I don't know the data on brain size other than the bigger your body is, your brain may be bigger. Not sure that has anything to do with cognition.
You are wrong about women scoring 5% lower on cognitive tests. Some subtests are somewhat lower for women. Some subtests are lower for men.
However, the means and the medians for cognitive tests are about the same for males and females. The shape of the distribution may vary for males and females. Women score higher than men on some tests. Men score higher than women.
BTW the man who *claims* to have invented Scrabble had a hard time beating his wife. She beat him most of the time. (She probably invented the game.)
Do the math on the below chart. Generally women score about 5% less than men on cognitive tests. Women and men are genetically different. They are biologically optimized for different but equally important roles. Because there is such a wide variation at the individual level everyone should be given an equal opportunity, but an equal outcome at the group level isn't likely unless the test is rigged.
