Score One for Larry?

My former professor Larry Summers (I guess I'm allowed to call him that, even though I only audited his course) got himself in a heap of trouble when he was president of Harvard for some remarks he made about why there were so few tenured women in math and science at top universities. To anyone who took the trouble to read his actual remarks, which were and still are available online, it should have been obvious (though to some it evidently wasn't) that he was not saying that there was a "math gene" that women lacked.

Rather, he says that men show more variability than women on a variety of traits. In other words, there are more men than women at both the very top and the very bottom on a lot of scales (and an even higher proportion at the very, very top and very, very bottom). I don't know what the actual evidence is on traits like height, but there is a plausible genetic argument in the fact that men have only one X chromosome, so unusual traits are less likely to be averaged out (an extreme example would be hemophilia and other sex-linked diseases).

Fast-forward to March 29 of this year, and a Nicholas Kristof column in the New York Times lamenting how boys are falling behind girls. There is a particular problem in reading, Kristof says. But, he adds:

There is one important exception: Boys still beat out girls at the very top of the curve, especially in math. 

In the high school class of 2009, a total of 297 students scored a perfect triple-800 on the S.A.T., 62 percent of them boys, according to Kathleen Steinberg of the College Board. And of the 10,052 who scored an 800 in the math section, 69 percent were boys.

So boys are doing worse, but there are more at the upper end. If it's a result of greater variation among boys we'd expect to see more at the lower end as well, but that's hard to judge, since boys are doing worse overall and since people at the bottom end don't take the SAT. Score one for Larry?

Not so fast. When we look at the actual SAT scores for 2009 seniors, it gets more complicated. First of all, boys continue to have a substantial edge in math: the mean is 534 for boys and 499 for girls (on a 200-800 scale). And this is not just because the boys at the high end pull up the average: 88 percent of boys scored above 400, for example, but only 82 percent of girls.

More surprisingly, given what Kristof says about reading, boys do slightly better on the critical-reading section of the SAT (mean of 503 versus 498; 69 percent above 450 versus 67 percent of girls). Only on the writing section do girls do better, with a mean of 499 versus 486.

But let's look still further before we discard the Summers hypothesis. The College Board also publishes calculations of the most commonly used measure of variation, the standard deviation. And on all three tests, boys show more variation than girls. For math, the standard deviation is 118 for boys, 112 for girls.

This may not sound like much of a difference, but you'd be surprised. Let's look at people 4 standard deviations above the mean, about the top .003%. These are the sort of people you might expect to find as professors of math and science at top research universities.

I did some rough calculations assuming the means were the same for men and women, and this much difference in variation translates into 2.4 times as many men as women. That suggests that even were there no discrimination and no difference in average ability between men and women, this small difference in variability would result in large differences in the numbers of men and women professors. So, yeah, score one for Larry. (I can also make a technical argument that the College Board's calculation actually understates the difference between boys and girls in amount of variation, but let's not get into it.)

On the other hand, with no difference in variability but the difference in means as shown above, there are 3.7 times as many men as women. So if there really is a "math gene," then things are much worse, while if the difference in means reflects socialization then big improvements may be possible.

Finally, let's take another look at Kristof's claims. Among college-bound kids, there's not much evidence of a big disadvantage for boys. But about 15% more girls than boys took the SAT. Since the ones who didn't take it are more likely to be those who would have scored low, the average scores of boys are misleadingly high, though I don't know by how much. More importantly, the problem of boys not keeping up appears to be a problem at the lower end. It's the ones who don't go to college that we need to worry about. It's not that boys are low-achieving, it's that there are more low-achieving boys.