Regression Toward The Mean

In statistics, regression toward the mean (also known as regression to the mean) is the phenomenon that if a variable is extreme on its first measurement, it will tend to be closer to the average on a second measurement, and—a fact that may superficially seem paradoxical—if it is extreme on a second measurement, will tend to have been closer to the average on the first measurement. To avoid making wrong inferences, the possibility of regression toward the mean must be considered when designing experiments and interpreting experimental, survey, and other empirical data in the physical, life, behavioral and social sciences.

The conditions under which regression toward the mean occurs depend on the way the term is mathematically defined. Sir Francis Galton first observed the phenomenon in the context of simple linear regression of data points. However, a less restrictive approach is possible. Regression towards the mean can be defined for any bivariate distribution with identical marginal distributions. Two such definitions exist. One definition accords closely with the common usage of the term “regression towards the mean”. Not all such bivariate distributions show regression towards the mean under this definition. However, all such bivariate distributions show regression towards the mean under the other definition.

Historically, what is now called regression toward the mean has also been called reversion to the mean and reversion to mediocrity.

In finance, the term Mean reversion has a different meaning. Jeremy Siegel uses it to describe a financial time series in which "returns can be very unstable in the short run but very stable in the long run." More quantitatively, it is one in which the standard deviation of average annual returns declines faster than the inverse of the holding period, implying that the process is not a random walk, but that periods of lower returns are systematically followed by compensating periods of higher returns.