Who's Counting: It's Mean to Ignore the Median


Aug. 6, 2006 — -- Believe it or not, the difference in the way the Democrats and Republicans react to the performance of the U.S. economy is clarified by a mathematical distinction studied in elementary school. The distinction is between the mean, which the Republicans emphasize, while the Democrats prefer the median. Before turning to the economy, let me review a little fourth-grade arithmetic.

The mean of a set of numbers is simply the average, and it is obtained by adding the numbers in the set and dividing by how many there are. The median of a set of numbers is obtained by listing the numbers in the set in ascending order and locating the middle number in the list. The same set of numbers can have a mean and a median, which differ significantly.

A real estate agent, for example, informs you that the mean price for a house in a certain neighborhood is $500,000 and implies that there are many houses above that price in the neighborhood. Not necessarily. If most of the houses sell for somewhere between $100,000 and $200,000, and there are a few multimillion dollar mega-mansions, the mean price of a house in the neighborhood might well be $500,000, even if the median is, say, $160,000.

Or a salesman tells you that the median commission on the nine sales he made that week is $80 and suggests that he therefore made $720 on these sales. Maybe, but he could have made millions of dollars on these sales if one of the sales was enormous, or he might have made little more than $400 if four of the sales were near $0 and the other five were $80 or slightly more.

The relevance of this distinction is apparent in the just-released figures on the U.S. economy for 2004, the latest year for which there is complete data. The Republicans chortle that the economy grew at a healthy rate of 4.2 percent. (It's slowed since then.) The Democrats point to data from the Census Bureau for the same year (and earlier as well), indicating that the real median family income fell and that poverty increased.

Economists Thomas Piketty and Emmanuel Saez, who have long studied income distribution, have recently looked at the data and calculated that during this one-year period the real incomes of the richest 1 percent, those making at least $315,000 annually, grew by almost 17 percent. Furthermore, this growth in income not only eluded the lower- and middle-income classes, but by and large passed up the upper middle class as well.

The income increases of even those whose incomes were greater than 95 percent of other Americans were quite minimal. The huge increases in income went to those with already huge incomes. In fact, half of the increased income going to the top 1 percent of households went to the top tenth of the top 1 percent!

And the minimum wage? The lowest in real terms that it's been since the 1950s. And the income of the typical college graduate? Down in 2004.

This rich-get-richer dynamic is not something new. The flocking tendency of Internet surfers, vacation motorists, market investors, as well as a host of physical, social and financial measures suggest a general phenomenon, usually described by what are called power laws in mathematics.

Along various social dimensions, the dynamics underlying these laws may lead naturally to the development of stark inequalities, which increasingly seem to reign not just here but throughout the world. The United Nations issued a report a few years ago, for example, saying the net worth of the world's three richest families -- the Gateses, the Sultan of Brunei and the Waltons of Wal-Mart -- exceeded the gross domestic product of the 43 poorest nations.

Still, this lopsidedness is neither necessary nor inevitable, and it bodes ill for civil society. Almost 2,400 years ago Aristotle, seeing the discord between ancient Greece's rich and poor, applied his idea of the golden mean to call for an equitable (but not equal) income distribution. For purposes of stability, he favored establishing a strong middle class and government policies to assist in this establishment.

A little game from the field of behavioral finance illustrates the class resentment Aristotle described. The so-called "ultimatum game" generally involves two players: One is given a certain amount of money, say $100, by an experimenter, and the other is given a kind of veto. The first player may offer any nonzero fraction of the $100 to the second player, who can either accept or reject it. If he accepts it, he is given whatever amount the first player has offered, and the first player keeps the balance. If he rejects it, the experimenter takes the money back.

Viewing this in rational game-theoretic terms, one would argue that it's in the interest of the second player to accept whatever is offered, since any amount, no matter how small, is better than nothing. This is not what happens, however. The offers generally range from 5 percent to 50 percent of the money involved, but, when deemed too small, the offers are often rejected. Better to receive nothing, the resentful rejectors say, than to be humiliated. Notions of fairness and equity, as well as anger and revenge, seem to play a role.

Almost as if on cue, just as I finished writing this, the House passed a modest minimum wage bill but typically linked it to a significant cut in the estate tax. Once again, one for you, 10 for me. (It seems the bill will not make it through the Senate, however.)

So was Aristotle a card-carrying Democrat? No. I think he was just expressing some increasingly uncommon common sense. It's mean to ignore the median.

Professor of mathematics at Temple University, John Allen Paulos has written such best-sellers as "Innumeracy" and "A Mathematician Plays the Stock Market." His "Who's Counting?" column on ABCNEWS.com appears the first weekend of every month.

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