92 Chapter 3 Numerical Summaries of Data After the lottery win, the mean is Mean = 1,025,000 + 31,000 + 34,000 + 44,000 + 56,000 5 = 238,000 To find the median, we arrange the numbers in order, obtaining 31,000 34,000 44,000 56,000 1,025,000 The median is the middle number: Median = 44,000 The extreme value of 1,025,000 has influenced the mean quite a lot, increasing it from 38,000 to 238,000. In comparison, the median has been influenced much less, increasing only from 34,000 to 44,000. Because the median is not much influenced by extreme values, we say that the median is resistant. DEFINITION Astatistic is resistant if its value is not affected much by extreme values (large or small) in the data set. We can summarize the results of Example 3.5 as follows. SUMMARY The median is resistant, but the mean is not. CAUTION The relationship between the mean and median and the shape of the data set holds for most data sets, but not all. The mean and median can help describe the shape of a data set The mean and median measure the center of a data set in different ways. The mean is the point at which a data set balances (see Figure 3.1). The median is the middle number, so that half of the data values are less than the median and half are greater. It turns out that when a data set is symmetric, the mean and median are equal. When a data set is skewed, however, the mean and median are often quite different. When a data set is skewed to the right, there are some large values in the right tail. Because the median is resistant while the mean is not, the mean is generally more affected by these large values than the median is. Therefore, for a data set that is skewed to the right, the mean is often greater than the median. Figure 3.4 illustrates the idea. For most data sets that are skewed to the left, the mean will be to the left of, or less than, the median. For most data sets that are skewed to the right, the mean will be to the right of, or greater than, the median. When a data set is approximately symmetric, the balancing point is near the middle of the data, so the mean and the median will be approximately equal. Median Mean Mean = Median Mean Median (a) (b) (c) Figure 3.4 (a) When a data set is skewed to the right, the mean is generally greater than the median. (b) When a data set is approximately symmetric, the mean and median will be approximately equal. (c) When a data set is skewed to the left, the mean is generally less than the median.
navidi_monk_essential_statistics_1e_ch1_3
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