### Gender-blind math admissions may be biased against men

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

## 8 Comments:

The fact that now, among those studied, the variance in mathematical aptitude is greater in men than in women has literally nothing to say about inherent (that is biological, immutable) sex differences. That over the last several decades women's performance has markedly improved suggests instead that the most likely explanation is cultural (you cannot deny the results of the study would most likely have shown an even greater difference in variance, and likely median performance, if the study were conducted 75 years ago; I doubt you are suggesting that over this time there has been some evolutionary event which has mitigated this). While it is certainly possible that there is some physical, eternal difference between the mathematical capabilities of men and women, there are plenty of other equally plausible explanations at hand. It is not reasonable or justified to believe that the results of this study have demonstrated a fundamental gender difference. And your policy suggestion precludes an investigation into possible cultural causes for the difference in variance by justifying the current (biased) allocation of resources so that they reinforce your intuitions. This is not science, but rather a post hoc explanation for a phenomenon that is almost certainly far more complex than you allow.

Milo - That argument seems backward to me. Yes, there has been a cultural shift, and if anything, girls are catered to in schools far more than boys these days.

However, if the variance is due to cultural disparities, and not due to inherent traits, wouldn't we see higher variance in the discriminated class?

For example, if we're discriminating against left handed people by not teaching them how to throw a curve ball or encouraging them to learn it, then certainly my discriminatory efforts would be non-uniform. Some left-handers would learn to throw the curve in a non-discriminating environment, while some would be victimized with heavier doses of discrimination.

That unevenness of discrimination would cause *more* variance in the discriminated class, not less.

In fact, you've made the case there is more likely to be cultural discrimination against boys than against girls!

Maybe that's indeed true. Perhaps reality is less politically correct than even this study shows. Maybe the male average is truly higher, but it's 'forced' to be equal by the schools.

I believe that international studies show that girls often (maybe typically) outscore boys and -- if I recall correctly -- the discrepancy favors girls more in countries that have more women in scientific-technical careers. Implication: Girls apply themselves to math as a function of imagining themselves in math-related careers.

I don't see why that would be the implication. The complete reverse (girls better at math hence more girls end up in math-related careers) seems just as likely if not more likely. Either way, the correlation (if it exists) doesn't really help tell you what's causing what.

Hey man, it is not regression TO the mean. That means that all points would be at the mean the next time. What I think you meant is regression TOWARD the mean, and this is not an illustration of that- you need to go back and examine what regression toward the mean really means, http://en.wikipedia.org/wiki/Regression_toward_the_mean

You need to measure people multiple times for this to happen which was not what you are talking about.

Here is an explanation of the difference in variance every bit as sound as that offered in this post (without the axiomatic assumption that men are better at math): women are so superior to men in mathematical reasoning that they, inherently, show less variance (they are all natural curve ball throwers, to borrow the metaphor). It has only been gender discrimination that has suppressed their performance until recently. Now that this is (slowly, and only perhaps) diminishing, women are performing more closely to their inherent capacity--though they are not there yet! Once the effects of sexism have been eradicated, women's median performance will eventually be higher than men's. Obviously, the best policy would be to stop teaching math to men at all and invest rather in the biologically superior mathematical reasoning of women who will, when the effects of discrimination have vanished, show both higher peak performance and less variance.

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