Mathematical Oddities in Affirmative Action

The Supreme Court ruled on affirmative action at the University of Michigan last month, and its balanced decision on such an emotional issue was criticized by many. The issue is also conceptually tangled with many different aspects, including two obscure mathematical ones.

The first involves the well-known normal curve that tells us how frequently a quantity assumes certain values.

The curve — fat and bell-shaped in the middle but flatter and thinner at either end — describes quantities such as height. Most people are of middling height, fewer are somewhat above or below average height, and fewer still are very tall or very short. Other quantities whose frequency is described by the normal curve include water use in a given city between 2 a.m. and 3 a.m., thicknesses of machined parts coming off an assembly line, scores on many standardized tests (whatever it is that they purport to measure), the number of admissions to a large hospital on any given day, distances of darts from a bull's-eye, leaf sizes, nose sizes, and the number of raisins in boxes of breakfast cereal.

Oddly enough, the shape of normal bell-shaped curves may also have an unexpected relevance to affirmative action. One of its important consequences is that even a small difference between the averages of different population groups is accentuated at the extreme ends of these curves.

Close on Average, Very Different at the Extremes

To illustrate, assume that two population groups vary along some dimension — say height again. Assume further that the two groups' heights vary in a (nearly) normal manner. Then even if the average height of one group is only slightly greater than the average height of the other, people from the slightly taller group will constitute a large majority among the very tall. Likewise, people from the slightly shorter group will constitute a large majority among the very short.

This is true even though the bulk of the people from both groups are of roughly average stature. If Koreans, for example, have a mean height of 5 feet 8 inches, and Mexicans, say, have a mean height of 5-foot-7, then (depending on the exact variability of the heights) perhaps 90 percent or more of those over 6-foot-2 will be Korean. In general, any differences between two groups will always be greatly accentuated at the extremes. (The effect can be seen graphically if one of the two normal curves is slid slightly to the left or right; the difference this sliding produces in the tails of the curves will be disproportionately large.)

If scores on a standardized test are being measured rather than heights and if only the highest scorers on the test are admitted to an elite school, then two groups whose scores differ minimally on average might nevertheless have very different admission rates. Such disparities are not necessarily evidence of racism or ethnic prejudice although, without doubt, they sometimes are. One can and should debate whether the tests in question are appropriate for the purposes at hand (often they're not), but one shouldn't be surprised when normal curves behave normally.

Equality Not Always Logically Possible

Aside from having a very dubious rationale, schemes of strict proportional representation, whether in schools or the workplace, are often impossible to implement.

A thought experiment illustrates this. Imagine a company, United Differences (UD), operating in a community that is 25 percent black and 75 percent white and 5 percent homosexual, 95 percent heterosexual. Unknown to UD and the community is the fact that only 2 percent of the blacks are homosexual, whereas 6 percent of the whites are. (The numbers are fictitious and chosen for illustration only.) Making a concerted attempt to assemble a work force of 1,000 that "fairly" reflects the community, the company hires 750 whites and 250 blacks.

However, just five of the blacks (or 2 percent) would be homosexual, whereas 45 of the whites (or 6 percent) would be (totaling 50, 5 percent of all workers). Despite these efforts, the company could still be accused by its black employees of being homophobic since only 2 percent of the black employees (five of 250) would be homosexual, not the community-wide 5 percent. The company's homosexual employees could likewise claim that the company was racist since only 10 percent of their members (five of 50) would be black, not the community-wide 25 percent. White heterosexuals would certainly make similar complaints.

To complete the descent into absurdity, factor in several other groups — Hispanics, women, Norwegians even. Their memberships will likely also overlap to various unknown degrees. People will identify with varying intensity with the groups to which they belong. The backgrounds and training across these various cross-sections is extremely unlikely to be uniform. Statistical disparities will necessarily result. Racism and all other forms of group hatreds are unfortunately real enough without making them our unthinking first inference when confronted with such disparities.

Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy, and the just released A Mathematician Plays the Stock Market. His Who’s Counting? column on appears the first weekend of every month.