3-1 Measures of Center 91 CP EXAMPLE 2 Median with an Odd Number of Data Values Find the median of the first eleven wait times (mins) for “Space Mountain” at 10 AM: 50 25 75 35 50 25 30 50 45 25 20 SOLUTION First sort the data values by arranging them in ascending order, as shown below: 20 25 25 25 30 35 45 50 50 50 75 Because the number of data values is an odd number (11), the median is the data value that is in the exact middle of the sorted list. The median is therefore 35.0 minutes. Note that the median of 35.0 minutes is different from the mean of 39.1 minutes found in Example 1. YOUR TURN. Find the median in Exercise 17 “Diamonds at the Diamonds.” EXAMPLE 3 Median with an Even Number of Data Values Repeat Example 2 after including the twelfth wait time for “Space Mountain” at 10 AM. That is, find the median of these wait times (minutes). 50 25 75 35 50 25 30 50 45 25 20 50 SOLUTION First sort the data values by arranging them in ascending order, as shown below: 20 25 25 25 30 35 45 50 50 50 50 75 Because the number of data values is an even number (12), the median is found by computing the mean of the two data values in the middle of the sorted list, which are 35 and 45. The median is therefore 135 + 452>2 = 40.0 minutes. YOUR TURN. Find the median in Exercise 7 “Celebrity Net Worth.” DEFINITION The mode of a data set is the value(s) that occur(s) with the greatest frequency. Important Properties of the Mode ■ The mode can be found with qualitative data. ■ A data set can have no mode or one mode or multiple modes. Finding the Mode: A data set can have one mode, more than one mode, or no mode. ■ When two data values occur with the same greatest frequency, each one is a mode, and the data set is said to be bimodal. Mode The mode isn’t used much with quantitative data, but it’s the only measure of center that can be used with qualitative data (consisting of names, labels, or categories only). CP M: What the Median Is Not Harvard biologist Stephen Jay Gould wrote, “The Median Isn’t the Message.” In it, he describes how he learned that he had abdominal mesothelioma, a form of cancer. He went to the library to learn more, and he was shocked to find that mesothelioma was incurable, with a median survival time of only eight months after it was discovered. Gould wrote this: “I suspect that most people, without training in statistics, would read such a statement as ‘I will probably be dead in eight months’ - the very conclusion that must be avoided, since it isn’t so, and since attitude (in fighting the cancer) matters so much.” Gould went on to carefully interpret the value of the median. He knew that his chance of living longer than the median was good because he was young, his cancer was diagnosed early, and he would get the best medical treatment. He also reasoned that some could live much longer than eight months, and he saw no reason why he could not be in that group. Armed with this thoughtful interpretation of the median and a strong positive attitude, Gould lived for 20 years after his diagnosis. He died of another cancer not related to the mesothelioma. continued
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