Understanding averages

One reason for the hostile reaction to the the now-notorious “Google Memo” is, I think, that it is very difficult to think about how group differences might apply to individuals. Statistical claims about groups almost always feel as if they relate to a general and universal effect applying to each individual within those groups.

The memo argued that “differences in distributions of traits between men and women may in part explain why we don’t have 50% representation of women in tech and leadership”. If true, a hiring policy would be flawed if it assumed that a totally fair system would produce something close to a fifty/fifty balance of men and women at every level of the company.

I will not get into whether this claim (or the others) is correct or not. Scott Alexander makes a good case that the difference might be explained as a product of different interests between men and women, not differences in ability. Or maybe it really is a product of sexism. My own views here don’t really matter or add anything to the debate.

What is interesting to me is how we think about claims about group differences, and how we relate that to individuals. My basic claim is that group averages can reflect how many people within a group have a certain trait, rather than how strong the trait is in each individual member of the group, and often we mix these two things up.

Think about the claim that “Women generally have a stronger interest in people rather than things, relative to men.” Whether or not this is true, it’s hard not to imagine this as a comment on women as individuals. For a given man and a given woman who seem similar in other respects, this claim sounds like it’s saying that each man will be more “interested in things” than each woman.

But it doesn’t! This is what my friend “the Anonymous Mugwump” refers to as a difficulty in “applying averages”. We tend to take claims about groups as claims about each individual within those groups. That means that claims about men or women being more inclined towards one thing than another sound both obviously false and easily rejected, and very insulting. In the case above, the group averages don’t tell us anything at all about the man and the woman as individuals if they’re already selected for their individual traits — like being employed in a certain profession or enjoying a certain hobby.

For every one hundred people, there might be ten men and nine women who are equally “interested in things”. This would mean that on average men are substantially more likely to be “interested in things” than women, but that as individuals none of those women are any less interested than those men and that there are millions of women who are just as “interested in things” as any man is. Applying the average to the individuals would be a very silly (if easily made) mistake.

It’s even worse when it’s something that can’t be framed as just a difference of opinion. If someone says that women are, on average, more neurotic than men, or that the distribution of intelligence means that “at the higher end of the distribution (from where you get your STEM graduates) you would expect to find more men”, it sounds like a claim that any given woman is more likely to be neurotic or less likely to have an IQ of 140+ than a similar man.

But again, think of claims like this about how many people within the group have this trait, not how strong this trait is among each member of the group.

In every one hundred people, there might be ten women and nine men who score highly on the neuroticism trait. There might be five men with exceptionally low IQs and four women with exceptionally low IQs, and five men and four women with exceptionally high IQs (and ninety men and ninety-two women somewhere in between).

Group differences, in other words, tell us next to nothing about the traits of a given man or woman within those groups — there are lots of neurotic men and lots of genius women. Very simple information about any individual is going to tell you much much more than whatever the distribution of attributes is for a group they happen to be part of.

They can, though, help us understand why (to use Scott Alexander’s example) more veterinarians are women than men, or perhaps why more men become engineers than women. For every one hundred people, there are three women interested in being vets and two men, and two women and three men interested in becoming engineers.

Consider also the danger of looking at relative rates but not absolute levels. Suppose red-haired people were twice as likely to commit fraud as blondes, brunettes and baldies. As well as raising interesting questions about why — perhaps red-heads are discriminated against and have fewer legitimate opportunities to make money — this could engender prejudice against red-heads including the view that a given red-head is likely to commit fraud against you. But if the levels of fraud are so low in both red-heads and others, 99% of red-heads might never even think of committing fraud, but still be unfairly judged as would-be fraudsters.

This is all very important for reasons beyond the “Google Memo”. Different immigrant groups may have different propensities to commit crime, for various and mostly benign reasons, and some higher than the rate of natives. But, again, these are group differences and are often misinterpreted, justifying and fuelling prejudice against individual members of those groups.

In cases where the absolute level is low in both groups, like violent crime, higher rates (still at very low levels) from an immigrant group are used to justify harsh and harmful policies excluding some criminals, but many more non-criminals, from living and working here.

It might be useful for the security services to know that Muslims are more likely than non-Muslims to become terrorists in Britain — for example, for knowing where to direct surveillance and anti-extremism resources. But this useful group average is misinterpreted by many people as implying that individual Muslims are sympathetic to terrorism, and leads to very bad mistreatment.

Since people don’t really know how to interpret group differences, the ones who do have a responsibility to talk about them very carefully to avoid fuelling prejudice and unfair discrimination. But, where possible, we should also be trying to help people understand the statistics. Group differences may tell us why more of one group is represented in a profession than another, but for the individuals within or trying to enter those fields they probably tell us next to nothing.