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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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