784 CHAPTER 12 Statistics Most people have an intuitive idea of what is meant by an “average.” The term is used daily in many familiar ways. “This car averages 19 miles per gallon,” “The average test grade was 82,” and “The average height of a blooming sunflower is 5.5 ft” are three examples. In this section, we will introduce four different averages and discuss the circumstances in which each average is used. Measures of Central Tendency and Position SECTION 12.3 LEARNING GOALS Upon completion of this section, you will be able to: 7 Calculate measures of central tendency of a set of data, including the mean, median, mode, and midrange. 7 Calculate the measures of position of a set of data including percentiles and quartiles. Why This Is Important An average is one of the most common ways to represent a set of data. Often, a set of data will have a different value for each different type of average. Understanding the appropriate average to use in each circumstance will enable you to make accurate conclusions about the data. Measures of Central Tendency An average is a number that is representative of a group of data. There are at least four different averages: the mean, the median, the mode, and the midrange. Each is calculated differently and may yield different results for the same set of data. Often, the averages will result in a number near the center of the data; for this reason, averages are commonly referred to as measures of central tendency. The arithmetic mean, or simply the mean, is symbolized either by x (read “x bar”) or by the Greek letter mu, μ. The symbol x is used when the mean of a sample of the population is calculated. The symbol μ is used when the mean of the entire population is calculated. Unless otherwise indicated, we will assume that the data featured in this book represent samples; therefore, we will use x for the mean. The Greek letter sigma, ,Σ is used to indicate “summation.” The notation x, Σ read “the sum of x,” is used to indicate the sum of all the data. For example, consider the following data: 4, 6, 1, 0, 5. Then x 4 6 1 0 5 16. Σ = + + + + = Alessandro Landi/123RF c) Construct a frequency distribution containing 12 classes. Answers will vary. d) Construct a histogram from the frequency distribution in part (c). Answers will vary. e) Construct a frequency polygon of the frequency distribution in part (c). Answers will vary. 44. Social Security Numbers Repeat Exercise 43 for the last digit of the students’ Social Security numbers. Include classes for the digits 0–9. a)– e) Answers will vary. Recreational Mathematics 45. a) Count the F’s Count the number of F’s in the sentence at the bottom of the Recreational Math box on page 771. There are 6 F’s. b) Can you explain why so many people count the number of F’s incorrectly? Answers will vary. 46. Vitamins In what month do people take the least number of daily vitamins? February, since it has the fewest number of days Research Activity 47. Social Security Over the years many changes have been made in the U.S. Social Security System. a) Do research and determine the number of people receiving Social Security benefits for the years 1945, 1950, 1955, 1960, . . . , 2015, 2020. Then construct a frequency distribution and histogram of the data. b) Determine the maximum amount that self-employed individuals had to pay into Social Security (the FICA tax) for the years 1945, 1950, 1955, 1960, . . . , 2015, 2020. Then construct a frequency distribution and a histogram of the data.
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