As this example shows, the median (unlike the mode) doesn’t have to be a number in our original set of data. The two closest to the middle are 44 and 46, so we’ll choose the number halfway between those to be the median: 45. Since there are an even number of data values in our list, we can’t pick the one right in the middle. For example, let’s say we have the data 41, 44, 46, 53. If there’s not one value right in the middle, we pick the two closest, then choose the number exactly between them. For our easy example (with data values 11, 12, 13, 13, 14), that first 13 is right in the middle there are two values to the left and two values to the right. In practice, we find the median just like we described in the average height example: by lining up all the data values in order from smallest to largest and picking the value in the middle. Only 13 has no violations, so it’s the median according to the definition. Here’s a chart with the rest of the data, with red shading to show where the definition is violated: Since more than 50% of the data are greater than 11, the definition is violated it’s not the median. Is the first number on the list, 11, the median? There are no values less than 11 (that’s 0%), and there are four values greater than 11 (that’s 80%). Consider the following list of numbers: 11, 12, 13, 13, 14. If we lined everyone up in order by height and found the person right in the middle, that person’s height is called the median, or the value that is greater than no more than half and less than no more than half of the values. Let's revisit our example of trying to identify the height of the “average” person. In practice, though, it doesn’t really matter if no data value appears more than once, then the mode is not helpful at all as a measure of centrality. But according to some other definitions, the data would have no mode. What happens if there is no number in the data that appears more than once? In that case, by our definition, every data value is a mode. What’s the mode of the number of people in these residences? If there are two modes, the data are bimodal. This is the mode, or the value that appears most often. In our discussion of average heights, the first possible definition we offered was the height that more people share than any other. In this section, we’ll explore each of these methods of finding the “average.” The Mode All give potentially different results, and all are useful for different reasons. Centrality is just a word that describes the middle of a set of data. They are all methods of measuring centrality (or central tendency). Could we define the average height to be the number that you should guess to give you the smallest possible score?Įach of these three methods of determining the “average” is commonly used. After we check every height and award points accordingly, the person with the lower score wins (because a lower score means that person’s guess was, overall, closer to the actual values). The result is the number of points you earn for that person. You and your friend figure out how far off each of your guesses were from the actual value, then square that number. Once the guesses are made, you bring in every person and measure their height. Imagine a game where you and a friend are trying to guess the typical person’s height. What exactly do we mean when we describe something as "average"? Is the height of an average person the height that more people share than any other? What if we line up every person in the world, in order from shortest to tallest, and find the person right in the middle: Is that person’s height the average? Or maybe it’s something more complicated. Solve application problems involving mean, median, and mode.Contrast measures of central tendency to identify the most representative average.Figure 8.29 What does it mean to say someone has average height? (credit: modification of work “I’m the tallest” by Jenn Durfey/Flickr, CC BY 2.0) Learning ObjectivesĪfter completing this section, you should be able to:
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