07 September 2007

More men at the top, and at the bottom.

As has been documented by a lot of people, it seems that a lot of psychological traits have the following properties:

  • 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

f(z) = {1 \over \sqrt{2\pi}} \exp \left( {-z^2 \over 2} \right)

and that for men is

g(z) = {1 \over \sigma\sqrt{2\pi}} \exp \left( {-z^2 \over 2\sigma^2} \right)

If we look at the ratio f(z)/g(z), this is the ratio of women to men at skill level z. It's

{f(z) \over g(z)} = \sigma \exp \left( {z^2 \left( \sigma^{-2} - 1 \right) \over 2} \right)

and this equals 1 when

z = \pm {\sigma \sqrt{2 \log \sigma \over \sigma^2-1}}.

When z is closer to zero than this, women will predominate; when z is larger, men will predominate. It turns out that

{\sigma \sqrt{2 \log \sigma \over \sigma^2-1}} = 1 + {\sigma-1 \over 2} + O((\sigma-1)^2)

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.


Harald Hanche-Olsen said...

So it would be interesting to study the distribution of log(income) for the two genders and compare results. Wonder if it has alread been done? Or if there is raw data available so you could do the analysis yourself?

The Lucifer Principle said...

It would be interesting to see if the same applies to human races.

halfempty said...

Considerations about IQ and wealth that don’t take culture into account aren’t worth much. “Earning potential”, considered as some sort of intrinsic property of individuals, and actual wealth will be at best only loosely correlated if the children of wealthy people inherit their wealth. The effects of inheritance of wealth may well swamp the effects of difference in distribution of IQ.

As for gender, in many systems of inheritance, women can’t inherit; in some, women are not permitted to own or to control assets (this was the case in France until 1946). In such cases too, the effects of cultural constraints will overwhelm the effects of difference in distribution of IQ.

wcyee said...

Hi. I'm curious, how do you get equation #5? Thanks!

Theo said...

For the record, two articles that actually site empirical research:

1) Hyde, J.S. (2005) The gender similarities hypothesis. American Psychologist, 60(6), 581-592. Full text.

...the greater male variability hypothesis was originally proposed more than a century ago, and it survives today (Feingold, 1992; Hedges & Friedman, 1993). In the 1800s, this hypothesis was proposed to explain why there were more male than female geniuses and, at the same time, more males among the mentally retarded. Statistically, the combination of a small average difference favoring males and a larger standard deviation for males, for some trait such as mathematics performance, could lead to a lopsided gender ratio favoring males in the upper tail of the distribution reflecting exceptional talent. The statistic used to investigate this question is the variance ratio (VR), the ratio of the male variance to the female variance. Empirical investigations of the VR have found values of 1.00 -1.08 for vocabulary (Hedges & Nowell, 1995), 1.05-1.25 for mathematics performance (Hedges & Nowell), and 0.87- 1.04 for self-esteem (Kling et al., 1999). Therefore, it appears that whether males or females are more variable depends on the domain under consideration. Moreover, most VR estimates are close to 1.00, indicating similar variances for males and females. Nonetheless, this issue of possible gender differences in variability merits continued investigation.

2) Spelke, E. S. (2005). Sex differences in intrinsic aptitude for mathematics and science: A critical review. American Psychologist, 60, 950-958. Full text.

If the genetic contribution were strong, however, then males should predominate at the upper tail of performance in all countries and at all times, and the male-female ratio should be of comparable size across different samples. Contrary to this prediction, the preponderance of high-scoring males is far smaller in some countries (e.g., Deary et al., 2003) and altogether absent in others (Feingold, 1994). Moreover, the preponderance of boys with high scores on the SAT-M has declined substantially in U.S. samples. In one sample of students selected for high talent, it declined from 10.7:1 in the 1980s to 2.8:1 in the 1990s (Goldstein & Stocking, 1994). The performance of boys and girls on standardized tests likely reflects a complex mix of social, cultural, and biological factors.