ATtheend: Further to Diana's point upthread about the spread of the distributions, it's worth reading Lise Elliot's Pink Brain, Blue Brain (she's a neuroscientist who specialises in brain plasticity in infant brain development).
She explains the d-value (the difference in means of two populations divided by the product of the standard deviations) - roughly, whether you have two very broad distributions which mostly overlap, or two sharp, distinct distributions. Insofar as there are any measurable and statistically significant differences in cognitive performance between men and women, most of them have very small d-values - meaning that you can't look at a test score for, say, maths ability at age 6, or reading ability at age 6, and say "well, that child's almost certainly a boy/girl" on the basis of it (small d-value). Whereas if I said to you "This person is 6'1" - do you want to take a five pound bet on them being male, I'll give you a fiver if they're female", you'd really be onto very good odds if you took that bet (despite the fact that 6'1" is comfortably within one standard deviation of the average male heigh for a man - i.e. it's not an extreme height for a man).
Further to the point about d-values, Elliot also goes on to explain that because of brain plasticity, even if you can find a small but measurable difference in, say, reading ability at age 6 between the average performance of the boys and the average performance of the girls, you'd be unable to decide how much of that difference was down to nature and how much down to nurture - because we as adults (and she has lots of examples of psychology experiments which demonstrate this) start treating children differently from infancy onwards.
So really the "male brains" and "female brains" claims are not well substantiated in the scientific literature. Of course you can cherry-pick and find the odd study - but when it comes to a meta-analysis of all the studies, there really is very little difference. And, as Diana says, the populations of trans people are so very small its very hard to get statistically significant results.
For what it's worth, I think there may well be some very subtle differences in brain structure - but much more subtle than "lady brains". And that doesn't remove the central problem with "self-identification", namely that you can't use self-identification to separate out genuine trans people from people (Davina Ayrton being one) who are just taking the piss or outright dangerous sexual perverts.