Here’s an interesting post, from Robin Hanson at Overcoming Bias, that illustrates a nice non-sports application of regression to the mean.

A few years ago, in a speech, Harvard president Larry Summers quoted a possible explanation for why there are so few women in math-related fields.

The hypothesis is that when it comes to math ability, men and women are equal, on average. But men’s ability is more spread out – there are more math geniuses, and also more math idiots (“idiots” is my term, not his). That is, the variance of men is higher than the variance of women.

Even if the difference in variance is small, it could lead to very large differences at the extremes. Suppose that both men and women have average math ability of 5' 8". And suppose the variance among women is three inches, but among men it's four inches.

Now, suppose that math-related fields require a lot of math ability – say, 6' 5" or over. That's three standard deviations above the mean for women, but only 2.25 SDs for men.

That means 13 out of 10,000 women will reach 6' 5", but 122 men will.

So a relatively minor difference in variance leads to a huge disparity at the extremes – in this case, 9 men to 1 women. (As it turns out, the real ratio is only 7% higher for men, not 33% higher like in my example. Still, the M/F ratio is 3:1 at 4 standard deviations.)

Summers drew a lot of fire for his remarks, and he resigned a few months later.

(Aside: years earlier, if I recall correctly, someone had complained that there were many more blacks among the ranks of superstars than among average players, which showed that teams were biased in favor of whites when it didn't matter much. Bill James rebutted that accusation with a related observation – that if black players are, on average, just slightly better than whites, there will be a similar disparity at the extremes.

This is not the same argument – it relates to a difference in means, rather than variances – but it's pretty much the same idea.)

A couple of weeks ago, a new study came out that actually did the research, and found exactly what Summers had hypothesized. Most of the press reports got the story wrong, because they didn't understand that the key was that the variances were different. The reporters latched on to the fact that the means were the same, and incorrectly concluded that Summers was wrong.

Anyway, the point is regression to the mean. And Hanson points out another consequence of the male/female variance difference. Specificially, suppose you have a man and a woman, and they both score equally high on a math admissions test. If all you care about is the chance of choosing the better mathematician, and you can only admit one of them, which one should it be?

The statistical answer is: the man.

Why? Because there fewer talented women, relative to equally-talented men, so it's more likely the woman got her high score by luck.

A high score can be obtained by a less-brilliant student who got lucky, or a more-brilliant student who got unlucky. The ratio of less- to more- is higher for women. Therefore, the high-scoring woman is more likely to be closer to average. Therefore, you have to regress to the mean more for the woman than for the man.

Here's a baseball analogy. Suppose you have a group of twenty veterans. They're solid regulars, but none of them has showed signs of stardom in their five-year careers. And suppose you have a group of twenty draft choices, and you suspect that some of them are duds, but some of them are bona-fide superstars.

One player from each group hits .333 in April. Which do you think is the better hitter? Obviously, it's the rookie. The solid regular was probably just very lucky. The rookie was probably lucky too, but there's a chance that he's a star or superstar, in which case he might only have been a little lucky.

It's the same idea with the men and women -- not as extreme, because the difference in variance is small, but it's still there. The women are more likely to be the solid regulars, and the men have the potential to be the stars (or duds).

Of course, this doesn't really have any strong policy implications. Universities don't really have to hold women to a higher standard – they can just make the test longer, to reduce the effects of luck, or give multiple tests.

Still, it's a valid statistical consequence, and politically incorrect enough to be very interesting.

Labels: Larry Summers, regression to the mean