Suppose we have collected data on some measure deemed important to society. Examples are fireman entrance exams, standardized test scores in mathematics, income, IQ, and so forth. Higher measures are considered better. The raw statistics (whether as a whole or by “cuts”; say, age, etc.) indicate that the distribution of, let us call them “scores”, are shifted higher for whites/males than for blacks/women.
This rightwards-shift we can call a “gap.” Now, this being the universe in which we live, something caused this gap to be. This something cannot be “chance.” Chance isn’t a thing and thus cannot be a cause. Neither can “randomness” be a cause, and for the same reason. Instead, some real thing or, more likely, things caused each individual to have the score he did. Thus, the gap between collectives is a necessary outcome of that collective’s individuals, and, speaking exactly, nothing can be said to have caused the gap. Only the scores of the individuals themselves had causes.
It is logically possible that each individual had the same cause for his score. If we want to be fanciful, we can suppose that sunspots (via some mechanism) caused the scores to take the values they did. However, given our experience with these things in our own lives, it is more plausible that each individual’s score was brought about by a different combination of causes, some of which were different and some of which were the same across many or all individuals, but of varying degrees in each.
What can statistics tell us about these causes? Nothing. At least, nothing much and nothing directly.
But that hasn’t stopped people from claiming that the observed gap supports some theory. The two leading theories are that the gap is there because of some innate difference in ability between whites/males and blacks/females, or that the gap is there because of racism/sexism.
Now, innate ability makes sense as a cause: it could be that, all other things equal, whites/males are just better than blacks/females on standardized mathematical tests, say, where by “better” I mean that the probability that any white/male scores higher than any black/female is greater than fifty percent. The “all other things equal” is somewhat problematic, because in that phrase can lie plenty of indirect causes. But consider the analogy that innate ability favors those over six feet being better basketball “dunkers” than those under six feet. Even here we must speak of “all other things equal”, but we have no difficulty seeing that tallness is the major cause of being a better dunker.
But racism/sexism cannot be a direct cause; only actions as a result of these attitudes that can affect the score can be a cause. For example, a person might despise whites/males, but as long as she does not let this emotion influence her actions (along pathways that influence the score), then her racism/sexism is harmless (with respect to the score). The racism/sexism first has to “become active”; for example, by a teacher being more disapproving towards those in the group she disfavors.
We can envision many different ways racism/sexism may become active, each of differing strength, while the number of reasons for innate differences appear smaller (say, certain genetic combinations). Whether or not this is true, the data in front of us say nothing about it.
Again, we have collected data which shows a “gap”. Both theories, innate differences and racism/sexism, say that we should see a gap. Thus, the gap we see confirms both theories—as it would confirm any theory which predicted the gap. For example, the theory of cultural differences also predicts gaps. Even stronger, the data contain no information on which theory is confirmed to a greater degree.
This is important because the matter has long been political. The most common, even de facto, belief is that any gap is prima facie evidence of racism/sexism. Indeed, the burden of proof is on the organization that awards the scores to show that it, the organization, is not racist/sexist. Yet the observed gap could have also been caused by innate or cultural differences (or something else).
It is practically impossible that any organization can prove its innocence. This is because of what we noted above: the number of ways racism/sexism could have influenced an individual are legion. Further, these ways are often undefined, or if defined, they are unquantified. And even if the ways were defined and quantified, it is extraordinarily unlikely that these quantifications can be had for any individual (since racism/sexism operates over a long period of time and in many instances). Even worse, the organization’s inability to exculpate itself will be (incorrectly) taken as further evidence of racism/sexism.
Perhaps worst of all, if it is thought the innate difference theory is false, yet it is at least partly causative, then it will be impossible—not just unlikely, but impossible—to eliminate “racism/sexism”, because there will always be gaps caused by innate differences, and these will be ascribed falsely to racism/sexism.
Even though they are often used for the purpose, statistics cannot say which of many competing theories caused a gap in some socially important measure. At least, not without information that is external to the scores themselves. The best form of external evidence would be a controlled trial, in which all possible avenues of racism/sexism/cultural differences among individuals has been controlled (or eliminated). All experience of human nature argues that such a trial will never happen.