ECON 250 · Chapter 18 · what Galton actually found in 1886
Galton measured the heights of parents and their grown children, looking for a law of inheritance. What he found instead was a pull back toward the average, and he called it regression toward mediocrity. The name later transferred to the method he used to measure it — which is why we call the technique regression, even though the technique does no such thing.
Below, each dot is one family. The only thing you can change is how strongly parent and child heights are related.
Dashed: children exactly as tall as their parents. Solid: the fitted line. Red — the tallest tenth of parents. Blue — the shortest tenth. The open circles are each group's average parent and average child.
| parents average | their children average | moved toward the mean by | |
|---|---|---|---|
| tallest tenth | — | — | — |
| shortest tenth | — | — | — |
Things to try. Push r to 1. The fitted line lies on the dashed line and nobody moves — no regression to the mean at all. Now pull r to 0: every group's children average the same height, whatever their parents. Regression toward the mean is not a law of biology. It is simply what r below 1 looks like.
This is the objection Galton worried about. If tall parents have shorter children and short parents have taller ones, generation after generation, should the whole population not collapse to a single height?
It does not, and the numbers below say why. Compare the spread of the parents with the spread of the children:
| mean | standard deviation | |
|---|---|---|
| parents | — | — |
| children | — | — |
Near enough identical, at any value of r. The distribution is not shrinking. What regresses is the average of a group picked out for being extreme, not the population. The children of tall parents are shorter than their parents, but plenty of tall children are born to ordinary parents, and they refill the top of the distribution. Selection, not shrinkage.
Why this matters beyond heights. Take the ten students who scored highest on one test and they will usually do worse on the next — not because anything went wrong, but because an unusually high score takes some luck and luck does not repeat. The same arithmetic explains the athlete who has a quieter season after making the magazine cover, and why a policy brought in right after an unusually bad year tends to look effective. Whenever a group is selected for being extreme and then measured again, expect movement toward the mean before you reach for a cause.
Two different things share one word. Regression toward the mean is the phenomenon on this page. Regression is the method in Chapter 18 — fitting a line by least squares. Nothing in least squares regresses toward anything.