Wednesday, August 22, 2007

When "average" is anything but typical


The article was a hit piece on the Bush administration's economic performance, so I should like it, right? Yesterday the New York Times ran an article by David Cay Johnston headlined "2005 Incomes, on Average, Still Below 2000 Peak." Here are the first two paragraphs:

Americans earned a smaller average income in 2005 than in 2000, the fifth consecutive year that they had to make ends meet with less money than at the peak of the last economic expansion, new government data shows.

While incomes have been on the rise since 2002, the average income in 2005 was $55,238, still nearly 1 percent less than the $55,714 in 2000, after adjusting for inflation, analysis of new tax statistics show.

Kind of petty ante stuff, that. So I kept reading. Way, way, way down the piece I come to this:

Nearly half of Americans reported incomes of less than $30,000, and two-thirds make less than $50,000.

Now wait a minute, didn't he just say that the "average" income was in the $55,000 range? Yes, he did.

But he didn't mean "average" the way you or I usually mean it. The dictionary defines "average" as:
  • "the level, amount, or degree of something that is typical of a group or class of people of things"
or
  • "without any extraordinary, untypical, or exceptional characteristic."
No, that's not what Mr. Johnston was talking about. He was talking the arithmetic "mean": if you added up all incomes in the United States from the poorest welfare mom to Bill Gates and Warren Buffet and divided by 2, you'd get that $55,000 number. Because a few people make astronomical amounts while many, many people make very little, that mean comes out higher than the incomes of two thirds of everybody.

The Wikipedia article on "arithmetic means" explains this very clearly:

The arithmetic mean may be misinterpreted to imply that most people's incomes are higher than is in fact the case. When presented with an "average" one may be led to believe that most people's incomes are near this number. This "average" (arithmetic mean) income is higher than most people's incomes, because high income outliers skew the result higher (in contrast, the median income "resists" such skew). However, this "average" says nothing about the number of people near the median income (nor does it say anything about the modal income that most people are near).

Nevertheless, because one might carelessly relate "average" and "most people" one might incorrectly assume that most people's incomes would be higher (nearer this inflated "average") than they are. For instance, reporting the "average" net worth in Medina, Washington as the arithmetic mean of all annual net worths would yield a surprisingly high number because of Bill Gates. Consider the scores (1, 2, 2, 2, 3, 9). The arithmetic mean is 3.17, but five out of six scores are below this!

This stuff matters. How many of those people in Washington would know that the "average" income cited here was very atypical indeed? After all, in Congress they get paid $165,000 annually without taking into account any wealth they bring with them. They probably think they are fairly typical, though perhaps somewhat fortunate!

One reason most of us feel as if our rulers live on some different plane of life is that they do.

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