I thought about writing separate articles about how gender roles are bullshit, how gender is a binary, and how transgender people are sexist, but I decided to just put all these truths into one article, because all those things are wrong for the same reason: they attempt to make gender an identity.
Gender is which role a given human body is able to fill in the process of reproduction. "That's the definition of sex, not gender! You bigot!" You can use whatever word you want, but as I'm about to explain, gender is completely meaningless if you separate it from sex. There is no second concept here. In the human species, there are two of these. A lot of people point as a counterexample to intersex people who have a mixture of biological traits. That objection is like saying I can't say humans have ten fingers because some people are born with abnormalities. Think about this: the only reason someone would try to "identify" as a gender other than what they biologically are is if gender is very important to them (you can see this reflected in the fact that almost no one tries to "identify" as a race other than what they biologically are; that's because people don't place as much cultural significance on that). Since gender is purely a trait of the body, it has nothing to do with your psychology or what roles you ought to fill (except insofar as it impacts your actual physical fitness for the job) and if you think it does or should, then you are a sexist, because that's what sexism is: attaching undue importance to a person's gender. And it's for exactly the same reason that gender roles of all sorts are illegitimate, because men and women are the same kind of being, and are capable of the same tasks minus reproduction.
Before I end this article, I know somebody is going to throw some statistics at my last sentence up there and say "but looky here I've got facts and facts trump philosophy". I could just point out that citations have to take a back seat to actual reasoning, but instead I'm going to explain how your stats might very well be true: it's a self-fulfilling prophecy. We were all taught gender roles as kids, even if only implicitly, and people act the way they're raised to think is normal for them. And there you have it: any statistics you're using to show that males and females really do have different psychology is adequately explained by the prevalence of the belief you're using them to argue for. Note that while many political "liberals" claim to oppose gender roles, they usually still welcome the idea that there are biologically-caused differences in psychology between men and women; so yes, it is a prevalent belief.