Regression to the mean can be explained by a correlation coefficient with the absolute value smaller than 1.

University of Michigan - Ann Arbor

Regression towards the mean is a phenomenon that occurs when resampling a single random variable. 

The phenomenon arises when the first sample of the random variable appears extreme, while the second sample appears much closer to the mean. This result occurs when the first sample is extremely above and below the mean.

Regression to the Mean

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Father and son heights are an example of regression to the mean in this book. Above average height father will tend to have a son who is shorter, while still above average. 

Conversely, an above average height son will tend to have a father who is also above average, but shorter. 

While this seems like a contradiction, these are tendencies across an entire population that will always trend toward the mean height.

This reversability of prediction in regression to the mean is why there is no difference between cause and effect in this situaton.


The Book of Why: Regression to the Mean

Regression to the mean explained by correlation

While using a Galton board to present to inheritance, on the board on the left, Francis Galton noticed that the balls piled up in a normal distribution or a bell-shaped curve. In the board, in the middle, the A and B were first thought as different generations with the distribution widened each time through them. The board on the right had chutes to move the distribution closer together with his explanation for regression to the mean.

Francis Galton ended discovering a statistics law instead of a hereditary. Galton thought that regression to the mean was a casual process but eventually realized that statistics and causation were separate.

Origins of Regression to the Mean  

Regression to the mean acts as a severe threat to internal validity when participants are selected for a research study specifically because of their extreme initial scores. Statistically, individuals who achieve exceptionally high or low scores on a pretest are highly likely to score closer to the population average on a subsequent posttest simply due to natural, random variability. This makes it incredibly difficult to determine whether a shift in scores was genuinely caused by the experimental treatment or was merely the result of statistical regression.

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