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.

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