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Why you probably have an above-average number of feet

Rebecca Goldin Ph.D , March 10, 2008

Looking at three kinds of averages

The average kid on the block might have a lot of trouble understanding what an average is. Every time Garrison Keillor signs off his “News from Lake Wobegon,” as a place “where all the women are strong, all the men are good-looking, and all the children are above average,” he gets a laugh. But the mathematical meaning of average is not always the same as the colloquial meaning – and even within math, there are three different kinds of “averaging” that are commonly referred to.

The average of a bunch of numbers is the one that we all learn in school – if only because our grade might have been determined by the average of our scores on the tests. This average is formally called the *mean* of the numbers. It’s computed by adding up *n* numbers and then dividing by that *n. *But the mean can be really misleading; for example, most people earn below average salaries, but have an above-average number of feet, as an excellent BBC article on this topic points out.

The main point for salaries is that the average is easily affected by a few people making a ton of money. For example, suppose three people make $47,000 each, three people make $50,000 each, three people make $53,000 each, and one person makes $500,000 (all annually). Their combined average salary is $95,000, even though almost everyone makes a lot less than that. For salary, a more appropriate number might be the *median*, which is a number such that half of everyone makes more, and half of everyone makes less. In our mock situation with only ten people, the median would be $50,000 – which is a good estimate for most people in the sample. According to the Census Bureau, the median household income in 2007 was $50,233.00. The mean is over $60,000 because those top earners have a higher weight in the average.

A different problem happens when you look at the number of feet people have. In this case, almost everyone has two feet, but there are a few people who have just one or no feet. Suppose we have five people, say four have two feet and one has one foot . If you take the average number of feet, you will find that the average is 1.8 feet. It follows that almost everyone has an above-average number of feet. In this case, the mean is again an inappropriate number to look at – we would do better to think about the *mode,* which is the number that occurs most frequently in the data set. Since four people have two feet, “two” will occur most often in the data. Why is mode a better choice than median? Mode is a good choice to use for a data set with a small set of values – in this case, the possibilities are zero, one or two, while median is a better choice for data that can vary continuously, like income.

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