the attractiveness of their male dates, and a sample of the results is listed below (1 = not attractive; 10 = extremely attractive). Can the results be used to describe the attractiveness of the population of adult males? 5 8 3 8 6 10 3 7 9 8 5 5 6 8 8 7 3 5 5 6 8 7 8 8 8 7

In Exercises 21–24, find the mean and median for each of the two samples, then compare the two sets of results. Blood Pressure A sample of blood pressure measurements is taken from Data Set 1 “Body Data” in Appendix B, and those values (mm Hg) are listed below. The values are matched so that 10 subjects each have systolic and diastolic measurements. (Systolic is a measure of the force of blood being pushed through arteries, but diastolic is a measure of blood pressure when the heart is at rest between beats.) Are the measures of center the best statistics to use with these data? What else might be better? Systolic: 118 128 158 96 156 122 116 136 126 120 Diastolic: 80 76 74 52 90 88 58 64 72 82

In Exercises 21–24, find the mean and median for each of the two samples, then compare the two sets of results. Parking Meter Theft Listed below are amounts (in millions of dollars) collected from parking meters by Brinks and others in New York City during similar time periods. A larger data set was used to convict five Brinks employees of grand larceny. The data were provided by the attorney for New York City, and they are listed on the Data and Story Library (DASL) website. Do the limited data listed here show evidence of stealing by Brinks employees? Collection Contractor Was Brinks 1.3 1.5 1.3 1.5 1.4 1.7 1.8 1.7 1.7 1.6 Collection Contractor Was Not Brinks 2.2 1.9 1.5 1.6 1.5 1.7 1.9 1.6 1.6 1.8

In Exercises 21–24, find the mean and median for each of the two samples, then compare the two sets of results. Pulse Rates Listed below are pulse rates (beats per minute) from samples of adult males and females (from Data Set 1 “Body Data”in Appendix B). Does there appear to be a difference? Male: 86 72 64 72 72 54 66 56 80 72 64 64 96 58 66 Female: 64 84 82 70 74 86 90 88 90 90 94 68 90 82 80

In Exercises 21–24, find the mean and median for each of the two samples, then compare the two sets of results. Bank Queues Waiting times (in seconds) of customers at the Madison Savings Bank are recorded with two configurations: single customer line; individual customer lines. Carefully examine the data to determine whether there is a difference between the two data sets that is not apparent from a comparison of the measures of center. If so, what is it? Single Line 390 396 402 408 426 438 444 462 462 462 Individual Lines 252 324 348 372 402 462 462 510 558 600

Degrees of Freedom Five pulse rates randomly selected from Data Set 1 “Body Data” in Appendix B have a mean of 78.0 beats per minute. Four of the pulse rates are 82, 78, 56, and 84. a. Find the missing value. b. We need to create a list of n values that have a specific known mean. We are free to select any values we desire for some of the n values. How many of the n values can be freely assigned before the remaining values are determined? (The result is referred to as the number of degrees of freedom.)

Censored Data Data Set 15 “Presidents” in Appendix B lists the numbers of years that U.S. presidents lived after their first inauguration. As of this writing, five of the presidents are still alive and after their first inauguration they have lived 37 years, 25 years, 21 years, 13 years, and 5 years so far. We might use the values of 37+, 25+, 21+, 13+, and 5+, where the positive signs indicate that the actual value is equal to or greater than the current value. (These values are said to be censored at the current time that this list was compiled.) If you use the values in Data Set 15 and ignore the presidents who are still alive, what is the mean? If you use the values given in Data Set 15 along with the additional values of 37+, 25+, 21+, 13+, and 5+, what do we know about the mean? Do the two results dier by much?

Trimmed Mean Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. By deleting some low values and high values, the trimmed mean is more resistant. To find the 10% trimmed mean for a data set, first arrange the data in order, then delete the bottom 10% of the values and delete the top 10% of the values, then calculate the mean of the remaining values. Use the axial loads (pounds) of aluminum cans listed below (from Data Set 30 “Aluminum Cans” in Appendix B) for cans that are 0.0111 in. thick. An axial load is the force at which the top of a can collapses. Identify any outliers, then compare the median, mean, 10% trimmed mean, and 20% trimmed mean. 247 260 268 273 276 279 281 283 284 285 286 288 289 291 293 295 296 299 310 504

Harmonic Mean The harmonic mean is often used as a measure of center for data sets consisting of rates of change, such as speeds. It is found by dividing the number of values n by the sum of the reciprocals of all values, expressed as \(\frac{n}{\sum_{x}^{\frac{1}{x}}}\) (No value can be zero.) The author drove 1163 miles to a conference in Orlando, Florida. For the trip to the conference, the author stopped overnight, and the mean speed from start to finish was 38 mi>h. For the return trip, the author stopped only for food and fuel, and the mean speed from start to finish was 56 mi>h. Find the harmonic mean of 38 mi>h and 56 mi>h to find the true “average” speed for the round trip. Text Transcription: n/ x^/1/x

Large Data Sets from Appendix B. In Exercises 25-28, refer to the indicated data set in Appendix B. Use software or a calculator to find the means and medians. Tornadoes Use the F-scale measurements from the tornadoes listed in Data Set 22 “Tornadoes” in Appendix B. Among the 500 tornadoes, how many have missing F-scale measurements? (Caution: In some technologies, missing data are represented by a constant such as -9 or 9999.)

Large Data Sets from Appendix B. In Exercises 25-28, refer to the indicated data set in Appendix B. Use software or a calculator to find the means and medians. Earthquakes Use the magnitudes (Richter scale) of the 600 earthquakes listed in Data Set 21 “Earthquakes” in Appendix B. In 1989, the San Francisco Bay Area was struck with an earthquake that measured 7.0 on the Richter scale. That earthquake occurred during the warm- up period for the third game of the baseball World Series. Is the magnitude of that World Series earthquake an outlier when considered in the context of the sample data given in Data Set 21? Explain.

Large Data Sets from Appendix B. In Exercises 25-28, refer to the indicated data set in Appendix B. Use software or a calculator to find the means and medians. Body Temperatures Refer to Data Set 3 “Body Temperatures” in Appendix B and use the body temperatures for 12:00 AM on day 2. Do the results support or contradict the common belief that the mean body temperature is \(98.6^{\circ} \mathrm{F}\)? Text Transcription: 98.6 degrees F

Large Data Sets from Appendix B. In Exercises 25-28, refer to the indicated data set in Appendix B. Use software or a calculator to find the means and medians. Births Use the birth weights (grams) of the 400 babies listed in Data Set 4 “Births” in Appendix B. Examine the list of birth weights to make an observation about those numbers. How does that observation aect the way that the results should be rounded?

In Exercises 29-32, find the mean of the data summarized in the frequency distribution. Also, compare the computed means to the actual means obtained by using the original list of data values, which are as follows: (Exercise 29) 36.2 years; (Exercise 30) 44.1 years; (Exercise 31) 224.3; (Exercise 32) 255.1.

In Exercises 29-32, find the mean of the data summarized in the frequency distribution. Also, compare the computed means to the actual means obtained by using the original list of data values, which are as follows: (Exercise 29) 36.2 years; (Exercise 30) 44.1 years; (Exercise 31) 224.3; (Exercise 32) 255.1.

In Exercises 29-32, find the mean of the data summarized in the frequency distribution. Also, compare the computed means to the actual means obtained by using the original list of data values, which are as follows: (Exercise 29) 36.2 years; (Exercise 30) 44.1 years; (Exercise 31) 224.3; (Exercise 32) 255.1.

In Exercises 29-32, find the mean of the data summarized in the frequency distribution. Also, compare the computed means to the actual means obtained by using the original list of data values, which are as follows: (Exercise 29) 36.2 years; (Exercise 30) 44.1 years; (Exercise 31) 224.3; (Exercise 32) 255.1.

Weighted Mean A student of the author earned grades of A, C, B, A, and D. Those courses had these corresponding numbers of credit hours: 3, 3, 3, 4, and 1. The grading system assigns quality points to letter grades as follows: A = 4; B = 3; C = 2; D = 1; F = 0. Compute the grade-point average (GPA) and round the result with two decimal places. If the dean’s list requires a GPA of 3.00 or greater, did this student make the dean’s list?

Weighted Mean A student of the author earned grades of 63, 91, 88, 84, and 79 on her five regular statistics tests. She earned grades of 86 on the final exam and 90 on her class projects. Her combined homework grade was 70. The five regular tests count for 60% of the final grade, the final exam counts for 10%, the project counts for 15%, and homework counts for 15%. What is her weighted mean grade? What letter grade did she earn (A, B, C, D, or F)? Assume that a mean of 90 or above is an A, a mean of 80 to 89 is a B, and so on.

Geometric Mean The geometric mean is often used in business and economics for find- ing average rates of change, average rates of growth, or average ratios. To find the geometric mean of n values (all of which are positive), first multiply the values, then find the nth root of the product. For a 6-year period, money deposited in annual certificates of deposit had annual interest rates of 5.154%, 2.730%, 0.488%, 0.319%, 0.313%, and 0.268%. Identify the single percentage growth rate that is the same as the five consecutive growth rates by computing the geometric mean of 1.05154, 1.02730, 1.00488, 1.00319, 1.00313, and 1.00268.

Quadratic Mean The quadratic mean (or root mean square, or R.M.S.) is used in physi- cal applications, such as power distribution systems. The quadratic mean of a set of values is obtained by squaring each value, adding those squares, dividing the sum by the number of val- ues n, and then taking the square root of that result, as indicated below: Quadratic mean = \(\sqrt{\frac{\sum x^{2}}{n}}\) Find the R.M.S. of these voltages measured from household current: 0, 60, 110, -110, -60, 0. How does the result compare to the mean? Text Transcription: Sqrt sum x^2/n

Median When data are summarized in a frequency distribution, the median can be found by first identifying the median class, which is the class that contains the median. We then assume that the values in that class are evenly distributed and we interpolate. Letting n denote the sum of all class frequencies, and letting m denote the sum of the class frequencies that precede the median class, the median can be estimated as shown below. (lower limit of median class) + (class width) \(\left(\frac{\left(\frac{n+1}{2}\right)-(m+1)}{\text { frequency of median class }}\right)\) Use this procedure to find the median of the frequency distribution given in Table 3-2 on page 88. How far is that result from the median found from the original list of McDonald’s lunch service times listed in Data Set 25 “Fast Food” in Appendix B? Text Transcription: ((n+1/2)-(m+1)/frequency of median class)