Variation in male and female Olympic performance II

In my previous post, I looked at what would happen if men and women had the same average athletic ability but men were more variable. I also looked at what would happen if men and women were equally variable but had different average abilities.

Now I want to look at something different. What if men and women have equal abilities in a given area, equal mean and variance, but more men are interested in that area? What effect does the greater competition? In this scenario, we would expect the male athletes to be better, but would the difference between men and women increase or decrease as you get to higher levels of competition?

Suppose ability for men and women are both normally distributed with mean 0 and variance 1. Then the performance of the best person out of n who try out for a sport is the nth order statistic of the standard normal. The median of this random variable is y(n) = Φ−1( 0.51/n ).  (See this paper for details.) The following table lists some values of y(n).

n y(n)
10 1.499
100 2.462
1,000 3.198
10,000 3.811
100,000 4.346
1,000,000 4.827

This means, for example, that if 100 people tried out, the best person is as likely to have ability above 2.462 as ability below that value.

Suppose 10 times as many men as women are interested in a sport. If there’s little competition, say 100 men versus 10 women, we’d expect the best man to have ability somewhere around 2.462 and the best woman to have ability around 1.499, a difference of  0.963. As the competition increases, the performance of the best man and the best woman increase, but the gap between them decreases. If 1,000,000 men are interested in a sport and 100,000 women, the differences in their abilities would be around 4.827 − 4.346  = 0.481, about half as much as difference as there was with less competition.

So according to these estimates, if men and women have equal ability in a sport but proportionately more men are interested in that sport, the difference between the best men and the best women will decline as the competition increases.

The same reason could be applied to show what advantage a large country would have over a smaller country if the citizens of both countries are equally talented and equally likely to want to compete in a sport.

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