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I have frequently seen that the Pearson correlation coefficient "describes the direction and strength of a relationship between two variables".

What is actually meant by "strength"?

The first thought that comes to mind is that "strength" refers to how certain we are about the value of one variable, given that we know the value of the other variable. I am looking for a more formal statistical statement.

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    $\begingroup$ I really like that phrasing, perhaps refined as, “If we know the value of one marginal variable, we can better pin down the value of the other.” Taken to the extreme with a correlation of $\pm 1$, if we know the value of one marginal variable, we know exactly what the other is. And if the correlation is zero, under certain circumstances, we don’t know anything about one variable by knowing the other, except for the marginal distributions. $\endgroup$
    – Dave
    Commented Nov 25, 2021 at 3:08

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I really like that phrasing, perhaps refined as, “If we know the value of one marginal variable, we can better pin down the value of the other.”

Taken to the extreme with a correlation of $\pm 1$, if we know the value of one marginal variable, we know exactly what the other is. And if the correlation is zero, under certain circumstances (first to come to mind is joint normality), we don’t know anything about one variable by knowing the other, except for the marginal distributions.

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