In general, with wide variation at the individual level, women’s brains are 10% smaller than men’s, and they score 5% less on cognitive tests. That is reflected in traditional medical school entrance exams. The only way to have women on average score higher is to rig the exam, or handicap the boys for many years in the government schools.
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.)