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I have searched a lot, and I can only find tables that show critical values up to n=30. Can someone provide, or point me to, a simple method of estimating this value for different $\alpha$?

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For values over thirty the approximation (for a two-tailed test) is $$\frac{\Phi^{-1}\left(1-\tfrac{\alpha}{2}\right)}{\sqrt{n-1}}$$ so for example with $\alpha = 0.05$ and $n=100$ the numerator is about 1.96 and the denominator about 9.95, giving a critical value of about 0.197.

This comes from $\rho$ having approximately a normal distribution for large $n$, with mean $0$ and variance $1/(n − 1)$, assuming independence of the observations.

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See Wikipedia: Spearman's rank correlation coefficient#Determining significance:

"One can test for significance using $$t = r \sqrt{\frac{n-2}{1-r^2}},$$ which is distributed approximately as Student's $t$ distribution with $n − 2$ degrees of freedom under the null hypothesis."

Here $r$ is the sample estimate of Spearman's rank correlation coefficient. The reason critical values often aren't tabulated for $n > 30$ is that this approximation gets better as $n$ gets larger, and is very good for $n > 30$. The Stata statistical software package uses this formula to calculate $p$-values for all values of $n$.

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