# Not Everything is a Normal Distribution

Bryan Caplan muses on EconLog about how to reconcile two assertions: His take that the “greatest minds in history” were always similarly great, and the “Flynn effect” which means that average IQ levels have gone up considerably over the past century or so. The puzzle for him is: “How could IQ go up while genuine intelligence stagnates?” I think there are several points in his post worth commenting on, but let me address only one of them here.

Bryan Caplan distinguishes IQ from “genuine intelligence,” which I think is warranted because the former measures certain mental capabilities that are related to intelligence, but not coextensive with it. Let’s start with IQ anyway as if it were all that’s to it.

It is at best a convenient idealization that the distribution for IQ is normal. It looks this way on a plot. This is the famous “Bell curve” that is named after the German mathematician Carl Friedrich Gauß because it was first discovered by the French mathematician Abraham de Moivre. You may also run a statistical test and find that you cannot reject the null hypothesis that it is normal. But despite all this: It is not true. The reason is quite simple. A normal distribution has a positive weight for values below zero. However, it is simply not possible that anyone could ever score a negative result on an IQ test. The worst that could happen would be zero.

And the same also applies on the other side. There is certainly some upper bound for how high a measured IQ can go. It takes some time to just read the questions or view the images. There is also a limit for how fast you can check boxes. And even if you could do away with all this, there must be a finite maximum speed for how fast the human brain can process information. It may be hard to pin down what the upper bound is exactly, yet IQ has to remain finite.

But then the normal function has positive weights to infinity. So it cannot be a fair description for rather high levels either, which is what the question turns around. A normal distribution may work fine for the central part of the distribution. You can then draw all kinds of conclusions if you play around with the mean and/or the variance. But that does not have to carry over to the borderline cases where a normal distribution obviously misses something. And there is in principle also no contradiction here that you could shift the mean around, leave the variance fixed, and still have the same weight at the upper end. That’s only a problem if you have indeed a normal distribution where the latter is determined by the former.

This would already be an argument under the assumption that IQ is the same as “genuine intelligence.” But that does not have to be the case. A simple explanation might be that to be a “greatest mind in history” it perhaps matters that you have a high IQ — you find it easy to do mental operations and that helps you with making some outstanding discoveries — but the exact level might not matter all that much. It could be other things, ie. that you luck out and work on something where there is indeed something great to discover. And that again may depend on what others have discovered before you or not. There might simply be only so many great breakthroughs within reach at some point in time. More cannot be forced into being. And if you compare the level of achievement inherent in such discoveries, it is easily conceivable that it looks the same across all ages.

Now if that is so, you basically feed IQ into a limiting function which makes the resulting distribution even less like a normal distribution. Someone might become a “greatest mind in history” with an IQ of 150, 200, or 250, but their achievement could be just the same. The person with an IQ of 150 would maybe take longer to hit gold. However, that does not matter for the greatness of his discovery.

Actually you often have it in history that something is “in the air,” and several people have the same idea in parallel, take Leibniz and Newton with differential calculus. As it seems, Newton was much faster and probably had a higher IQ. But Leibniz could do it, too. The reason it had not happened before, although there were others who had an inkling, was maybe not that others before had been less intelligent. There are also no obvious constraints, all you need is paper and a pencil.

Why it happened practically in parallel was perhaps the coincidence that several other mathematicians like Pascal, Descartes, or Fermat, had already stretched the envelope and had come close, and also that people were interested in physics at the time, which naturally leads to questions about derivatives and differential equations. If you had had 100 times as many Newtons and Leibnizes, even with higher IQs, it might have also happened around the same time and not before, and the difference could have been only a slight speed-up with more competition. It would always have been differential calculus, though, the same outcome.

Or to take a parallel example: If you looked at how fast people can run a sprint over 100 meters, you could also set up something like an RQ (running quotient) that measures relevant capabilities and lumps them together. I would guess that that would also have approximately a normal distribution over a population and a good predictive value for actual speed. Still the same argument applies as above that it does not have to work at the extremes. There cannot be runners with negative speeds, and there is also some upper bound for us humans that cheetahs can only laugh about. Since human height has increased over the past century, and longer legs are probably useful for speed and so are more muscles, I would expect that average RQ has experienced a massive “Flynn effect,” too.

Still if you feed this roughly normal distribution into a similar limiting function because there is some hard speed limit for humans, the outcome at the upper end might only budge a little. The world record in 1921 stood at 10.4 seconds, and it is now at 9.572 seconds. That’s less than a second between the two values while the average over the population has perhaps gone up by a multiple. Arguing on the basis of a normal distribution may not be appropriate if it is not a fair description of what you are after. The mean may go further up, the variance may remain the same, but the world record, it seems safe to say, will probably never fall below 9 seconds.

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