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I have two variables, and I can calculate e.g. the Pearson correlation between them, but I would like to know something analogous to what a t-test would give me (i.e. some notion of how significant the correlation is).

Does such a thing exist?

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    $\begingroup$ Use the Fisher Transformation or a t-test based on a scaled tangent tranformation: en.wikipedia.org/wiki/… $\endgroup$ – whuber Jan 17 '11 at 19:11
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Yes, you can get a $p$-value for testing the null hypothesis that the Pearson correlation is zero. See http://en.wikipedia.org/wiki/Pearson%27s_correlation#Inference.

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    $\begingroup$ For anyone else interested: this can be done in R with cor.test() $\endgroup$ – Xodarap Jan 17 '11 at 21:58
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As an alternative, this beautiful paper (a bit technical/mathsy, so beware) allows you to test for any correlation (not just $\rho=0$) using a Non-informative Bayesian Decision Theoretic approach.

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