Sex differences and probability distributions
Matthew Martin 12/04/2015 10:33:00 AM
d = -0.20 has been observed for sex differences in trust (Feingold, 1994). This size of sex difference is considered “small” and indicates 58% of women are higher than average man in trust (based on Cohen's U3).But here's the important caveat:
d = +0.50 has been observed for sex differences in spatial rotation skills (Silverman et al., 2007). This size of sex difference is considered “moderate” and indicates 69% of men are higher than average woman in spatial rotation skills.
d = +0.80 has been observed for sex differences in physical aggression (Archer, 2004). This size of sex difference is considered “large” and indicates 79% of men are higher than average woman in physical aggression.
d = -1.00 has been observed for sex differences in tender-mindedness (Feingold, 1994). This size of sex difference indicates 84% of women are higher than average man in tender-mindedness.
d = +2.00 has been observed for sex differences in throwing distance among children (Thomas & French, 1985). This size sex difference indicates 98% of boys throw farther than the average girl.
As I have noted in earlier posts (see here and here), sex differences with larger d values are not “more real” than smaller-sized sex differences. All men do not have to be taller than all women for a sex difference in average height to be “real” and have important social consequences (Prentice & Miller, 1992). Sex differences with larger d values also are not necessarily more attributable to evolution or biology, and smaller sex differences are not necessarily more cultural or due to learning than larger sex differences. Nature does not work that way.One issue I have is the way he presents the data. A statement like "69% of men are higher than average woman in spatial rotation skills" make it sound like he's saying there's a 69 percent probability that a man is better at spatial rotation than women, which would be a misinterpretation—in fact for most of these examples, the two distributions overlap more than they separate, and the headline statistic overstates the relevant probabilities.
Schmitt's post doesn't quite give us enough info to calculate the relevant probabilities, but if we assume everything is normally distributed and that the standard deviation is always the same for men and women, we can crunch some numbers. So let's do that:
"58% of women are higher than average man in trust"The relevant probability here is this: if we randomly sample one man and one woman, what is the probability that the woman is more trusting? Crunching the numbers with our added assumptions yields a 56 percent probability that the woman is more trusting. Ok, that's not that different. Either way, about even odds. Here's what the two distributions look like graphed:
"69% of men are higher than average woman in spatial rotation skills"That's a 64 percent probability that the man has better spatial rotation skills than the woman.
"79% of men are higher than average woman in physical aggression"That's a 72 percent probability that the man is more aggressive.
"84% of women are higher than average man in tender-mindedness"The difference between the headline figure and the relevant probabilities gets a bit more pronounced in the tails. This stat actually corresponds to a 76 percent probability that the woman is more tender-minded than the man.
"98% of boys throw farther than the average girl"This makes it sound like a man is all-but-certain to through farther than any woman, but in fact there's a 7 percent chance our randomly sampled woman will throw farther than our randomly sampled man. And as in the rest of the examples, this still understates the degree of overlap, as lots of women can throw farther than lots of men:
The statistics overstate the case for sex differences. Even the computed probabilities comparing a randomly sampled man and woman understate the degree of overlap between the two distributions, where, even though the averages differ, lots of men and women fall to the left and right of each other. And that's to say nothing of the fact that most of these differences are socially constructed rather than innate. If Schmitt's point was that men and women, as groups, are not the same as each other, then my point is this: we need to make sure we aren't making the same mistake about the people within our arbitrarily defined groups. No actual woman is identical to the average across all women. No man is the average man. With very few exceptions, the differences between the two distributions are dwarfed by the overlaps.