90 CHAPTER 3 Describing, Exploring, and Comparing Data CP Median The median can be thought of loosely as a “middle value” in the sense that about half of the values in a data set are less than the median and half are greater than the median. The following definition is more precise. DEFINITION The median of a data set is the measure of center that is the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude. Important Properties of the Median ■ The median does not change by large amounts when we include just a few extreme values, so the median is a resistant measure of center. ■ The median does not directly use every data value. (For example, if the largest value is changed to a much larger value, the median does not change.) Calculation and Notation of the Median The median of a sample is sometimes denoted by x∼ (pronounced “x-tilde”) or M or Med; there isn’t a commonly accepted notation, and there isn’t a special symbol for the median of a population. To find the median, first sort the values (arrange them in order) and then follow one of these two procedures: 1. If the number of data values is odd, the median is the number located in the exact middle of the sorted list. 2. If the number of data values is even, the median is found by computing the mean of the two middle numbers in the sorted list. C Redefining Units of Measure In 1983, the meter was redefined to be the distance traveled by a beam of light in 1>299,792,458 of a second. In 1967, the time unit of one second was redefined to be the amount of time it takes an atom of cesium-133 to vibrate 9,192,631,770 times. From 1889 to 2018, the kilogram was defined to be the mass of the International Prototype Kilogram—a chunk of platinumiridium metal that was stored in a vault in Paris. The kilogram has now been redefined in terms of the Planck constant—a physical constant in nature. The actual definition is complicated, but the new definition completes the goal of having the basic units of the International System of Units (S.I.) defined by constants of nature instead of physical objects. As of this writing, these are the only countries that have not yet adopted the S.I. system as their official system of weights and measures: Liberia, Myanmar, and the United States! I m r t t b EXAMPLE 1 Mean Data Set 33 “Disney World Wait Times” in Appendix B includes wait times (minutes) for six popular rides. Find the mean of the first eleven wait times for “Space Mountain” at 10 AM: 50 25 75 35 50 25 30 50 45 25 20 SOLUTION The mean is computed by using Formula 3-1. First add the data values, and then divide by the number of data values: x = Σx n = 50 + 25 + 75 + 35 + 50 + 25 + 30 + 50 + 45 + 25 + 20 11 = 430 11 = 39.1min The mean of the wait times for “Space Mountain” is 39.1 minutes. YOUR TURN. Find the mean in Exercise 5 “Super Bowl Jersey Numbers.”
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