Section 3.1 Measures of Center 93 SUMMARY In most cases, the shape of a histogram reflects the relationship between the mean and median as follows: Shape Relationship Between Mean and Median Skewed to the right Mean is noticeably greater than median Approximately symmetric Mean is approximately equal to median Skewed to the left Mean is noticeably less than median For an exception to this rule, see Exercise 64. Which is a better measure of center, the mean or the median? The short answer is that neither one is better than the other. They both measure the center in different, but appropriate, ways. When the data are highly skewed or contain extreme values, some people prefer to use the median, because the median is more representative of a typical value. However, the mean is still an appropriate measure of center, and is sometimes preferable, even when the data are highly skewed (see Exercises 53 and 54). The following table summarizes the features of the mean and median. Advantages Disadvantages Mean Takes every value into account Highly influenced by extreme values: not resistant Median Not much influenced by extreme values: resistant Depends only on middle value or middle two values Check Your Understanding 1. Compute the mean and median of the following sample: 74 87 36 97 60 58 46 2. Compute the mean and median of the following sample: 69 17 75 96 74 80 3. Someone surveys the families in a certain town and reports that the mean number of children in a family is 2.1. Someone else says that this must be wrong, because it is impossible for a family to have 2.1 children. Comment. 4. A data set has a mean of 6 and a median of 4. Would you expect this data set to be skewed to the right or skewed to the left? 5. A data set has a mean of 5 and a median of 7. Would you expect this data set to be skewed to the right or skewed to the left? Answers are on page 104. Critical thinking about the mean and median We can compute the mean and median for any list of numbers. However, they do not always produce meaningful results. The mean and median are useful for numbers that measure or count something. They are not useful for numbers that are used simply as labels. Example 3.6 illustrates the idea. EXAMPLE 3.6 Determining whether the mean and median make sense Following is information about the five starting players on a certain college basketball team: Their heights, in inches, are 74, 76, 79, 80, and 82. Their uniform numbers are 15, 32, 4, 43, and 26. Will we obtain meaningful information by computing the mean and median height? How about the mean and median uniform number? Explain.
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