R = h2 S

From 2017: What Scientific Term of Concept Ought to Be More Widely Known? by Gregory Cochran.

The Breeder's Equation
R = h2 S.

R is the response to selection, S is the selection differential, and h2 is the narrow-sense heritability. This is the workhorse equation for quantitative genetics. The selective differential S, is the difference between the population average and the average of the parental population (some subset of the total population). Almost everything is moderately to highly heritable, from height and weight to psychological traits.

Consider IQ. Imagine a set of parents with IQs of 120, drawn from a population with an average IQ of 100. Suppose that the narrow-sense heritability of IQ (in that population, in that environment) is 0.5. The average IQ of their children will be 110. That’s what is usually called regression to the mean.

Do the same thing with a population whose average IQ is 85. We again choose parents with IQs of 120, and the narrow-sense heritability is still 0.5. The average IQ of their children will be 102.5—they regress to a lower mean.

You can think of it this way. In the first case, the parents have 20 extra IQ points. On average, 50% of those points are due to additive genetic factors, while the other 50% is the product of good environmental luck. By the way, when we say "environmental,” we mean “something other than additive genetics.” It doesn’t look as if the usual suspects—the way in which you raise your kids, or the school they attend—contribute much to this "environmental" variance, at least for adult IQ. We know what it’s not, but not much about what it is, although it must include factors like test error and being hit on the head with a ball-peen hammer.

The kids get the good additive genes, but have average "environmental" luck—so their average IQ is 110. The luck (10 pts worth) goes away.

The 120-IQ parents drawn from the IQ-85 population have 35 extra IQ points, half from good additive genes and half from good environmental luck. But in the next generation, the luck goes away… so they drop 17.5 points.

The next point is that the luck only goes away once. If you took those kids from the first group, with average IQs of 110, and dropped them on a friendly uninhabited island, they would eventually get around to mating—and the next generation would also have an IQ of 110. With tougher selection, say by kidnapping a year’s worth of National Merit Finalists, you could create a new ethny with far higher average intelligence than any existing. Eugenics is not only possible, it’s trivial.

So what can you explain with the breeder’s equation? Natural selection, for one thing. It probably explains the Ashkenazi Jews—it looks as if there was (once) an unusual reproductive advantage for people who were good at certain kinds of white collar jobs, along with a high degree of reproductive isolation.

It also explains why the professors’ kids are a disproportionate fraction of the National Merit Finalists in a college town—their folks, particularly their fathers, are smarter than average—and so are they. Reminds me of the fact that Los Alamos High School has the highest test scores in New Mexico. Our local high school tried copying their schedule, in search of the secret. Didn’t work. I know of an approach that would, but it takes about fifteen years.

But those kids, although smarter than average, usually aren’t as smart as their fathers: partly because their mothers typically aren’t theoretical physicists, partly because of regression towards the mean. The luck goes away.

That's the reason why the next generation has trouble running the corporation Daddy founded: regression to the mean, not just in IQ.