Regression to the mean, defined

Regression to the mean, in full

Francis Galton named the effect while studying inherited height, in an 1886 paper for the Journal of the Anthropological Institute titled Regression towards Mediocrity in Hereditary Stature. Unusually tall parents had children who were tall but, on average, closer to the population mean than the parents themselves, and unusually short parents saw the mirror image. The children were not being pulled anywhere. The parents' extremeness was partly heritable and partly the luck of their own draw, and only the heritable part carried over.

Any measurement splits into a stable component and a noise component. Selecting the top of a distribution selects for both at once, so the group you picked holds more than its share of favourable noise. Noise does not persist, by definition. Measure the same group again and the stable part stays while the noise is redrawn, so the new result sits closer to the middle. The larger the noise relative to the signal, the larger the regression.

It is not a corrective force, and it is not the gambler's fallacy wearing a lab coat. The two say different things. Regression says the next measurement of an extreme case will probably be less extreme, which is true. The gambler's fallacy says an independent chance event will lean the opposite way to compensate, which is false, and is taken apart in our guide to whether lottery numbers are due. Where a result is pure noise, regression is total: the best forecast for next time is simply the mean.

Category: Probability
Also called: Regression toward the mean, Regression towards mediocrity
In the glossary: All 75 terms, A–Z

A worked example

The same idea with real numbers attached.

Over 300 draws of a 6-from-49 game, each number is expected to appear 300 x 6/49, about 36.7 times. Suppose 17 turned up 52 times, well clear of that. Since the excess is noise and nothing else, the best forecast for the next 300 draws is 36.7 again, not higher and not lower. The count regresses fully to the mean, which is a different claim from saying 17 is now due for a quiet spell.

Sources

Every dated or checkable claim above, with where it came from.

Francis Galton described and named the effect in an 1886 paper on hereditary stature, reporting that the children of exceptionally tall or short parents fell on average closer to the population mean than their parents did.
Francis Galton, "Regression towards Mediocrity in Hereditary Stature", Journal of the Anthropological Institute 15 (1886), 246–263

Gambler's fallacyProbabilityThe gambler's fallacy is the mistaken belief that a run of one outcome makes the opposite outcome more likely next, as though independent chance events kept a memory and corrected themselves.Law of large numbersProbabilityThe law of large numbers states that as the number of independent trials grows, the average of the results converges to the expected value, while the absolute counts of each outcome need not even out.Expected valueProbabilityExpected value is the long-run average of a random quantity, found by multiplying each possible outcome by its probability and adding the results, so a 1-in-100 chance at 50 has an expected value of 0.50.Independent eventsProbabilityTwo events are independent when the outcome of one leaves the probability of the other unchanged, which means their joint probability is the plain product P(A and B) = P(A) x P(B).Confirmation biasPattern-seeingConfirmation bias is the tendency to notice, seek out and remember evidence that fits a belief already held, while giving less weight or less attention to evidence that would count against it.

Where it comes up on this site

Are lottery numbers due?Open →Every game ranked by oddsOpen →How the daily numbers workOpen →

Regression to the mean: frequently asked

No, though they are easy to confuse. Regression is a real effect about re-measuring a case you picked for being extreme; the gambler's fallacy is a false belief that independent events settle up. Selection and noise on one side, an imaginary force on the other.

It does not cool off. It goes back to its base rate, because it never left it.

More probability terms

Combination Cryptographically secure PRNG Digital root Expected value Factorial Gambler's fallacy

The rest of the vocabulary

Lottery mechanics, probability, numerology and astrology, defined in one A–Z.

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