2-4 Measures of Center

Transcription

1 -4 Measures of Center 59-4 Measures of Center Remember that the main objective of this chapter is to master basic tools for measuring and describing different characteristics of a set of data. In Section -1 we noted that when describing, exploring, and comparing data sets, these characteristics are usually extremely important: center, variation, distribution, outliers, changes over time. The mnemonic of CVDOT ( Computer Viruses Destroy or Terminate ) is helpful for remembering those characteristics. In Sections - and -3 we saw that frequency distributions and graphs, such as histograms, are helpful in investigating distribution. The focus of this section is the characteristic of center. Definition A measure of center is a value at the center or middle of a data set. There are several different ways to determine the center, so we have different definitions of measures of center, including the mean, median, mode, and midrange. We begin with the mean.

2 60 C HAPTER Describing, Exploring, and Comparing Data Mean The (arithmetic) mean is generally the most important of all numerical measurements used to describe data, and it is what most people call an average. Six Degrees of Separation Social psychologists, historians, political scientists, and communications specialists are interested in the Small World Problem : Given any two people in the world, how many intermediate links are necessary to connect the two original people? Social psychologist Stanley Milgram conducted an experiment in which subjects tried to contact other target people by mailing an information folder to an acquaintance who they thought would be closer to the target. Among 160 such chains that were initiated, only 44 were completed. The number of intermediate acquaintances varied from to 10, with a median of 5 (or six degrees of separation ). The experiment has been criticized for including very social subjects and for no adjustments for many lost connections from people with lower incomes. Another mathematical study showed that if the missing chains were completed, the median would be slightly greater than 5. Definition The arithmetic mean of a set of values is the measure of center found by adding the values and dividing the total by the number of values. This measure of center will be used often throughout the remainder of this text, and it will be referred to simply as the mean. This definition can be expressed as Formula -1, which uses the Greek letter (uppercase Greek sigma) to indicate that the data values should be added. That is, x represents the sum of all data values. The symbol n denotes the sample size, which is the number of values in the data set. Formula -1 mean 5 x n The mean is denoted by x (pronounced x-bar ) if the data set is a sample from a larger population; if all values of the population are used, then we denote the mean by m (lowercase Greek mu). (Sample statistics are usually represented by English letters, such as x, and population parameters are usually represented by Greek letters, such as m.) Notation x n N x 5 x n Sx m 5 N denotes the addition of a set of values. is the variable usually used to represent the individual data values. represents the number of values in a sample. represents the number of values in a population. is the mean of a set of sample values. is the mean of all values in a population. EXAMPLE Monitoring Lead in Air Listed below are measured amounts of lead (in micrograms per cubic meter, or mg> m 3 ) in the air. The Environmental Protection Agency has established an air quality standard for lead: 1.5 mg> m 3. The measurements shown below were recorded at Building 5 of the World Trade Center site on different days immediately following the destruction caused by the terrorist attacks of September 11, 001. After the col-

3 -4 Measures of Center 61 lapse of the two World Trade Center buildings, there was considerable concern about the quality of the air. Find the mean for this sample of measured levels of lead in the air SOLUTION The mean is computed by using Formula -1. First add the values, then divide by the number of values: x 5 Sx n The mean lead level is mg> m 3. Apart from the value of the mean, it is also notable that the data set includes one value (5.40) that is very far away from the others. It would be wise to investigate such an outlier. In this case, the lead level of 5.40 mg> m 3 was measured the day after the collapse of the two World Trade Center towers, and the excessive levels of dust and smoke provide a reasonable explanation for such an extreme value. One disadvantage of the mean is that it is sensitive to every value, so one exceptional value can affect the mean dramatically. The median largely overcomes that disadvantage. Median Definition The median of a data set is the measure of center that is the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude. The median is often denoted by x (pronounced x-tilde ). To find the median, first sort the values (arrange them in order), then follow one of these two procedures: 1. If the number of values is odd, the median is the number located in the exact middle of the list.. If the number of values is even, the median is found by computing the mean of the two middle numbers. Figure -10 demonstrates this procedure for finding the median. Class Size Paradox There are at least two ways to obtain the mean class size, and they can have very different results. At one college, if we take the numbers of students in 737 classes, we get a mean of 40 students. But if we were to compile a list of the class sizes for each student and use this list, we would get a mean class size of 147. This large discrepancy is due to the fact that there are many students in large classes, while there are few students in small classes. Without changing the number of classes or faculty, we could reduce the mean class size experienced by students by making all classes about the same size. This would also improve attendance, which is better in smaller classes. EXAMPLE Monitoring Lead in Air Listed below are measured amounts of lead (in mg> m 3 ) in the air. Find the median for this sample SOLUTION First sort the values by arranging them in order: continued

4 6 C HAPTER Describing, Exploring, and Comparing Data Mean: Median: Find the sum of all values, then divide by the number of values. Sort the data. (Arrange in order.) Odd number of values Even number of values Median is the value in the exact middle. Add the two middle numbers, then divide by. Mode: Midrange: Value that occurs most frequently highest value + lowest value FIGURE -10 Procedures for Finding Measures of Center Because the number of values is an even number (6), the median is found by computing the mean of the two middle values of 0.73 and Median Because the number of values is an even number (6), the median is the number in the exact middle of the sorted list, so the median is mg> m 3. Note that the median is very different from the mean of mg> m 3 that was found from the same set of sample data in the preceding example. The reason for this large discrepancy is the effect that 5.40 had on the mean. If this extreme value were reduced to 1.0, the mean would drop from mg> m 3 to mg> m 3, but the median would not change. EXAMPLE Monitoring Lead in Air Repeat the preceding example after including the measurement of 0.66 mg> m 3 recorded on another day. That is, find the median of these lead measurements: SOLUTION First arrange the values in order: Because the number of values is an odd number (7), the median is the value in the exact middle of the sorted list: 0.73 mg> m 3. After studying the preceding two examples, the procedure for finding the median should be clear. Also, it should be clear that the mean is dramatically affected by extreme values, whereas the median is not dramatically affected. Because the median is not so sensitive to extreme values, it is often used for data sets with a relatively small number of extreme values. For example, the U.S. Census Bureau recently reported that the median household income is $36,078 annually. The median was used because there is a small number of households with really high incomes.

5 -4 Measures of Center 63 Mode Definition The mode of a data set, often denoted by M, is the value that occurs most frequently. When two values occur with the same greatest frequency, each one is a mode and the data set is bimodal. When more than two values occur with the same greatest frequency, each is a mode and the data set is said to be multimodal. When no value is repeated, we say that there is no mode. EXAMPLE Find the modes of the following data sets. a b c SOLUTION a. The number 1.10 is the mode because it is the value that occurs most often. b. The numbers 7 and 55 are both modes because they occur with the same greatest frequency. This data set is bimodal because it has two modes. c. There is no mode because no value is repeated. In reality, the mode isn t used much with numerical data. But among the different measures of center we are considering, the mode is the only one that can be used with data at the nominal level of measurement. (Recall that the nominal level of measurement applies to data that consist of names, labels, or categories only.) For example, a survey of college students showed that 84% have TVs, 76% have VCRs, 60% have portable CD players, 39% have video game systems, and 35% have DVD players (based on data from the National Center for Education Statistics). Because TVs are most frequent, we can say that the mode is TV. We cannot find a mean or median for such data at the nominal level. Just an Average Guy Men s Health magazine published statistics describing the average guy, who is 34.4 years old, weighs 175 pounds, is about 5 ft 10 in. tall, and is named Mike Smith. The age, weight, and height are all mean values, but the name of Mike Smith is the mode that corresponds to the most common first and last names. Other notable statistics: The average guy sleeps about 6.9 hours each night, drinks about 3.3 cups of coffee each day, and consumes 1. alcoholic drinks per day. He earns about $36,100 per year, and has debts of $563 from two credit cards. He has banked savings of $3100. Midrange Definition The midrange is the measure of center that is the value midway between the highest and lowest values in the original data set. It is found by adding the highest data value to the lowest data value and then dividing the sum by, as in the following formula. highest value 1 lowest value midrange 5

6 64 C HAPTER Describing, Exploring, and Comparing Data Mannequins Reality Health magazine compared measurements of mannequins to measurements of women. The following results were reported as averages, which were presumably means. Height of mannequins: 6 ft; height of women: 5 ft 4 in. Waist of mannequins: 3 in.; waist of women: 9 in. Hip size of mannequins: 34 in.; hip size of women: 40 in. Dress size of mannequins: 6; dress size of women: 11. It becomes apparent that when comparing means, mannequins and real women are very different. The midrange is rarely used. Because it uses only the highest and lowest values, it is too sensitive to those extremes. However, the midrange does have three redeeming features: (1) It is easy to compute; () it helps to reinforce the important point that there are several different ways to define the center of a data set; (3) it is sometimes incorrectly used for the median, so confusion can be reduced by clearly defining the midrange along with the median. EXAMPLE Monitoring Lead in Air Listed below are measured amounts of lead (in mg> m 3 ) in the air from the site of the World Trade Center on different days after September 11, 001. Find the midrange for this sample SOLUTION The midrange is.910 mg> m 3. The midrange is found as follows: highest value 1 lowest value Unfortunately, the term average is sometimes used for any measure of center and is sometimes used for the mean. Because of this ambiguity, we should not use the term average when referring to a particular measure of center. Instead, we should use the specific term, such as mean, median, mode, or midrange. When encountering a reported value as being an average, we should know that the value could be the result of any of several different definitions. In the spirit of describing, exploring, and comparing data, we provide Table -8 which summarizes the different measures of center for the cotinine levels listed in Table -1 in the Chapter Problem. Recall that cotinine is a metabolite of nicotine, so that when nicotine is absorbed by the body, cotinine is produced. A comparison of the measures of center suggests that cotinine levels are highest in smokers. Also, the cotinine levels in nonsmokers exposed to smoke are higher than nonsmokers not exposed. This suggests that secondhand smoke does have Table -8 Comparison of Cotinine Levels in Smokers, Nonsmokers Exposed to Environmental Tobacco Smoke (ETS), and Nonsmokers Not Exposed to Environmental Tobacco Smoke (NOETS) Smokers ETS NOETS Mean Median Mode 1 and Midrange

7 -4 Measures of Center 65 an effect. There are methods for determining whether such apparent differences are statistically significant, and we will consider some of those methods later in this text. Round-Off Rule A simple rule for rounding answers is this: Carry one more decimal place than is present in the original set of values. When applying this rule, round only the final answer, not intermediate values that occur during calculations. Thus the mean of, 3, 5, is , which is rounded to 3.3. Because the original values are whole numbers, we rounded to the nearest tenth. As another example, the mean of 80.4 and 80.6 is (one more decimal place than was used for the original values). Mean from a Frequency Distribution When data are summarized in a frequency distribution, we might not know the exact values falling in a particular class. To make calculations possible, we pretend that in each class, all sample values are equal to the class midpoint. Because each class midpoint is repeated a number of times equal to the class frequency, the sum of all sample values is S( f? x), where f denotes frequency and x represents the class midpoint. The total number of sample values is the sum of frequencies Sf. Formula - is used to compute the mean when the sample data are summarized in a frequency distribution. Formula - is not really a new concept; it is simply a variation of Formula -1. Formula - First multiply each frequency and class midpoint, then add the products. T x 5 Ss f? xd Sf c sum of frequencies (mean from frequency distribution) For example, see Table -9 on the following page. The first two columns duplicate the frequency distribution (Table -) for the cotinine levels of smokers. Table -9 illustrates the procedure for using Formula - when calculating a mean from data summarized in a frequency distribution. In reality, software or calculators are generally used in place of manual calculations. Table -9 results in x , but we get x if we use the original list of 40 values. Remember, the frequency distribution yields an approximation of x, because it is not based on the exact original list of sample values.

8 66 C HAPTER Describing, Exploring, and Comparing Data Table -9 Finding the Mean from a Frequency Distribution Cotinine Level Frequency f Class Midpoint x f? x Not At Home Pollsters cannot simply ignore those who were not at home when they were called the first time. One solution is to make repeated callback attempts until the person can be reached. Alfred Politz and Willard Simmons describe a way to compensate for those missing results without making repeated callbacks. They suggest weighting results based on how often people are not at home. For example, a person at home only two days out of six will have a > 6 or 1> 3 probability of being at home when called the first time. When such a person is reached the first time, his or her results are weighted to count three times as much as someone who is always home. This weighting is a compensation for the other similar people who are home two days out of six and were not at home when called the first time. This clever solution was first presented in Totals: Sf 40 S(f x) x 5 Ssf? xd Sf Weighted Mean In some cases, the values vary in their degree of importance, so we may want to weight them accordingly. We can then proceed to compute a weighted mean, which is a mean computed with the different values assigned different weights, as shown in Formula -3. Ssw? xd Formula -3 weighted mean: x 5 Sw For example, suppose we need a mean of three test scores (85, 90, 75), but the first test counts for 0%, the second test counts for 30%, and the third test counts for 50% of the final grade. We can assign weights of 0, 30, and 50 to the test scores, then proceed to calculate the mean by using Formula -3 as follows: x 5 5 Ssw? xd Sw s0 3 85d 1 s d 1 s d As another example, college grade-point averages can be computed by assigning each letter grade the appropriate number of points (A 5 4, B 5 3, etc.), then assigning to each number a weight equal to the number of credit hours. Again, Formula -3 can be used to compute the grade-point average. The Best Measure of Center So far, we have considered the mean, median, mode, and midrange as measures of center. Which one of these is best? Unfortunately, there is no single best answer to that question because there are no objective criteria for determining the most representative measure for all data sets. The different measures of center have different advantages and disadvantages, some of which are summarized in Table -10.

9 -4 Measures of Center 67 Table -10 Comparison of Mean, Median, Mode, and Midrange Takes Every Affected by Advantages Measure How Value into Extreme and of Center Definition Common? Existence Account? Values? Disadvantages Mean x 5 Sx most always yes yes used throughout n familiar exists this book; works average well with many statistical methods Median middle value commonly always no no often a good choice used exists if there are some extreme values Mode most frequent sometimes might not no no appropriate for data data value used exist; may be at the nominal level more than one mode Midrange high 1 low rarely always no yes very sensitive used exists to extreme values General comments: For a data collection that is approximately symmetric with one mode, the mean, median, mode, and midrange tend to be about the same. For a data collection that is obviously asymmetric, it would be good to report both the mean and median. The mean is relatively reliable. That is, when samples are drawn from the same population, the sample means tend to be more consistent than the other measures of center (consistent in the sense that the means of samples drawn from the same population don t vary as much as the other measures of center). An important advantage of the mean is that it takes every value into account, but an important disadvantage is that it is sometimes dramatically affected by a few extreme values. This disadvantage can be overcome by using a trimmed mean, as described in Exercise 1. Skewness A comparison of the mean, median, and mode can reveal information about the characteristic of skewness, defined below and illustrated in Figure -11. Definition A distribution of data is skewed if it is not symmetric and extends more to one side than the other. (A distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half.)

10 68 C HAPTER Describing, Exploring, and Comparing Data FIGURE -11 Skewness Mean Mode Median (a) Skewed to the Left (Negatively Skewed): The mean and median are to the left of the mode. Mode = Mean = Median (b) Symmetric (Zero Skewness): The mean, median, and mode are the same. Mode Mean Median (c) Skewed to the Right (Positively Skewed): The mean and median are to the right of the mode. Data skewed to the left (also called negatively skewed) have a longer left tail, and the mean and median are to the left of the mode. Although not always predictable, data skewed to the left generally have a mean less than the median, as in Figure -11(a). Data skewed to the right (also called positively skewed) have a longer right tail, and the mean and median are to the right of the mode. Again, although not always predictable, data skewed to the right generally have the mean to the right of the median, as in Figure -11(c). If we examine the histogram in Figure -1 for the cotinine levels of smokers, we see a graph that appears to be skewed to the right. In practice, many distributions of data are symmetric and without skewness. Distributions skewed to the right are more common than those skewed to the left because it s often easier to get exceptionally large values than values that are exceptionally small. With annual incomes, for example, it s impossible to get values below the lower limit of zero, but there are a few people who earn millions of dollars in a year. Annual incomes therefore tend to be skewed to the right, as in Figure -11(c). Using Technology The calculations of this section are fairly simple, but some of the calculations in the following sections require more effort. Many computer software programs allow you to enter a data set and use one operation to get several different sample statistics, referred to as descriptive statistics. (See Section -6 for sample displays resulting from STATDISK, Minitab, Excel, and the TI-83 Plus calculator.) Here are some of the procedures for obtaining such displays. STATDISK Choose the main menu item of Data, and use the Sample Editor to enter the data. Click on Copy, then click on Data once again, but now select Descriptive Statistics. Click on Paste to get the data set that was entered. Now click on Evaluate to get the various descriptive statistics, including the mean, median, midrange, and other statistics to be discussed in the following sections. Minitab Enter the data in the column with the heading C1. Click on Stat, select Basic Statistics, then select Descriptive Statistics. The results will include the mean and median as well as other statistics. Excel Enter the sample data in column A. Select Tools, then Data Analysis, then select Descriptive Statistics and click OK. In the dialog box, enter the input range (such as A1:A40 for 40 values in column A), click on Summary Statistics, then click OK. (If Data Analysis does not appear in the Tools menu, it must be installed by clicking on Tools and selecting Add-Ins.) TI-83 Plus First enter the data in list L1 by pressing STAT, then selecting Edit and pressing the ENTER key. After the data values have been entered, press STAT and select CALC, then select 1-Var Stats and press the ENTER key twice. The display will include the mean x, the median, the minimum value, and the maximum value. Use the down-arrow key to view the results that don t fit on the initial display.

11 -4 Measures of Center 69-4 Basic Skills and Concepts In Exercises 1 8, find the (a) mean, (b) median, (c) mode, and (d) midrange for the given sample data. 1. Tobacco Use in Children s Movies In Tobacco and Alcohol Use in G-Rated Children s Animated Films, by Goldstein, Sobel, and Newman (Journal of the American Medical Association, Vol. 81, No. 1), the lengths (in seconds) of scenes showing tobacco use were recorded for animated movies from Universal Studios. The first six values included in Data Set 7 from Appendix B are listed below. Is there any problem with including scenes of tobacco use in animated children s films? Harry Potter In an attempt to measure the reading level of a book, the Flesch Reading Ease ratings are obtained for 1 randomly selected pages from Harry Potter and the Sorcerer s Stone by J. K. Rowling. Those values, included in Data Set 14 from Appendix B, are listed below. Given that these ratings are based on 1 randomly selected pages, is the mean of this sample likely to be a reasonable estimate of the mean reading level of the whole book? Cereal A dietitian obtains the amounts of sugar (in grams) from 1 gram in each of 16 different cereals, including Cheerios, Corn Flakes, Fruit Loops, Trix, and 1 others. Those values, included in Data Set 16 from Appendix B, are listed below. Is the mean of those values likely to be a good estimate of the mean amount of sugar in each gram of cereal consumed by the population of all Americans who eat cereal? Why or why not? Body Mass Index As part of the National Health Examination, the body mass index is measured for a random sample of women. Some of the values included in Data Set 1 from Appendix B are listed below. Is the mean of this sample reasonably close to the mean of 5.74, which is the mean for all 40 women included in Data Set 1? Drunk Driving The blood alcohol concentrations of a sample of drivers involved in fatal crashes and then convicted with jail sentences are given below (based on data from the U.S. Department of Justice). Given that current state laws prohibit driving with levels above 0.08 or 0.10, does it appear that these levels are significantly above the maximum that is allowed? Motorcycle Fatalities Listed below are ages of motorcyclists when they were fatally injured in traffic crashes (based on data from the U.S. Department of Transportation). Do the results support the common belief that such fatalities are incurred by a greater proportion of younger drivers?

12 70 C HAPTER Describing, Exploring, and Comparing Data 7. Reaction Times The author visited the Reuben H. Fleet Science Museum in San Diego and repeated an experiment of reaction times. The following times (in hundredths of a second) were obtained. How consistent are these results, and how does the consistency affect the usefulness of the sample mean as an estimate of the population mean? Bufferin Tablets Listed below are the measured weights (in milligrams) of a sample of Bufferin aspirin tablets. What is a serious consequence of having weights that vary too much? In Exercises 9 1, find the mean, median, mode, and midrange for each of the two samples, then compare the two sets of results. 9. Customer Waiting Times Waiting times (in minutes) of customers at the Jefferson Valley Bank (where all customers enter a single waiting line) and the Bank of Providence (where customers wait in individual lines at three different teller windows): Jefferson Valley: Providence: Interpret the results by determining 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? 10. Regular Coke> Diet Coke Weights (pounds) of samples of the contents in cans of regular Coke and diet Coke: Regular: Diet: Does there appear to be a significant difference between the two data sets? How might such a difference be explained? 11. Mickey D vs. Jack When investigating times required for drive-through service, the following results (in seconds) are obtained (based on data from QSR Drive-Thru Time Study). McDonald s: Jack in the Box: Which of the two fast-food giants appears to be faster? Does the difference appear to be significant? 1. Skull Breadths Maximum breadth of samples of male Egyptian skulls from 4000 B.C. and 150 A.D. (based on data from Ancient Races of the Thebaid by Thomson and Randall-Maciver): 4000 B.C.: A.D.: Changes in head sizes over time suggest interbreeding with people from other regions. Do the head sizes appear to have changed from 4000 B.C. to 150 A.D.?

13 -4 Measures of Center 71 In Exercises 13 16, refer to the data set in Appendix B. Use computer software or a calculator to find the means and medians, then compare the results as indicated. T 13. Head Circumferences In order to correctly diagnose the disorder of hydrocephalus, a pediatrician investigates head circumferences of -year-old males and females. Use the sample results listed in Data Set 3. Does there appear to be a difference between the two genders? T 14. Clancy, Rowling, Tolstoy A child psychologist investigates differences in reading difficulty and obtains data from The Bear and the Dragon by Tom Clancy, Harry Potter and the Sorcerer s Stone by J. K. Rowling, and War and Peace by Leo Tolstoy. Refer to Data Set 14 in Appendix B and use the Flesch-Kincaid Grade Level ratings for 1 pages randomly selected from each of the three books. Do the ratings appear to be different? T 15. Weekend Rainfall Using Data Set 11 in Appendix B, find the mean and median of the rainfall amounts in Boston on Thursday and find the mean and median of the rainfall amounts in Boston on Sunday. Media reports claimed that it rains more on weekends than during the week. Do these results appear to support that claim? T 16. Tobacco> Alcohol Use in Children s Movies In Tobacco and Alcohol Use in G- Rated Children s Animated Films, by Goldstein, Sobel, and Newman (Journal of the American Medical Association, Vol. 81, No. 1), the lengths (in seconds) of scenes showing tobacco use and alcohol use were recorded for animated children s movies. Refer to Data Set 7 in Appendix B and find the mean and median for the tobacco times, then find the mean and median for the alcohol times. Does there appear to be a difference between those times? Which appears to be the larger problem: scenes showing tobacco use or scenes showing alcohol use? In Exercises 17 0, find the mean of the data summarized in the given frequency distribution. 17. Old Faithful Visitors to Yellowstone National Park consider an eruption of the Old Faithful geyser to be a major attraction that should not be missed. The given frequency distribution summarizes a sample of times (in minutes) between eruptions. 18. Loaded Die The author drilled a hole in a die and filled it with a lead weight, then proceeded to roll it 00 times. The results are given in the frequency distribution in the margin. Does the result appear to be very different from the result that would be expected with an unmodified die? 19. Speeding Tickets The given frequency distribution describes the speeds of drivers ticketed by the Town of Poughkeepsie police. These drivers were traveling through a 30 mi> h speed zone on Creek Road, which passes the authors college. How does the mean compare to the posted speed limit of 30 mi> h? 0. Body Temperatures The accompanying frequency distribution summarizes a sample of human body temperatures. (See the temperatures for midnight on the second day, as listed in Data Set 4 in Appendix B.) How does the mean compare to the value of 98.6 F, which is the value assumed to be the mean by most people? T -4 Beyond the Basics 1. Trimmed Mean Because the mean is very sensitive to extreme values, we say that it is not a resistant measure of center. The trimmed mean is more resistant. To find the Table for Exercise 17 Time Frequency Table for Exercise 18 Outcome Frequency Table for Exercise 19 Speed Frequency Table for Exercise 0 Temperature Frequency

14 7 C HAPTER Describing, Exploring, and Comparing Data 10% trimmed mean for a data set, first arrange the data in order, then delete the bottom 10% of the values and the top 10% of the values, then calculate the mean of the remaining values. For the weights of the bears in Data Set 9 from Appendix B, find (a) the mean; (b) the 10% trimmed mean; (c) the 0% trimmed mean. How do the results compare?. Mean of Means Using an almanac, a researcher finds the mean teacher s salary for each state. He adds those 50 values, then divides by 50 to obtain their mean. Is the result equal to the national mean teacher s salary? Why or why not? 3. Degrees of Freedom Ten values have a mean of Nine of the values are 6, 78, 90, 87, 56, 9, 70, 70, and 93. a. Find the tenth 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? 4. Censored Data An experiment is conducted to test the lives of car batteries. The experiment is run for a fixed time of five years. (The test is said to be censored at five years.) The sample results (in years) are.5, 3.4, 1., 51, 51 (where 51 indicates that the battery was still working at the end of the experiment). What can you conclude about the mean battery life? 5. Weighted Mean Kelly Bell gets quiz grades of 65, 83, 80, and 90. She gets a 9 on her final exam. Find the weighted mean if the quizzes each count for 15% and the final counts for 40% of the final grade. 6. Transformed Data In each of the following, describe how the mean, median, mode, and midrange of a data set are affected. a. The same constant k is added to each value of the data set. b. Each value of the data set is multiplied by the same constant k. 7. The harmonic mean is often used as a measure of central tendency 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 (No value can be zero.) Four students drive from New York to Florida (100 miles) at a speed of 40 mi> h (yeah, right!). Because they need to make it to statistics class on time, they return at a speed of 60 mi> h. What is their average speed for the round trip? (The harmonic mean is used in averaging speeds.) 8. The geometric mean is often used in business and economics for finding average rates of change, average rates of growth, or average ratios. Given n values (all of which are positive), the geometric mean is the nth root of their product. The average growth factor for money compounded at annual interest rates of 10%, 8%, 9%, 1%, and 7% can be found by computing the geometric mean of 1.10, 1.08, 1.09, 1.1, and Find that average growth factor. 9. The quadratic mean (or root mean square, or RMS) is usually used in physical applications. In power distribution systems, for example, voltages and currents are usually referred to in terms of their RMS values. The quadratic mean of a set of values is n S 1 x

15 -5 Measures of Variation 73 obtained by squaring each value, adding the results, dividing by the number of values n, and then taking the square root of that result, expressed as quadratic mean 5 Å Sx Find the RMS of these power supplies (in volts): 110, 0, 60, Median When data are summarized in a frequency distribution, the median can be found by first identifying the median class (the class that contains the median). We then assume that the values in that class are evenly distributed and we can interpolate. This process can be described by n 1 1 slower limit of median classd 1 sclass widthd a b sm 1 1d frequency of median class where n is the sum of all class frequencies and m is the sum of the class frequencies that precede the median class. Use this procedure to find the median of the data set summarized in Table -. How does the result compare to the median of the original list of data, which is 170? Which value of the median is better: the value computed for the frequency table or the value of 170? n -5 Measures of Variation Study Hint: Because this section introduces the concept of variation, which is so important in statistics, this is one of the most important sections in the entire book. First read through this section quickly and gain a general understanding of the characteristic of variation. Next, learn how to obtain measures of variation, especially the standard deviation. Finally, try to understand the reasoning behind the formula for standard deviation, but do not spend much time memorizing formulas or doing arithmetic calculations. Instead, place a high priority on learning how to interpret values of standard deviation. For a visual illustration of variation, see Figure -1 on page 74, which includes samples of bolts from two different companies. Because these bolts are to be used for attaching wings to airliners, their quality is quite important. If we consider only the mean, we would not recognize any difference between the two samples, because they both have a mean of x in. However, it should be obvious that the samples are very different in the amounts that the bolts vary in length. The bolts manufactured by the Precision Bolt Company appear to be very similar in length, whereas the lengths from the Ruff Bolt Company vary by large amounts. This is exactly the same issue that is so important in many different manufacturing processes. Better quality is achieved through lower variation. In this section, we want to develop the ability to measure and understand variation. Another ideal situation illustrating the importance of variation can be seen in the waiting lines at banks. In times past, many banks required that customers wait in separate lines at each teller s window, but most have now changed to one single main waiting line. Why did they make that change? The mean waiting time didn t change, because the waiting-line configuration doesn t affect the efficiency of the