Regression to the mean

Take the tallest children in a class. Their parents tend to be taller than average — but not as tall as the children themselves. Measure the same children again next week; their scores will drift toward the class mean. This is regression to the mean, and it requires no explanation beyond probability.

What to notice

• Orange dots are parents in the top quartile (X > +0.67σ). Red dot is their children’s mean.
• The red dot always lies below the orange cluster — children’s mean Y = r·(mean X of parents) < mean X of parents, since r < 1.
• r near 1: strong inheritance — children’s mean barely moves. r near 0: children’s mean collapses to zero regardless of parents.
• The blue regression line $E[Y\mid X=x]=rx$ captures the effect exactly.

Why it happens

The math is simple: in a bivariate normal with correlation r, the conditional expectation is $E[Y\mid X=x]=rx$. For any r < 1, extreme values of X predict less-extreme values of Y. There’s no force pulling children toward mediocrity — the effect is entirely a consequence of imperfect correlation.

The practical danger

Regression to the mean produces many false causal stories:

• A student scores unusually low on a test, gets tutoring, scores higher — tutoring gets the credit.
• A sports team has an exceptional season, regresses the next year — the coach gets blamed.
• A company tries an unusual intervention when sales are worst, sales recover — intervention gets the credit.

In each case the change would have happened without any intervention. Galton named the phenomenon in 1886 studying human height. The word “regression” — now used for all linear prediction — comes from his description of this pull.