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This question is taken from Freedman

In a certain class, midterm scores average out to $60$ with an SD of $15,$ as do scores on the final. The correlation between midterm scores and final scores is about $0.50.$

Estimate the average final score for the students whose midterm scores were

(a) 75 (b) 30 (c) 60

The estimated average score of student in final who scored 30 in mid term is

\begin{align} & \text{average} - r\cdot 2\text{sd} \\[8pt] = {} & 60 - 0.5\cdot2\cdot15 \\[8pt] = {} & 45 \end{align}

But the estimated average score in mid terms who scored 45 in final is not 30

\begin{align} & \text{average} - r\cdot1\cdot\text{sd} \\[8pt] = & 60-0.5\cdot1\cdot15 \\[8pt] = {} & 22.5 \end{align}

This is bizzare. Am I doing the calculation wrong? If not, what explains this strange result?

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That is to be expected, but many people have considered it paradoxical.

Here's one way to look at it. Some people who score $2$ SDs below the mean on the midterm are really that weak, and some are having a bad day and are not really that weak. And some who score $2$ SDs above the mean on the midterm are that strong, but some are doing better than they usually do. And likewise on the final. But the ones who are doing worse than they usually do on the final exam are not the same one who are doing worse than usual on the midterm, and the ones who perform better than they usually do on the final exam are not the same ones who do better than they usually do on the midterm.

The average $y$-value for a given $x$-value is given by $y-60 = \dfrac 1 2 (x-60).$

The average $x$-value for a given $y$-value is given by $x-60 = \dfrac 1 2 (y-60).$

These are two different lines in the $(x,y)$-plane. If you expected them to be the same line, consider first what happens if the correlation is $0.$ Then the average $y$-value for a given $x$-value is given by $y=60$ and the average $x$-value for a given $y$-value is $x=60.$ Next consider what happens if the correlation is $0.01.$ The you get a line of slope slightly more than $0$ when you predict the average $y$-value for a given $x$-value and a line of large positive slope when you predict the average $x$ value for a given $y$-value. As the correlation grows, the two lines get closer to each other. When the correlation is $1,$ and only then, both lines are the same.

This is the phenomenon of "regessing to the mean", and that's actually why regression is called regression.

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