12.3 Measures of Central Tendency and Position 787 The last average we will discuss is the midrange. The midrange is the value halfway between the lowest (L) and highest (H) values in a set of data. It is determined by adding the lowest and highest values and dividing the sum by 2. A formula for determining the midrange follows. Midrange The midrange of a set of data can be calculated using the following formula. Midrange lowest value highest value 2 = + Example 5 Determine the Midrange Determine the midrange of the nurses’ ages given in Example 1. Solution The ages of the nurses are 30, 21, 51, 37, and 51. The lowest age is 21, and the highest age is 51. Midrange lowest highest 2 21 51 2 72 2 36 years = + = + = = 7 Now try Exercise 23. The “average” of the ages 30, 21, 51, 37, 51 can be considered any one of the following values: 38 (mean), 37 (median), 51 (mode), or 36 (midrange). Which average do you feel is most representative of the ages? We will discuss this question later in this section. MATHEMATICS TODAY Average Wealth of U.S. Households According to the Federal Reserve’s Survey of Consumer Finances, the 2022 mean net worth of U.S. households was about $1,063,770. Net worth is the total amount of your assets (such as bank account balances, retirement accounts, your home, your car) minus your debt. The mean net worth was inflated by the nation’s super wealthy, including the net worth of people such as Jeff Bezos, Bill Gates, and Warren Buffett. The 2022 median U.S. household net worth was about $192,900. In your opinion, would you say the 2022 average U.S. household net worth was $1,063,770 or $192,000? Why This Is Important This example shows that even though the same data are used, different averages can vary widely. When making conclusions about a set of data using the average, it is important to know which average was used. Example 6 Measures of Central Tendency The salaries, in thousands of dollars, of eight social workers are 43, 28, 31, 38, 45, 63, 63, and 76. For this set of data, determine the (a) mean, (b) median, (c) mode, and (d) midrange. Then (e) list the measures of central tendency from lowest to highest. Solution a) x x n 43 28 31 38 45 63 63 76 8 387 8 48.375 = Σ = + + + + + + + = = The mean is 48.375 b) Ranking the data from the smallest to largest gives 28, 31, 38, 43, 45, 63, 63, 76 Since there are an even number of pieces of data, the median is halfway between 43 and 45. The median 43 45 2 88 2 44. = + = = c) The mode is the piece of data that occurs most frequently. The mode is 63. d) The midrange L H 2 28 76 2 104 2 52. = + = + = = e) The averages from lowest to highest are the median, mean, midrange, and mode. Their values are 44, 48.375, 52, and 63, respectively. 7 Now try Exercise 27. Data Sets: Mean, Median, Mode At this point, you should be able to calculate the four measures of central tendency: mean, median, mode, and midrange. Now let’s examine the circumstances in which each should be used.
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