Maths in a minute: Regression to the mean

Usain Bolt celebrates his victory over 100m and new world record at the Beijing Olympics. Image: Jmex60.

Sometimes you just can't argue with the evidence. If a large sample of very ill people got better after dancing naked at full moon, then surely the dance works. Less contentiously, if the country's best-performing schools produce worse results over time, then surely something is wrong with the education system.

But hang on a second. Before you jump to conclusions, you need to rule out a statistical phenomenon called regression to the mean. The idea is that if you choose a set of measurements because they are quite extreme, and then do the same measurements a little while later, the result is likely to be less extreme.

Think of Usain Bolt. If you measured his performance over 100m the day after he ran a world record, the time you'd get would probably not be another world record, but something slower. This is because his world record performance on the previous day was not entirely down to his physical ability but also to all sorts of other factors - his mood, the condition of the track, the passion of the crowd - which to all intents and purposes are random. On his next run some or all of these factors are probably absent, so his performance will be closer to his personal average, or mean.

Similarly, if you select a group of very ill people to test a drug (or dance) on, then on your next measurement they are likely to feel better simply because their ill-being has regressed to the mean (never mind the placebo effect). You can't automatically assume it was because of the drug. And if you select a group of schools because of their outstanding performance, you're likely to see worse results next time around. You can't necessarily blame the government.

Regression to the mean was first noticed by a cousin of Charles Darwin, Sir Francis Galton, in the 19th century. You can read about him and his discovery in the maths magazine The Commutator. And for an example of how measurements of school performance can potentially give misleading results, read this article by Daniel Read of Warwick Business School.