What is regression to the mean? Why extremes don't last

Regression to the mean is the tendency for an unusually high or low result to be followed by one closer to the average, whenever the result depends partly on luck. A student who scores far above their usual on one test will probably score lower on the next, not because they got worse but because some of the first score was good fortune that won't repeat. Nothing pulls the result back. The luck just doesn't come twice.

In short

• When a measurement mixes a stable part (skill, health, true ability) with luck, extreme results are partly extreme luck, and the next result tends to be less extreme.
• The more luck matters, the stronger the pull: a result that is half luck tends to move halfway back to the average.
• It fools people into crediting or blaming whatever happened between the two results, which is why studies need a comparison group.

What is regression to the mean, in plain terms?

Think of almost any result as two parts added together: a stable part that stays the same from one try to the next, and a chance part that changes every time. A golfer's score is skill plus the day's wind, bounces and nerves. A blood pressure reading is a person's typical pressure plus the effects of that morning's coffee, stress and the cuff.

Now pick out the most extreme results, the best scores or the highest readings. Those cases got there partly because their stable part was high and partly because their chance part was high too. On the next try the stable part is the same, but the chance part is a fresh draw, and on average it is ordinary. So the group's next result is, on average, closer to the middle. Regression to the mean is not a force; it is what happens when you select on a result that luck helped produce.

How does it work? Dice, step by step

A game makes the idea exact. Each player has a skill level from 1 to 6, fixed once and for all by one roll of a die. Each round, the player's score is that skill plus a fresh roll of a second die for luck. Scores run from 2 to 12, and the average is 7. Skill and luck vary by the same amount, so each accounts for half of the scores' variance, the usual measure of spread.

What should a player who scores 12 in round one expect in round two? A 12 can happen only one way: skill 6 and luck 6. In round two the skill is still 6, but the luck die averages 3.5, so the expected score is 9.5. The player who scored 2 must have skill 1 and luck 1, and can expect 1 + 3.5 = 4.5 next time. A score of 11 comes from skill 5 with luck 6, or skill 6 with luck 5, equally likely, so the expected skill is 5.5 and the expected next score is 9.

Round 1 score	How far above or below 7	Expected skill	Expected round 2 score	Distance from 7 now
12	+5	6	9.5	+2.5
11	+4	5.5	9	+2
10	+3	5	8.5	+1.5
7	0	3.5	7	0
4	−3	2	5.5	−1.5
2	−5	1	4.5	−2.5

Every row moves exactly halfway back to 7. That is no coincidence: the expected next result is the average plus r times the distance from it, where r is the share of the variance that comes from the stable part, here 0.5. It is also the correlation between round one and round two. If scores were pure skill, r would be 1 and nobody would regress. If they were pure luck, r would be 0, and every player, best or worst, would expect 7 next time. In real life the mix is somewhere between, which is why the pull is partial. The dice make luck visible, and their spread behaves just as probability predicts.

Note what doesn't happen. The players with high skill stay above average: the 12-scorer's expected 9.5 is still well above 7. And the spread of scores in round two is just as wide as in round one, because new players get lucky to replace the old ones. The scores don't squeeze toward the middle over time. Only the same people's next result, picked out because it was extreme, tends back.

Where did the idea come from?

The Victorian scientist Francis Galton found the pattern in human height. In an 1886 paper, he compared the heights of 928 adult children with those of their 205 sets of parents, averaging each pair of parents into a "mid-parent" after scaling women's heights up to the male equivalent. Tall parents had tall children, but on average the children's height deviated from the population average by only about two-thirds as much as their parents' (Galton, 1886). Very short parents had children who were short, but less so.

Galton called it "regression towards mediocrity", meaning towards the middle. It works in both directions in time: the parents of very tall children are, on average, less tall than their children. That symmetry is the giveaway that it is a statistical effect, not a biological drive towards the average, and heights in the population don't converge. Galton's work on it also gave statistics the word "regression" for fitting lines to data, and led towards the idea of correlation.

How does regression to the mean fool people?

It produces cause-and-effect stories out of nothing. The psychologist Daniel Kahneman described the moment he understood it, teaching Israeli Air Force flight instructors that praise works better than punishment. A senior instructor objected: when he praised a cadet for a clean maneuver, the next attempt was usually worse, and when he screamed at a cadet for a bad one, the next was usually better. Kahneman realized both observations were regression. Exceptional flights are followed by more ordinary ones whatever the instructor says, so, as he put it in his Nobel autobiography, people are "statistically punished for rewarding others and rewarded for punishing them".

The same mistake shows up widely:

• The "Sports Illustrated cover jinx". Athletes reach the magazine's cover after an exceptional run, which usually includes some luck, so a dip afterwards is the likely outcome, jinx or no jinx (Wikipedia).
• The rookie who disappoints in year two. The best first-season performers were selected partly for luck.
• Treatments that seem to work. People try a remedy when their back pain, cold or headache is at its worst, and feel better soon after, as they would have anyway. This is one large part of what passes for the placebo effect.
• Policies aimed at the worst cases. Schools with the lowest scores one year, or road junctions with the most crashes, tend to improve the next year even if nothing is done.

How does a control group catch it?

By letting regression happen to both groups and comparing them. Blood pressure is the classic medical case: pick people for a study because their pressure is high on one reading, and their average will be lower when measured again, with no treatment at all, because a single reading carries a good deal of chance variation.

Suppose, as an illustration, a clinic enrolls everyone whose first reading is 160 or above, and the group averages 170. Give half a new drug at random and half a placebo. A month later the placebo group averages 160, and the drug group 152. The placebo group's 10-point fall is mostly regression plus whatever else happened to everyone. Only the 8-point gap between the groups is evidence about the drug. Without the control group, the drug would have been credited with an 18-point fall.

Two other habits help: take the average of several measurements instead of selecting on one, and be wary of any before-and-after comparison where the "before" was picked because it was extreme. A related mistake is to study only the cases that came through a filter, which is survivorship bias, and both are traps for anyone weighing evidence from performance data.

Check yourself

• Without the table, explain why the player who scores 12 in round one should expect about 9.5 in round two.
• Why does the spread of scores not shrink over time, even though extreme players regress?
• In the blood pressure example, which number is evidence about the drug, and why not the 18-point fall?

Sources

• Regression towards mediocrity in hereditary stature — Francis Galton, Journal of the Anthropological Institute, 1886
• Daniel Kahneman, biographical — The Nobel Prize, 2002
• Regression toward the mean — Wikipedia
• Sports Illustrated cover jinx — Wikipedia