I'm somewhat mathematically illiterate, I'd appreciate some help in order to satisfy a curiosity.
Concise summary attempt: chess ratings are derived from past performance and therefore thought to be good predictors of future performance, odds of the match according to the ratings of the opponent. Men are "on average", better chess players than women, but it would still be "predicted" that odds of a match between a man and a woman would be "dictated" only by their ratings. But a psychological finding was that girls underperformed in matches against boys. The hypothesized psychological explanation is that girls, being stereotypically worse than boys (and, in fact, worse on average), are excessively stressed, intimidated in such matches, and perform worse than when competing against other girls. A commenter on a blog suggested that this explanation isn't needed, that the finding can be explained just by regression towards the mean. This explanation doesn't seem to make sense to me, just as (intuitively, and I could be wrong) I wouldn't expect a similar finding in a random small group of chess players which also had a smaller average score -- the individual ratings would still be more predictive than "belonging" to this group, I'd bet. I also think I've read once that regression towards the mean could the explanation in the opposite direction, that women aren't "really" worse than men on average, that it's just a mathematical artifact of there being way fewer female chess players. It couldn't work both ways, could it?
Not so concise. :-/
Following there's the post I wrote before, with perhaps some still relevant points, additional explanation of my "reasoning", and hopefully not terrible to read.
Chess has a fairly interesting rating system, that is roughly (perhaps "roughly" underscores it) an estimate of performance, and that "predicts" odds of next performance, according to the rating of the opponent. Chess players gain or lose points according to the rating difference of the adversary, and the result. The higher are the odds against the winner (per higher rating of the adversary), the higher the winner's ratings are increased. One can even score points by drawing with a significantly stronger opponent (perhaps with "any" stronger opponent, but only a negligible increase the closer the ratings are). And likewise, the higher were the odds favoring someone who loses, the more the loser's rating is decreased.
A psychological study has found that girls, in gender-mixed tournaments (which I guess are the majority[*]) underperform when playing against boys, according to this system. The theoretical psychological explanation is "stereotype threat", that psychological stress makes people perform significantly worse in situations that evoke stereotypes about the inferior performance of their group, like "girls aren't good at chess", or even mildly improve, in positive stereotypes.
One of the commenters on the blog suggested that it's just the expected from regression towards the mean, given that women, in general, have lower ratings than men (that's a fact, not questioned here).
But can it really explain this finding? Would it also be expected that by sorting random groups of players (can be all men, compared only against men, or computers/chess engines), only adding up to a similar a lower average rating of that group (or perhaps preferably a similar rating distribution), and looking at their matches, find that individuals on this group are "more likely" to have underperformed, as expected from their ratings?
It strikes me just as odd as saying that somehow Kasparov has a "higher chance" of underperforming if I, a weak chess non-player, enter the same tournament, and we're wearing the same shirt color. And at the same time I'd not play any better, even if everybody with other shirt colors are bad enough to have together a lower average than the embarrassing "Kasparov and I" average. That is, a random, smaller group, only with a lower average rating wouldn't obviously affect the performance of any individual player (good or bad). But would it somehow be saying anything about odds of individuals within that group? Perhaps something like boys, heavy-metal fans only, would be expected to still often have shorter hair length when compared to girls? While that doesn't seem that unlikely, I'm not sure it's quite analogous with the expectations of the rating mechanism.
[*] the proportion can have a role explaining the disparity, as the ratings of female-only tournaments could be inflated. But again I'm not sure it's that much significative, I think it would be similar for any proportionally weak "beginner/weaker player tournaments". And there would be many more weaker boys than girls in absolute numbers, if that levels things up. I'm not sure of anything really. I just think that such think would have an effect at national levels, that is, weak countries with inflated endogenous ratings, but I've never heard of that. It's also perhaps worth mentioning that the full range of potential explanations isn't in question here (it even sort of goes against the site's rules), the scope should be whether just random below-average groupings would allow us to predict a lower average performance for individuals within that group, even when that wouldn't be expected without such grouping, somehow. Or whether somehow these are mathematically different propositions, an invalid analogy.
For me it's just utterly counter intuitive, seems plain wrong, but in the other hand, like many other people, I also thought that it wouldn't make a difference to chose the other door or not on the Monty Hall dilemma.
I'm sorry for the length of the post.