I am having my mind turned upside down with a problem I am dealing with. So imagine we have a situation where we have pairs of points where x=heights of fathers and y=heights of the sons of these fathers. The mean of $X$ is equal to the mean $Y$ and $SD(Y) = SD(X)$, and $0<R<1$. Now I am supposed to show that the expected height of a son whose father is shorter than avg. is also less than average, but by a smaller degree. First off I need help understanding the "regression effect" i.e. "Regression Towards the Mean." If the standard deviations are the same why is this happening? How am I supposed to show this? Any help is appreciated. Thanks guys.
2026-04-03 10:05:00.1775210700
Linear Regression Problem ("Regression Towards the Mean")
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Note: I am not going to do any calculations in my answer. I shall just try to explain the term: regressing towards the mean.
As you said- for this data set it means that shorter fathers have taller sons (loosely), and taller fathers tend to have shorter sons.
Note that if such an effect didn't happen then on an average successive generations would get taller and taller, or shorter and shorter. This doesn't happen in nature- hence the observed effect. This is what is meant by regressing towards the mean.
And, why is this happening mathematically? How can you show it? Hint is to use the various relations you know about $SD(X)$, $SD(Y)$ and $R$. Since $0<R<1$, we know there is positive correlation- which isn't perfect. Hence the expected values of the means will differ, but by a factor depending upon $R$. Play around with the relations, and you will see excatly what :).
I admit this is a very loose answer- and probably should just be a comment, except that it's too long to be a comment.