dcw planet

Recently, Larry Summers, the president of Harvard University, has found himself in hot water for suggesting that the under-representation of women in the highest echelons of science might be due to innate biological differences between the sexes. In the past, others have been similarly chastised for suggesting that the the over-representation of blacks in the highest echelons of many professional sports might arise from innate biological differences among races.

The most amusing aspect of these debates is that almost all the participants are completely off the mark. How different groups perform at a given task on average is entirely irrelevant to predicting which group will contribute the most to that tiny portion of the population who are the very best at the task. Whether men or women are, on average, better at science has no effect on whether the best scientists are men or women. Whether blacks or whites are, on average, better at competitive athletics has no effect on whether the best competitive athletes are black or white. How can this be?

Below is a picture of the bell curve, named for its shape, beloved by statisticians and bemoaned by test-takers everywhere.



It is a picture that shows how often a particular value will be measured. When the line over a value is high, that value is measured often; when the line over a value is low, that value seldom occurs. The shape of the curve, which fits the measured distributions of many attributes and test scores remarkably well, says that medium values occur often, while higher and lower values occur less frequently. Very high and very low values occur, of course, very infrequently.

Most of the area under a bell curve, representing the great mass of the population, lies near the middle value, called the average or mean. The very highest values, obtained by the people who are the very best at the measured skill, are represented by the small area under the far-right tail of the bell curve.

Before we proceed to determine who dominates that far-right tail, you must know that you need two numbers to characterize a bell curve. First, of course, you need to know the middle value, which statisticians represent by the Greek letter μ (mu). But you also need to know how spread out the values are around the middle; visually, that corresponds to how wide or narrow the bell curve is. Statisticians call the width of a bell curve its standard deviation, and represent it by the Greek letter σ (sigma).



Take, for example, measurements of IQ. The average IQ is μ=100, and the standard deviation σ=15. A score over 115 (one standard deviation above the mean) will be measured for around one person in ten. If instead we had σ=5, a score of 115 (now three standard deviations above the mean) would be measured only for around one person in 100,000. As σ decreases, the bell curve becomes narrower and it becomes more difficult to get a score far from the mean. Conversely, as σ increases, the bell curve becomes wider and it becomes easier to get an extreme score.



Now consider two different groups of people, which we will call pink and blue. Suppose we measure the distributions of some characteristic in the two populations, and get the bell curves shown above. I don't know what the characteristic is; perhaps it's height or weight, or perhaps it's interest in science or the score on some math test. You'll notice immediately that the pink population scores, on average, lower than the blue population. It has a lower μ. What you might not notice right away is that the pink population's bell curve is also a little wider than the blue population's. It has a higher σ. That fact isn't important for determining whether a typical pink or blue person is likely to have a higher score. But it is decisive for determining whether the highest scores will belong to pink or blue people. Notice that, despite that fact that the blue average is higher than the pink average, pink dominates the very highest scores, due to its higher standard deviation. Below is a close-up of the far-right tails of the distributions, to make this easier to see.



But surely, you might argue, having a higher μ helps. If blue's average were high enough, there would be more blues than pinks in the region where there are now more pinks than blues. And that is "sort of" true, but only "sort of". It turns out that, as long as you go far enough out in the tails of a bell curve, the population with the higher standard deviation will always dominate. By changing the means, you can change the score where the cross-over occurs, but you can't change the fact that the higher-σ group will eventually, at some point in the far-out tails of the distribution, overtake the lower-σ group.

I don't know whether the pink and blue lines sketched above accurately represent the distributions of scientific ability in the sexes, or of athletic ability in whites and blacks. Such measurements are fraught with peril for the career of any statistician who might think to undertake them. And whatever one chooses to measure, whether it is the right measure of ability is entirely debatable. But I do find it amusing that, for any measure, the suggestion that one sex or race might have a higher μ is enough to elicit a torrent of vitriolic responses, while the suggestion that one group might have a higher σ usually elicits yawns. For the question at hand, it is really σ that counts!

By the way, the mathematical properties of bell curves were first investigated in detail by the German mathematician Carl Friedrich Gauss in the early 19th century. So we should really all have this down by now.