## Sexual differences in intelligence differences

There are some interesting patterns of sex differences in intelligence scores. One pattern is that men tend to score higher on the spatial/mathematical component than women, a fact that got the president of Harvard pushed out of his job a few years ago for suggesting that it explains the membership differences in STEM. Women, in turn, tend to score a little higher on the verbal component than men. Perhaps this pattern can be readily explained by evolutionary theory.

But, a different pattern is a little more mystifying: men have a little greater standard deviation than women, so women are a little more clustered around the mean of 100 IQ, and, when you go to each tail end, you find more men than women. This has been known for a hundred years. Wendy Johnson et al ("Sex Differences in Variability in General Intelligence: A New Look at the Old Question," 2014) cited data from 1914 that indicated 15.45 as an IQ standard deviation for males versus 14.52 for females. Standard deviations calculated since then have always varied somewhat, but there is always a small gap between the sexes that gives males a little larger variance (see also Strand et al's "Sex differences in Cognitive Abilities Test scores- A UK national picture," 2006).

Even a small gap makes a big difference at the tail ends, and it may help to explain embarrassing sex ratios in high-IQ societies. If the general standard deviation of IQ generally is 15, and if the qualifying Q value for Mensa is 0.98 (top 2%), then the required z-score is 2.053749, which means Mensa requires an IQ greater than 2.053749*15=30.806. I am using this z-score calculator.

But, as men and women have different standard deviations, they have different z-scores. The z-score for male Mensa qualification is 30.806/15.45 = 1.9939, which corresponds to a Q value of 0.023081, meaning the top 2.3% of males qualify for Mensa. The z-score for female Mensa qualification is 30.806/14.52 = 2.1216, which corresponds to a Q value of 0.016936, meaning the top 1.7% of females qualify for Mensa.

And this means, based on statistical data alone, we would expect 2.3/1.7 = 1.4 times as many men as women in Mensa. The actual reported male to female disparity is a little more than that, being 2:1 (66% male to 33% female). So, the sex ratio may still be a problem, but it is only half as much of a problem as it may appear if it is wrongly assumed that the standard deviations between the sexes are equal.

The further along the right tail end we travel, the larger the discrepancies get. Mensa requires a measly 1 in 50, but to qualify for the Prometheus Society you need to be in the top 1 in 30,000, or z=3.9879, or Q=0.00003333, or IQ greater than 100+3.9879*15=159.8. But for males the required z-score is 59.8/15.45b=3.87055, which is a Q value of 0.000054, which means 1 in 18,400 men can qualify. For females, the required z-score is 59.8/14.52 = 4.11845, which is a Q-value of 0.000019, which means 1 in 52,400 women would qualify. We would statistically expect 2.8 times as many men as women in the Prometheus Society. The embarrassing sex ratio of Mensa would actually be the best case scenario for the Prometheus Society!