What is regression toward the mean?

Francis Galton, answered from the texts and cited to the page.

Francis Galton answers

Regression toward the mean is, at bottom, a law of deviation — a statement about how far the offspring of exceptional parents depart, on average, from the general level of the population. The clearest way to put it is this. Take stature as the case, since that is where I first established the law on human data. The population settles about a mean — in my observations, sixty-eight and a half inches for men — and every individual's stature is most precisely described not as a raw measurement from the ground but as a deviation upward or downward from that central level.¹

Now observe what happens across generations. If a parent stands at a deviation D above the mean, the offspring will stand, on average, not at D but at one-third of D — that is, closer to the general level than the parent was.² The exceptional parent produces less exceptional children; the children of the very tall are tall, but not so tall as their parents were; and the children of the very short are short, but not so short.

Each generation is pulled, as it were, back toward the common centre. I first noticed this in experiments with seeds, and was prepared to distrust it until the human stature data confirmed it beyond reasonable question.³ The law is not a biological peculiarity of seeds or of men; it is a consequence of the structure of inheritance itself. The child inherits partly from his parents and partly, so to speak, from the general stock — from the accumulated ancestry behind the parents — and that ancestry is, on average, more mediocre than any exceptional individual drawn from it.

What the law yields, once you have it, is a predictive instrument: the regression line, which states the mean stature of offspring for any given parental stature. From that instrument grew what I called co-relation — the measure of how far any two variable characters are yoked together — and from co-relation grew the whole apparatus of modern statistical measurement.⁴

The law of frequency of error, which Quetelet applied to the proportions of the human body, supplies the mathematical scaffolding: the distribution of statures in a stable population follows the astronomers' law of error, with equal grades of stature separating successive classes in the ranked row, the classes diminishing in number as one moves toward the extremes.⁵

The one clarification the later mathematics pressed on me is worth stating plainly. I first half-read regression as a biological tendency — a reversion of offspring toward some racial type. It is in truth a mathematical property of any two imperfectly correlated quantities: whenever the correlation between parent and child is less than perfect, which it always is, regression toward the mean follows as a necessity of the arithmetic, with no biological mechanism required beyond the imperfection of transmission itself. The science sharpened the idea; the idea was sound.

Sources

1. the law of Regression in Stature refers primarily to Deviations, that is, to measurements made from the level of mediocrity to the crown of the head, upwards or downwards as the case may be, and not from the ground to the crown of the head. (In the population with which I am now dealing, the level of mediocrity is 68½ inches (without shoes).)

   Natural Inheritance

2. the Deviation of the Sons from P are, on the average, equal to one-third of the deviation of the Parent from P, and in the same direction.

   Natural Inheritance

3. If this remarkable law of Regression had been based only on those experiments with seeds, in which I first observed it, it might well be distrusted until otherwise confirmed.

   Natural Inheritance

4. it would be a poor prerogative to inherit say the fifth part of the peculiarity of some gifted ancestor, but the chance of 1 to 5, of inheriting the whole of it, would be deservedly prized.

   Natural Inheritance

5. the difference, according to the law of frequency, between them and the 63rd man would be the same as that between the 63rd and the 75th, the 75th and the 84th, the 84th and the 90th. The intervening men between these divisions, whose numbers are 13, 12, 9, and 6, form a succession of classes, diminishing as we see in numbers, but each separated from its neighbours by equal grades of stature.

   Hereditary Genius