# Higher variance in men

In this blog post I want to discuss the observation that got Harvard president Larry Summers into trouble. Summers mentioned in a 2005 talk that men generally have a higher variance in cognitive ability and opined that this might explain the under representation of women in top level science. This led to intense negative press and was likely a factor in his resignation in the following year.

In mathematical ability this ratio of standard deviations for boys and girls is generally around 1.2. This can be observed in the math SAT (where you have to correct for the fact that more girls take the test) and in almost all countries that participate in PISA. This seems like a small difference but it quickly has a very noticeable tail effect. For example, if we put the mean at 100 and the female standard deviation at 15 to make the numbers IQ-like, that going two standard deviations beyond the mean puts girls at 130 and boys at 136. Because an IQ of 130 is roughly 1:50 but an IQ of 136 is roughly 1:100, we can expect to see twice as many boys as girls at the ability level around 136.

Of course being right didn’t help Larry Summers any.

But higher variance for men in all traits that are relevant for fitness also makes evolutionary sense. To see that, it is easiest to instead look at women. Higher variance in this case means that dice with more pips are thrown for fitness relevant traits. So if women had high variance in fitness relevant traits they would risk losing out, but also have the chance to win big. Unfortunately for women that is a bad gamble, because the number of kids they can have in a lifetime is limited by nature. Losing would mean 0 kids, but winning might mean 12 kids instead of 8 (in an environment long ago with nasty childhood mortality on top).

For men this gamble makes more sense, because they can have up to 1000 kids! Which much more nicely balances the chance to have 0. So if there is a way for evolution to increase the standard deviation in men for fitness relevant traits, we should expect to see this.

And in fact there is such a way! The X chromosome only occurs once in a male human cell, but twice in female cells. To handle the superfluous X chromosome, female cells randomly inactivate one of the X chromosomes during embryogenesis. This happens early but not early enough that women are not mosaics of different activated X chromosomes. So the traits encoded on the X chromosomes are averaged over two Xs in women, and are not averaged at all in men!

This increases the standard deviation for men of all traits encoded on the X chromosome compared to women. Unsurprisingly, the X chromosome is found to be enriched for the most fitness relevant traits like height and intelligence.

So we do have the observation, an evolutionary reason and a biological mechanism!

But we are already used to even the most clear cut fact in IQ research being extremely controversial.