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I know that the Pearson's correlation co-efficient the Pearson's correlation co-efficient $R$ is defined as: $$R=\frac{\mathrm{Cov}(X,Y)}{S_X S_Y}$$ and that $-1 \leq R \leq +1$ as a result of the Schwarz inequality, but I can get no intuition whatsoever as to why $R^2$ is a good statistic representing the noisy nature of the linear dependence of $X$ and $Y$. Is there a more rigorous derivation which defines it from variance of the error signal?

I mean if I take a dataset $(X,Y)$ and fit a line $\hat{Y}= aX +b$ using OLS to it. Shouldn't the variance of the random variable $Z=Y-\hat{Y}$ itself be the best estimate as to the goodness of the fit/"noisy nature of linear dependence", why does $R^2$ come into the picture here?

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Can't comment so forgive my brevity but R^2 is represented as 1 - SSE (sum of squared errors) where the residual is defined by equation you used to represent Z. With regards to the variance of the residuals the assumption for OLS is simply constant variance and mean zero.

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