I first heard about the hypothesis that men have variability of cognitive skills compared to women when I was about 10. I read it in a local newspaper actually. The town I grew up in had 2 local newspapers. One was free and was distributed into the letterboxes like spam flyers are distributed these days. The newspaper would be shoved into the letterbox even if you didn’t want it. You didn’t even have a choice over the fact whether you are getting it and it was less of a newspaper and more like a giant billboard full of shitty ads. Many people used that one to wrap smelly food in it instead of reading it. Now the second one was a premium one. You had to pay to get the second one and boy it was worth it: amongst the mundane provincial news about a lost cat, jokes stolen from back-then internet and a crossword, one would find science & technology tidbits like the gem above. The local newspaper had a section that speculated about the greater picture: about the universe, the stars and yes, in one particular instance, about the nature of intelligence and gender differences therein.

So I read that thing and, being an impressionable 10 year old, went in my head “heh, kinda makes sense, I guess”. If my memory doesn’t betray me, and it almost certainly does, the article also stipulated there are no differences in average intelligence, i.e. women, on average, are just as smart as men. Thus, at the time, it did not even occur to me that what I read was somehow uncharitable, inflammatory and sexist. When I returned to school I didn’t start teasing girls with my new revelation. Oh I almost definitely teased the girls, just not with the level of sophistication found in the local newspaper. I didn’t shout “your chances of being cleverer than threshold X are lower than mine” to the girls that I did not like. I stuck to a more simple and verified “you’re fat” technique and it worked pretty well.

And so that was it for me as long as the claim was concerned. I lived a happy 10 or so years, never having a particularly great interest in the subject, moved out from town, moved to Britain, graduated from a university and got enrolled onto a maths PhD program and then during one of those never-ending lunches with other PhD students I just blurted it out: you know you probably expect more maths PhDs to be male because of the difference in variance. It was a beginning of a shitstrom: the claim was perceived, by some, as intentionally inflammatory.

And the shitstorm had a big impact on me: I soon read about Lawrence Summer affair, where the guy publicly suggested pretty much what I suggested:

Summers said that “even small differences in the standard deviation [between genders] will translate into very large differences in the available pool substantially out [from the mean]”

And then this happened, according to wiki:

Summers apologized repeatedly. Nevertheless, the controversy is speculated to have contributed to his resigning his position as president of Harvard University the following year, as well as costing Summers the job of Treasury Secretary in Obama’s administration.

And I read about a dozen of other similar cases. And I went: “holy fuck!…”. People can have their careers ruined by saying what I believed for years. This is totally not OK but also it’s really not worth even getting into this debate. However, now that I am wiser and older, I am going to say: “fuck it, let’s do this anyway!”. That’s the spirit.

Funnily enough, I am not even concerned about the validity of the claim: I don’t think it’s a big deal either way. What I really want to write about is that the claim is not discriminatory and should not cause the uproar that it causes. True or false, people who throw such hypothesis up in the air are not being sexist.

Now, let’s set the scene first with some hopefully not provocative line of reasoning. Evolutionary men and women went through different paths: they have different selection pressures which resulted in different bodies and different brains. Differences in our brains is substantial: men, on average, have 16%1 more neurons then women. Remember that this is just a physical difference that one can arrive at after dissecting lots and lots of brains of dead people. Seriously, let’s not get angry at this point at the researchers whose job is to dissect brains of dead people. Like let’s just not yell at them if they happen to have found something different between the two genders: “What? You found a difference again. No, no, no, no no. Your job is to count the neurons until there are no sex differences are found, go and do the recount!”.

I can already feel the excitement of those horrible horrible men (and women!) who think that women should have never gotten a right to vote or shouldn’t be allowed to drive or some other total bullshit. Now don’t you even start, you horrible people, and let me just finish.

If you ever dabbled in artificial neural networks in machine learning you know this fact: number of neurons have nothing on the architecture of the neural net. Tons of research effort is being spent on discovering new ways of connecting those neurons up in some intricate fashion in order to get the algorithm to actually produce something worthwhile. “Just increase the number of neurons in your artificial network” is not how big advances in deep learning are made: you will not break any records with such attitude. It’s all about how those neurons are connected to each other. In this manner a neural network with far fewer neurons but a superior architecture can vastly outperform another one. And this is pretty much what happened in humans: despite fewer neurons women can and have outperformed men in a variety of cognitive tasks: in the analogy of deep learning, they have a different architecture.

However one consequence about having a different architecture is that there must be a difference. There is just no way you can downsize your artificial neural network change the architecture, retrain and go: “hm… looks like it made no difference”. Whilst it is theoretically possible that for every given input the output will be exact same as prior to the changes, it is practically impossible. Going back to human brains: whether it’s map reading2 or multitasking3 - something, somewhere has got to change after reconfiguring the brain so substantially. And this 16% difference will result in some statistically significant real life performance differences too. Finally it doesn’t have to be differences in mean performance on a given test, it could well be a difference in variance as well.

Another important thing to point is how not egalitarian this process is. The evolution doesn’t care about things like equal opportunities and fairness: it doesn’t have those values, it’s a stupid process. When brains of men and women started to diverge it did not stop there for a minute, thinking “What have I done? Now those assholes will use this to justify their nasty prejudices!”. We must all agree that it is OK to say that we have evolved to have different heights, body fat percentage and, yes, of course, different brains. So saying there is a population difference should not be something to get fired about: it’s just inevitable given the physical differences in our brains.

“Ok, ok, ok, so when can talk about the differences”, my imaginary intellectual opponent might concede, “but what you are essentially saying is women can’t do maths and that’s not only wrong, but racist and you are a pig and oh my god I can’t believe I am having this conversation”. Yeah… I don’t think highly of my intellectual opponents, one might say.

Well… no. I just said more men than women can’t do maths. Higher variance works both ways: not only you have more men who excel at maths, you also have more men who are god awful at maths. If the only thing I cared about in my child was that he/she wasn’t terrible at maths, having a daughter would be my best bet.

Not only that, jumping from a statistical difference on a population level down to individuals makes no sense. Yes, the higher variability implies that amongst the top mathematicians you are likely to find more males, but that doesn’t mean anything on the individual level. You are also more likely to find that top mathematicians come from Northern Hemisphere. That’s not because Australians can’t do maths. That’s because 88% of the population lives in the Northern Hemisphere. And when an Australian comes along and applies for a post-doc position you’d be a complete moron for not interviewing him/her based on the fact that he/she is from Australia.

Variance difference functions pretty much the same way. Let’s do some maths with imaginary numbers. Let’s say men and women score 100 on some cognitive test, on average, but with 18 and 15 variance, respectively. You noticed that everyone who works as a mathematician has at least 130 score. So, assuming, 130 score is an absolute minimum, we notice that about 4.8% of the men qualify and only about 2.3% women. That’s twice as many men compared to women in the candidate pool. And that’s assuming there’s no difference in interests, motivation and other factors that could also favour men. You’d still be a moron about not interviewing women though.

However I think I understand where the uproar is coming from. The people who get behind getting men like me fired don’t see the claim as some sort of innocent foray into intellectual curiosity, they see it as a threat to the status of women. If the scientific community starts widely accepting the claim about higher variability soon these nasty people will come out, denying the women equal opportunities and justifying all of it using the mainstream scientific consensus. These people who are already discriminating against women will be given a super weapon to beat women with.

I see how this is terrifying. But I also think that this fear has gotten out of hand because it makes the society extremely uncharitable towards curious people with no such agenda in mind.

My first response to such fear would be is to use science against such veiled attempts. If a person is spotted p-hacking and peddling some other statistical bullshit to justify some discriminatory narrative - then the research should be brought into question and such unscrupulous researcher should face the consequences. If people in power are spotted quoting some statistical difference in order to deny individuals a chance, they should be and must be shot down, no questions asked.

However I have the impression that the researchers who deal with politically charged subjects are the most careful statisticians out there. Because they know what they write will be called into question and thus they employ the most stringent statistical techniques precisely so that their integrity is not challenged.

And it’s a great great shame such diligent researchers are banished, shamed and flamed.

My second response is: “Jesus Christ seriously don’t shoot the messenger”. The messages come from the physical reality that scientists are trying to understand and model. They didn’t make men score with higher variability even if they wished for them to do so: it’s outside of their control. The reality sent us a message. The reality doesn’t send us messages in some pre-packaged, well-explained and reasoned academic papers that you can just harvest from some academic trees and put them on arxiv. No, the reality sends us this really time-consuming and next to impossible to decipher messages and then somebody worked for months trying to read it and all they got was some “die you piece of shit” reaction in return.

And I will talk about the consequences of such intellectual terrorism. Now that scientists are unwilling to sacrifice their careers publishing things that might be interpreted as inflammatory we have got ourselves into this situation where any research about gender differences is not trustworthy. So we get into this paradoxical situation where studies upon studies keep coming up favourable towards women yet the world stays surprisingly inattentive. It is too easy to be caught up in this narrative that the patriarchy and macho culture is so pervasive and it’s only getting worse: look we’ve got yet another study how women outperform men in industry X yet the world still mostly employ men in the industry X: there must be some evil macho men cackling in the background denying women the equal opportunity yet again. Alternatively you are in an massive echo chamber and you don’t even know. Step out of it, it’s nicer out here: sexism isn’t as widespread as it might appear to you.

Finally, the claim about variability doesn’t strike me as particularly offensive given how small the quoted differences usually are. I am not up to date with the scientific literature on the subject, but the difference in standard deviation is often about 10% or so. That’s tiny! Whilst such a tiny difference manifests itself somewhat prominently at the tails, in every day situations it is a distraction: individual differences totally dominate gender differences.


Going back to the veracity of the claim for a second. The most common counter I heard is something along the lines that girls get better grades than boys in maths in both high schools and universities. However such fact alone doesn’t contradict the variability hypothesis if you do the above calculation but with a threshold of 130 replaced by 110. One gets 28.9% (boys) vs 25.3% (girls). The difference in proportions is not as big anymore because one must remember the exponential decay of Gaussian distribution forces massive differences at the tail. Whilst at the level of 110 it’s still true that more boys qualify, the differences in grades could be caused by other gender differences, such as diligence.

  1. technically one must count neurons in some parts of the brain only because bigger bodies need bigger brain, however the overall argument holds. For the curious, here’s a much more technical treatment of the subject:

    This study’s objective was to investigate morphometric gender differences of the cerebral cortex in six males and five females, 12 to 24 years old. Though human brains lack sexual dimorphism on routine neuropathologic examinations, gender-specific brain weight, functional, and morphologic differences exist, suggesting that cortical differences may be found. Yet the cerebral cortex may be exempt from gender differences, as demonstrated by the fact that normal males and females perform comparably on intelligence tests. Stereologic morphometry on standardized histologic sections from 30 bilateral cortical loci determined cortical thickness, neuronal density, and derived neuronal number estimates. The mean +/- SD cortical thickness of the 60 loci examined was similar in males and females with right and left hemispheric gender ratios being balanced. In contrast, the average neuronal density of the same 60 loci was significantly higher in the male group than in the female group, and the corresponding mean male-to-female ratios were 1.18 in the right and 1.13 in the left hemisphere, which differ significantly from each other and from the balanced cortical thickness ratios. Estimates of neuronal numbers – the product of neuronal thickness times density – were 13% higher in males than in females, with mean male-to-female ratios of 1.13 in both hemispheres. The data provide morphologic evidence of considerable cerebral cortical dimorphism with the demonstration of significantly higher neuronal densities and neuronal number estimates in males, though with similar mean cortical thickness, implying a reciprocal increase in neuropil/neuronal processes in the female cortex.

    Abstract of https://www.ncbi.nlm.nih.gov/pubmed/10073431

  2. apparently done better by men, on average, but I have no stake in the claim 

  3. apparently done better by women, on average, but I have no stake in the claim