# Variation, Within and Between

Occasionally, you will encounter someone who says this:

Variation between human races is greater than variation within human races

If you do, you know you’ve encountered someone who has been indirectly exposed to the work of Richard Lewontin.

There are two forms of “Lewontin’s fallacy.” One is the original claim that Lewontin made. It is demonstrably untrue, which is obvious once examined with graduate-level statistical knowledge. A later, weaker version is simply nonsensical. I’ll address these in order.

Lewontin’s Original Fallacy

In 1972, Lewontin published an article called The apportionment of human diversity, using blood group proteins. The work is pretty typical for its time, except it extremely political correct connotations, and so eventually took on a life of its own. Rather than discuss the original article, which has been thoroughly debunked, bizarrely focuses on blood proteins anyway, here’s an analogy. (I’m too tired to do matrix algebra now.)

Say someone comes to you, and says this:

“The racial groups that map to what we consider ‘East Asian’ or ‘Caucasian’ do not exist. There is no attribute of either race you can find, in which the majority of variation is between races, rather than within races. Hair, skin tone, skeletal shape, and so on all vary within both populations, so that means there is only one population.

In other words, the groups “East Asian” and “Caucasian” are entirely social groups. It is impossible to write a machine learning system to tell an East Asian apart from a European, if you don’t include purely social constructs like name, clothing style, and so on.

The obvious refutation (which mathematically requires matrix algebra) is to ask why in the world you would use only one dimension of variation (like height, or skin tone) to classify individuals as part of multiple populations.

You can just use multiple indicators, together. That way if there has been a murder, say, and the corpse has been stripped of clothing and identification and has been dumped, you can use multiple indicators together to determine the race of the victim.

If there is DNA evidence, you can do the same.

Indeed, you can do the same with “races” such as “German” and “French”!

If for some reason you’re transported back to the 1970s, and all you have is blood proteins, you can do the same.

The solution to Lewontin’s fallacy is to use multiple indicators together, and not just one.

These days, it seems crazy to suggest it would be impossible to tell the race of an individual from DNA. There’s even a popular PBS show about the concept! But in the 1970s, some people really were that ignorant.

The Remnant that Remains

There’s no reason to take Lewontin’s original fallacy seriously, but sometimes you’ll hear a variation of it

Variation in intelligence between human races is greater than the mean difference of intelligence of the races

This is like saying moisture is taller than speed. It makes no sense.

In some areas of life, differences in variation between groups is the fact that matters most. For instance, on many measures (say IQ, or time orientation) males have greater variation than females, while both tend to have the same average. From this you would expect you would see many more male violent criminals than female violent criminals, and also more male CEOs of large companies than female CEOs of large companies. There is little if any difference in the average of these traits between the sexes. There is substantial difference in the variation of these traits between the sexes, though.

In other areas, averages matter. For instance, the average IQ of American whites from the south-eastern United States is lower than the average IQ of American whites from the northern states. From this you might wonder if large companies have a disproportionately small number of CEOs from the American South, while white southerners have responded to this “dixie ceiling” by organizing politically to obtain political goods that they cannot gain in the marketplace.

I have never seen anyone talk, in a popular setting, about a comparison between a variation on the one hand and an average on the other. Typically one or the other is relevant to the conversation, and bizarre second-order comparisons (what is the variability in height of Australians compared t the average height of South Americans) are simply uninterpretable. But if you’ve never worked with variation as a real thing (through calculating a standard deviation to solve a problem, say), the remnant of the fallacy is a good-guess by an ignorant laymen of what Lewontin may have been talking about.

Conclusion

The phrase “Variation between human races is greater than variation within human races” is meaningless. It either refers to an empirical incorrect claim from the 1970s, on the impossibility of using “blood proteins” to predict race, or an incoherent claim that compares averages against variation.