Section 3.1 Measures of Center 91 EXAMPLE 3.4 Using technology to compute the mean and median Use technology to compute the mean and median of the recovery times in Example 3.3. Solution Enter the data into L1, then use the 1-Var Stats command. Figure 3.2 presents the TI-84 Plus display. The mean is ¯x = 17.5 and the median (denoted “Med”) is 17. Step-by-step instructions for computing the mean and median with the TI-84 Plus are presented in the Using Technology section on page 96. Figure 3.2 TI-84 Plus display showing the mean and median for the data in Example 3.3 Figure 3.3 presents MINITAB output. The mean and median are highlighted in red. Step-by-step instructions for computing the mean and median in MINITAB are presented in the Using Technology section on page 96. Variable N Mean SE Mean StDev Minimum Q1 Median Q3 Maximum Time 8 17.5 1.832 5.182 12.00 13.00 17.00 20.25 27.00 Figure 3.3 Objective 3 Compare the properties of the mean and median Comparing the Properties of the Mean and Median Both the mean and the median are frequently used as measures of center. It is important to know how their properties differ. The mean is more influenced by extreme values than the median is One important difference between the mean and the median is that the formula for the mean uses every value in the data set, but the formula for the median depends only on the middle number or the middle two numbers. This is particularly important for data sets in which one or more numbers are unusually large or unusually small. In most cases, these extreme values will have a large influence on the mean, but little or no influence on the median. Example 3.5 illustrates this principle. EXAMPLE 3.5 Determining that the mean is more influenced by extreme values than the median is Five families, named Smith, Jones, Gonzales, Brown, and Jackson, live in an apartment building. Their annual incomes, in dollars, are 25,000, 31,000, 34,000, 44,000, and 56,000. The Smith family, whose income is 25,000, wins a million-dollar lottery, so their income increases to 1,025,000. Find the mean and median income both before and after the Smiths win the lottery. Which measure of center is more influenced by the large number, the mean or the median? Solution We compute the mean and median before the lottery win. The mean income is Mean = 25,000 + 31,000 + 34,000 + 44,000 + 56,000 5 = 38,000 The median is the middle number: Median = 34,000
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