Regression to the Mean

Daniel Kahneman recounts an “aha!” moment while teaching Israeli Air Force flight instructors. After his lecture on the benefits of rewarding improved performance, a seasoned instructor objected. “On many occasions I have praised flight cadets for clean execution of some aerobatic maneuver. The next time they try the same maneuver they usually do worse. On the other hand, I have often screamed into a cadet’s earphone for bad execution, and in general he does better on his next try. So please don’t tell us that reward works and punishment does not, because the opposite is the case.”

The instructor’s observation was astute and correct: praise was often followed by deterioration, punishment by improvement. But the sequence did not establish his causal conclusion that punishment was more effective. What he had witnessed was regression to the mean, the tendency for extreme measurements to be followed by measurements closer to the average, when the extremes partly reflect temporary variation rather than persistent differences. If a cadet’s unusually good performance partly reflected favorable circumstances, those circumstances need not repeat on the next attempt. Performance could then decline without any change in skill or instruction. Likewise, a cadet who did unusually poorly was likely suffering from a run of bad luck combined with his true skill level; his next performance, with luck allowed to average out, would tend to be better. Without a comparison, the instructor could not separate that fluctuation from any effect of his actions.

This insight into regression is a fundamental statistical principle first identified in the late nineteenth century by Francis Galton, a cousin of Charles Darwin. Galton was studying the inheritance of human traits, and height proved far easier to quantify than “genius.” He noticed that exceptional parents—the very tall or the very short—had children who were, on average, closer to the population mean than their parents were. In his 1889 book Natural Inheritance he wrote: “It is theoretically a necessary fact … that the Stature of the adult offspring must on the whole, be more mediocre than the stature of their Parents.” He initially called this “regression towards mediocrity,” a term that evolved into regression to the mean.

Galton’s mathematical understanding made all the difference. He saw that height is determined by a combination of inborn heredity and innumerable external forces—nutrition, health, pure chance. Someone who is six foot one is tall partly because of height-promoting genes and partly because external factors happened to push upward as well. The very tallest people are those for whom both the genetic and the chance components are favorable. Their children will inherit some of those genes, but there is no reason the same random good fortune will strike again. Thus the children of remarkably tall parents are, on average, still taller than the typical person, but not as tall as their parents. The same holds in reverse for the very short. In this example, incomplete inheritance and variation produce the pattern. More generally, regression occurs across imperfectly correlated measurements; it does not require biological inheritance.

Precisely the same logic explains why businesses, athletes, artists, and medical patients seem to slide back toward average over time. In the 1920s, the business theorist Horace Secrist observed that firms with the highest profits in one period tended to show less extreme profits ten years later. He wrote an entire book about the “triumph of mediocrity” in business competition, without adequately accounting for regression. Some firms’ initial profits may have included unusually favorable temporary conditions. Later results could be less extreme even if their management remained strong. That possibility needs to be distinguished from actual changes in competition or business quality.

The same phenomenon pervades everyday life. A person decides to diet precisely when their weight has drifted to the top of its usual range—those are the moments that prompt the thought, “I’ve got to do something.” Even if the diet has no real effect, a subsequent weigh-in is likely to show a loss simply because the person is regressing toward their personal average. The classic mistake is to credit the diet rather than the statistical tendency. More rigorous diet studies avoid this pitfall not by looking only at heavy participants, but by randomly assigning two groups to different diets and comparing them; randomization makes the groups comparable in expectation, so regression alone should not systematically favor one group.

In criminal justice, the “Scared Straight” program took juvenile offenders on tours of prisons where inmates terrified them about life inside. Early results appeared dramatic: arrests dropped dramatically after participation. The participants were selected because they had already been in trouble. If unusually bad recent behavior partly reflected temporary influences, some improvement could occur without the program. When proper randomized controlled trials were finally conducted, Scared Straight was found to slightly increase antisocial behavior. The before-and-after comparison could not establish the program’s effect.

Medical researchers have fallen into the same trap. A 1976 study celebrated the benefits of bran for regulating “oral-anal transit time,” reporting that patients with abnormally fast transit slowed down toward a 48-hour norm, while those with slow transit sped up. But all of us have fast days and slow days, whatever our underlying digestive health. An unusually quick Monday is likely to be followed by a more typical Tuesday, bran or no bran. Without an appropriate comparison, the reported pattern cannot distinguish a treatment effect from regression.

Jeremy Siegel’s discussion of return to the mean concerns a separate claim about financial returns over different horizons. Averaging many observations can reduce variability without implying that poor returns must be followed by compensating gains. Likewise, a seasonal business’s bad season may reflect temporary conditions, persistent problems, or a predictable cycle. Regression needs to be assessed against the relevant process. In basketball, a scoring streak can arise by chance, but observing that possibility does not establish that every “hot hand” is an illusion. A fixed success probability is an assumption to test, not a consequence of regression.

Galton himself extended the insight beyond inheritance. In an 1890 critique of Alphonse Bertillon’s system for identifying criminals through multiple body measurements, he pointed out that the lengths of arms, legs, and other body parts are correlated. Adding a highly redundant measurement may yield little additional identification information—just as, when predicting an adult child’s height, knowing one parent’s height gives a great deal of information, but knowing the other parent’s height adds less, because the two are themselves correlated. This concerns the information added by a measurement. Regression concerns the expected value of one imperfectly correlated measurement conditional on another; the ideas are related but distinct.

The temptation to invent causal explanations for what is really just regression is powerful. The biometrician Walter F. R. Weldon noted in 1905 that many biologists thought regression was a “peculiar property of living things” that kept species true to type. But if regression of children on parents is taken as a biological law, then one must also confront the equally obvious fact that there is a regression of fathers on children—the fathers of abnormally tall children are, on average, less abnormal than their children. No one would argue that children exert a biological force that pulls their already-grown fathers toward mediocrity. The phenomenon is not biological, nor competitive, nor creative exhaustion; it is mathematical.

Regression is a tendency in conditional averages, not a guarantee about the next result. Durable skill, changing conditions, and correlated temporary influences can all matter. Before crediting an intervention, ask why the initial cases were selected and what would likely have happened without it. If praise follows unusually good results and punishment follows unusually poor ones, regression can make praise look ineffective and punishment look successful even when neither caused the change.