Chi Sox Nut
New member
- Joined:
- Dec 2, 2010
- Posts:
- 78
- Liked Posts:
- 16
- Location:
- Phoenix, AZ
In statistics, regression toward 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.[1][2][3] 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.[4] 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.
Mathematically:
Let X1, X2 be random variables with identical marginal distributions with mean μ. In this formalization, the bivariate distribution of X1 and X2 is said to exhibit reversion toward the mean if, for every number c, we have:
μ ≤ E[X2 | X1 > c] < E[X1 | X1 > c], and
μ ≥ E[X2 | X1 < c] > E[X1 | X1 < c]
This definition is "general" in the sense that every bivariate distribution with identical marginal distributions exhibits reversion toward the mean.
So?
