Regression to the Mean

You came first in your class on this month's exam, well above your usual score. Next month, you'll probably slip a few places. That isn't necessarily because you slacked off. A statistical pattern is at work: regression to the mean.

Why it happens

Every score can be thought of as "skill + luck." An exceptionally good result usually means decent skill plus luck on your side: you happened to review the right topics, you happened to be in great form.

Next time, luck won't single you out again; it goes back to average. So your score drifts back toward your usual level. Likewise, a student who did unusually badly will usually do a bit better next time.

Do the math

Say the class average is 70 and the correlation between two exams is 0.5. A student scores 90 this time, 20 points above average. A sensible expectation for next time is:

70 + 0.5 × 20 = 80.

The weaker the correlation (the more luck matters), the stronger the pull back to the mean; the stronger the correlation (mostly skill), the weaker the pull.

Regression to the mean in daily life

• The fund with the best return last year usually isn't top this year.
• Athletes who land on a magazine cover after a hot streak often "cool off" afterwards.
• A child gets scolded after a bad test and improves next time; gets praised after a great test and slips next time. Parents conclude scolding works better than praise, but it may just be regression to the mean.

In the simulation below, pick the people who did best on the first test and see how they do on the second.

Using this in forecasts

When forecasting what happens next to someone or something that just had an extreme result, don't simply copy the last result. Pull your estimate back toward the average. And don't mistake regression for the effect of whatever you did in between.