Ok, but that statement would look different. Contrast these statements:
Women are more nurturing and compassionate than men. Men are better able to provide physical protection of the home.
Everskyward is more compassionate than Cooter because she's a woman. Cooter is better than Everskyward at defending his home, since he is a man.
You are speaking of the second group of statements, while I have in mind the first.
To clarify your statements a bit:
On average, women are more nurturing and compassionate than men.
On average, men are better able to provide physical protection of the home.
You and Everskyward have both correctly pointed out that this statement does not allow you to conclude that Everskyward is more compassionate or that you are better at protecting the home.
But consider this statement:
Because on average women are more nurturing and compassionate than men, the statistical likelihood is greater that any one randomly chosen woman is more nurturing and compassionate than any one randomly chosen man.
That statement is a true fact, IF the first statement is true. Logic demands it. The problem with political correctness is when the first statement indeed has been scientifically proven to be true but people are silenced from using statistical likelihood to advantage for fear of hurting somebody's feelings.
The truth is our brains are hard wired to build stereotypes based on past encounters with individuals, and use that information to better predict how a future encounter with a similar individual will go. I have learned from past experience that very young humans are less able to restrain themselves when in social situations requiring silence, such as a cavern tour I went on yesterday where a baby whined and a boy kept interrupting the tour guide with stupid questions. No adult behaved that way, but I was not surprised the children did.
The next day we went on a brewery tour, and a young boy signed up with us. It crossed my mind this kid is going to be a pain, interrupting the tour guide with stupid questions. I thought this about him, perhaps unfairly, because of the base of data built in my memory from past experience. Maybe it was unfair of me to think this about him because maybe that particular kid would have been silent the whole time. As it turned out he wasn't; he kept interrupting the tour guide with questions out of time with the tour.
It is statistical likelihood that told me that kid would do that, NOT personal knowledge of that individual kid. All of us accept this syndrome all day long every day on a thousand little events in our lives. I bet every single one of you when boarding a commercial plane and seated near an infant thought, "I hope that baby doesn't cry the whole flight." PROFILING! STEREOTYPING! You should be ashamed of yourselves. There are some individual babies who don't make a peep on an airplane flight, therefore you shouldn't think these stereotyping thoughts in your head.
By the way, before anyone bashes me for dissing talkative kids and crying babies, I've raised children myself, I tolerate this and don't feel ill will toward them. I am merely pointing out that my brain can accurately predict their behavior.
Also, I am not trying to claim that we have scientifically proven women are more nurturing and men are better at protecting the home, I am saying IF that is true, then it would be understandable to stereotype. Is there a problem of people stereotyping when on average it ISN'T true? Absolutely. I believe this happens when propaganda has been used against groups, we have a ton of examples in history and is the reason we are careful to beware of stereotypes. But we shouldn't go too far in the other direction, and completely ignore real behavior. The little old grandma in a wheelchair being patted down by TSA while the young male muttering Allah Akbar is waved through comes to mind.