Regression toward the mean

Quincunx (Galton Box) - Galton 1889 diagram

Regression toward the mean is a statistical phenomenon that occurs when a variable that is extreme on its first measurement tends to be closer to the average on its second measurement. This concept is crucial in the fields of statistics, psychology, medicine, and economics.

Overview

The concept of regression toward the mean was first identified by Francis Galton in the 19th century. Galton observed that the heights of children of tall parents tended to be closer to the average height than their parents' heights. This phenomenon can be explained by the fact that extreme values are often influenced by a combination of factors, some of which are random and not likely to be repeated.

Mathematical Explanation

In statistical terms, regression toward the mean can be described using the regression equation: \[ Y = \alpha + \beta X + \epsilon \] where:

• \( Y \) is the dependent variable,
• \( X \) is the independent variable,
• \( \alpha \) is the intercept,
• \( \beta \) is the slope, and
• \( \epsilon \) is the error term.

When \( \beta \) is less than 1, the predicted value of \( Y \) will be closer to the mean of \( Y \) than the value of \( X \) is to the mean of \( X \).

Applications

Regression toward the mean has important implications in various fields:

Medicine

In clinical trials, patients with extreme symptoms often show improvement over time, partly due to regression toward the mean. This can complicate the interpretation of treatment effects.

Psychology

In psychological testing, individuals who score extremely high or low on a test are likely to score closer to the average on subsequent tests. This is important for understanding the reliability of psychological assessments.

Economics

In economics, regression toward the mean can explain why extreme economic performances (e.g., very high or very low growth rates) tend to be followed by more average performances.

Misinterpretations

Regression toward the mean is often misunderstood as a causal effect. It is important to recognize that it is a statistical artifact rather than a real change in the underlying variable.

Related Concepts

• Central limit theorem
• Law of large numbers
• Selection bias
• Random error

See Also

• Statistical regression
• Francis Galton
• Clinical trials
• Psychological testing
• Economic performance