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The SciPy implementation of Pearson's $r$ also gives a two-tailed $p$-value.

I understand that a $p$-value for a given correlation gives the probability of a correlation coefficient at least as big to be observed if the null hypothesis is true.

I find it hard to understand how this test can be two-tailed, however. What would be the meaning of a one-tailed $p$-value for non-correlation, then? Since $r$ is signed I think only a one-tailed p-value could satisfy the definition given above.

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    $\begingroup$ You may be able to answer the question yourself if you consider two of the possible null hypotheses that you might have when testing a correlation: 1) That r equals zero; 2) That r is not greater than 0. $\endgroup$
    – Ian_Fin
    Aug 11, 2016 at 12:57

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Ok, so this is actually quite easy:

  • A 2-tailed p-value gives the probability of a correlation at least as extreme as r to be observed if the true correlation is in fact zero.
  • A 1-tailed p-value gives the probability of a correlation at least as extreme as r to be observed if the true correlation is in fact zero, or of the opposite sign than r.
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