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I would like to establish the proof for the standard error of a correlation coefficient. Assume that we have two iid samples $\{X_i\}_{i=1}^N$ and $\{Y_i\}_{i=1}^N$ We know that the sample correlation is given by

$$\text{SCor(X,Y)}=\frac{\sum_i (X_i - \bar X ) ( Y_i - \bar Y) }{\sqrt{\sum_i (X_i - \bar X )^2 \times \sum_i (Y_i - \bar Y )^2}} $$

I would like to avoid distributional assumptions until they are absolutely required. I am struggling to disentangle the products in the expectation to find the first and second moments of the sample correlation. Ultimately of course I am searching for the square root of the variance of the sample correlation.

This post is based on the definition found here and I would like to contribute to that page if possible because this is an issue too rarely addressed. Please excuse me if this is too elementary.

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OP wrote: We know that the sample correlation ... is an unbiased estimator ... of Cor(X,Y) ................... Really? How do 'we' know this??? I know it not. –  wolfies Jul 9 '13 at 16:15
    
thank you for pointing this out, I was under the false impression that it was. I updated the post to reflect this. –  7zf Jul 9 '13 at 16:30
    
So, you have scor = numerator/Sqrt[denominator]. The numerator is a symmetric polynomial in power sums, so we can find the moments of the numerator; the same is true for denominator. However, Sqrt[denominator] is not a symmetric polynomial in power sums, so I don't believe you will be able to find a general solution for that. And the ratio is not a symmetric polynomial either ... so I don't believe you will be able to find a general solution for that either. –  wolfies Jul 9 '13 at 16:57
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