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The Pearson product-moment correlation coefficient is a measure of the linear relationship between two variables $X$ and $Y$, giving a value between +1 and −1.
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Derivation of the standard error for Pearson's correlation coefficient
I do not have the answer, but for me there is an error in the formula of the question.
It is: $$SE_r =\frac{1-r^2}{\sqrt{n-2}}$$ and
$$\newcommand{\Var}{\operatorname{Var}} \Var(r)=\frac{(1-r^2)^2}{ …