# Male and Female Chess Players - Expected Discrepancies at Tails of Distributions

I'm interested in the findings of this paper from 2009:

Why are (the best) women so good at chess? Participation rates and gender differences in intellectual domains

This paper attempts to explain why the very best male chess players appear to be so much better than the very best female players (females make up only 2% of the world's best 1000 players). Specifically, they claim that the large discrepancy between the best male and the best female chess players is entirely explained by 2 facts:

• There are 15 times more male than female chess players
• We expect this ratio to be exacerbated at the extreme ends of the distribution, entirely for statistical reasons. To quote the paper:

Even if two groups have the same average (mean) and variability (s.d.), the highest performing individuals are more likely to come from the larger group. The greater the difference in size between the two groups, the greater is the difference to be expected between the top performers in the two groups

And again,

This study demonstrates that the great discrepancy in the top performance of male and female chess players can be largely attributed to a simple statistical fact—more extreme values are found in larger populations.

And so, according to the authors, if only 6% of chess players are female, then we'd only expect 2% of them in the top 1000, so no other explanations regarding biological differences or social biases are required.

### My Question

I can't get my head around the idea that small differences in population size are exacerbated at the extreme ends of the distribution. In particular, what's wrong with this counter-example:

About 1 in 12 chess players are born in the month of January. So they make up a small fraction of all chess players. By these statistical methods, we'd expect them to be particularly under-represented at the highest level - maybe only 1 in 30 of the top players would be born in January. But of course you could apply this same logic to every month, and you ultimately reach an absurd conclusion.

It seems to me that if you divide a population into 2 groups, you'd expect the same ratio of performers at all ends of the scale.

As I'm contradicting the results of a published paper, I guess I must ask - what am I doing wrong?

• I cannot find the claims you cite. Could you identify where in the paper the 6%/2% differential is discussed?
– whuber
Commented Jan 14, 2015 at 2:14
• I have another alternative. Consider this question: why tall basketball players are more successful than short ones? The authors' logic would suggest that it's because there are more tall basketball players than short ones. Don't you see the flaw in their argument? Commented Jan 14, 2015 at 2:25
• @whuber In their data, 1 in 15 (6%) of the players are female, and they also state in the Discussion that only 1% of grandmasters are female (I was perhaps being generous with 2% above). But isn't this the whole point of the paper - explaining the exacerbated discrepancy at the highest level (1% at top and only 1 woman in the top 100 vs 6% in population) using statistics alone? That's how I interpreted their argument, but maybe that's not quite right.
– tom
Commented Jan 14, 2015 at 4:19
• For those who like to tinker around with simulations. Commented Jan 14, 2015 at 7:21
• "As I'm contradicting the results of a published paper, I guess I must ask - what am I doing wrong?" - publication is no guarantee of correctness... Commented Jan 14, 2015 at 8:22

I think you are misreading the paper, they do not claim what you say. Their claims are not based on number of top players, but on their ratings. If the statistical distribution of strength is the same among men and women, then the expected number of women among the top 100 is 6, if their proportion of the total population is 6%. Some citations from the paper:

A popular explanation for the small number of women at the top level of intellectually demanding activities from chess to science appeals to biological differences in the intellectual abilities of men and women. An alternative explanation is that the extreme values in a large sample are likely to be greater than those in a small one.

That is indeed true. You would expect the rating of the best man to be above the rating of the best woman. The paper goes on to try to compute by how much, a result which will depend very heavily on the assumed distribution.

In section 3, results, they go on to pair best man with best woman, same for next best, and so on, for the first 100 such pairs. Then they calculate the rating difference, and compare that to the expected rating difference given the fact that there are many more male than female players. All of this seems correct, and is very different from how you present it. It might well be that their analysis is little robust, and that a more thorough analysis could be done, but their basic idea is correct.

• +1 My reading of the paper is the same: that it focuses on rating differences rather than differences of proportions.
– whuber
Commented Jan 14, 2015 at 15:10
• That makes sense, thank you. I fear I was reading this paper with a preconceived idea of the question they were answering: this paper is often presented as an explanation for why there are so few women in the top echelons of chess, compared to the number of females overall. But it seems to tackle this question indirectly, by looking at rating differences of the top players.
– tom
Commented Jan 15, 2015 at 0:21