- men and women have approximately the same average for this trait, and
- both genders have an approximately normal distribution for this trait, but
- the distribution of men's values for this trait has a larger standard deviation than the women's values
This has the effect that men are overrepresented at both extremes. The canonical example is skill in science or mathematics; it's been claimed that women and men are on average equally good at mathematics, but most of the best mathematicians are male. This isn't a contradiction, because most of the worst mathematicians are male but we don't notice it. (It actually doesn't matter that the averages are the same; even if men were on average worse than women at math, if they had a larger standard deviation then they'd predominate at the higher levels.)
The article Is There Anything Good About Men?, which was an invited address by Roy Baumeister at the American Psychological Association, addresses this. This is thought to arise from the fact that men can have more offspring than women.
Let's say that the X-ability of women is normally distributed with mean 0 and standard deviation 1, and the X-ability of men is normally distributed with mean 0 and standard deviation σ. Then the probability density function for the mathematical skill of women is
and that for men is
If we look at the ratio f(z)/g(z), this is the ratio of women to men at skill level z. It's
and this equals 1 when
When z is closer to zero than this, women will predominate; when z is larger, men will predominate. It turns out that
and since σ probably isn't much larger than 1, men will predominate at more than about one standard deviation from the mean and women at less than one standard deviation from the mean. Furthermore, we have f(0)/g(0) = σ; again, since σ isn't that much greater than 1, the predominance of women over men at the center of the overall distribution is difficult to see.
Yet if σ = 1.1 -- meaning that men's skill have a standard deviation 1.1 times that of women -- then g(3)/f(3) = 1.99, so men will be twice as common as women at z=3 (which corresponds to 3 standard deviations above the mean for women, and 2.73 for men). (The same is true at three standard deviations below the mean.) At z=4, men are overrepresented by a factor of 3.6, and at z=5, by a factor of eight.
Another thing that occurred to me is the economic ramifications of this difference. It's well-known that there are more obscenely rich men than obscenely rich women. It seems to me that economic ability -- i. e. the ability to earn money -- could be proportional to, say, the exponential of (some constant times general intelligence); so for every ten IQ points you gain, your income goes up by 30%. (I made up these numbers.) This would mean that economic ability is lognormally distributed (the name is a bit counterintuitive, if you don't know it, but it means that the logarithm of economic ability is normally distributed). But the mean of a lognormally distributed variable is eμ+σ2/2, where μ and σ are the mean and standard deviation of the variable's logarithm. So if intelligence is normally distributed in both canonical genders, but men are more spread out than women in intelligence, then the mean of men's earning potential will be greater than that of women. I'm not saying that earning potential is directly tied to general intelligence (I know plenty of people who are smart but not rich) but it wouldn't surprise me to learn that earning potential is lognormally distributed and that something like what I've outlined here is at work.