In a mathematical context, regression to the mean is the statement that an extreme event is likely to be followed by a less extreme event.
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Regression and correlation were major breakthroughs in scientific thought. For Isaac Newton and his peers, the universe obeyed deterministic laws of cause and effect. Everything that happened had a reason. Yet not all science is so reductive. In biology, for example, certain outcomes—such as the occurrence of lung cancer—can have multiple causes that mix together in a complicated way. Correlation provided a way to analyse the fuzzy relationships between linked sets of data. For example, not everyone who smokes will develop lung cancer, but by looking at the incidence of smoking and the incidence of lung cancer mathematicians can work out your chances of getting cancer if you do smoke. Likewise, not every child from a big class in school will perform less well than a child from a small class, yet class sizes do have an impact on exam results. Statistical analysis opened up whole new areas of research—in subjects from medicine to sociology, from psychology to economics. It allowed us to make use of information without knowing exact causes. Galton’s original insights helped make statistics a respectable field: ‘Some people hate the very name of statistics, but I find them full of beauty and interest,’ he wrote. ‘Whenever they are not brutalized, but delicately handled by the higher methods, and are warily interpreted, their power of dealing with complicated phenomena is extraordinary.**
In 2002 the Nobel Prize in Economics was not won by an economist. It was won by the psychologist Daniel Kahneman, who had spent his career (much of it together with his colleague Amos Tversky) studying the cognitive factors behind decision-making. Kahneman has said that understanding regression to the mean led to his most satisfying ‘Eureka moment’. It was in the mid 1960s and Kahneman was giving a lecture to Israeli air-force flight instructors. He was telling them that praise is more effective than punishment for making cadets learn. On finishing his speech, one of the most experienced instructors stood up and told Kahneman that he was mistaken. The man said: ‘On many occasions I have praised flight cadets for clean execution of some aerobatic manœuvre, and in general when they try it again, they do worse. On the other hand, I have often screamed at cadets for bad execution, and in general they do better the next time. So please don’t tell us that reinforcement works and punishment does not, because the opposite is the case.’ At that moment, Kahneman said, the penny dropped. The flight instructor’s opinion that punishment is more effective than reward was based on a lack of understanding of regression to the mean. If a cadet does an extremely bad manœuvre, then of course he will do better next time—irrespective of whether the instructor admonishes or praises him. Likewise, if he does an extremely good one, he will probably follow that with something less good. ‘Because we tend to reward others when they do well and punish them when they do badly, and because there is regression to the mean, it is part of the human condition that we are statistically punished for rewarding others and rewarded for punishing them,’ Kahneman said.
Regression to the mean is not a complicated idea. All it says is that if the outcome of an event is determined at least in part by random factors, then an extreme event will probably be followed by one that is less extreme. Yet despite its simplicity, regression is not appreciated by most people. I would say, in fact, that regression is one of the least grasped but most useful mathematical concepts you need for a rational understanding of the world. A surprisingly large number of simple misconceptions about science and statistics boil down to a failure to take regression to the mean into account.
Take the example of speed cameras. If several accidents happen on the same stretch of road, this could be because there is one cause—for example, a gang of teenage pranksters have tied a wire across the road. Arrest the teenagers and the accidents will stop. Or there could be many random contributing factors—a mixture of adverse weather conditions, the shape of the road, the victory of the local football team or the decision of a local resident to walk his dog. Accidents are equivalent to an extreme event. And after an extreme event, the likelihood is of less extreme events occurring: the random factors will combine in such a way as to result in fewer accidents. Often speed cameras are installed at spots where there have been one or more serious accidents. Their purpose is to make drivers go more slowly so as to reduce the number of crashes. Yes, the number of accidents tends to be reduced after speed cameras have been introduced, but this might have very little to do with the speed camera. Because of regression to the mean, whether or not one is installed, after a run of accidents it is already likely that there will be fewer accidents at that spot. (This is not an argument against speed cameras, since they may indeed be effective. Rather it is an argument about the argument for speed cameras, which often displays a misuse of statistics.)
