Sample size for a Pearson correlation

How many participants to demonstrate an association between two continuous measures — a biomarker and a score, dose and response.

Method & formula

The correlation coefficient is transformed with Fisher's z, C = ½·ln((1+r)/(1−r)), which is approximately normal with variance 1/(n−3); the required sample is n = ((z_1−α/2 + z_1−β)/C)² + 3. With the defaults (r = 0.3, 80% power) that is 85 participants; a "small" correlation of 0.1 needs 783.

This sizes a test that the correlation is zero. If your real question is prediction or agreement, a correlation test is usually the wrong instrument — regression or Bland–Altman limits of agreement answer those.

Source: Fisher RA. On the 'probable error' of a coefficient of correlation deduced from a small sample. Metron 1921;1:3–32.

Inputs, explained

Expected correlation (r): The magnitude of the correlation you expect, between 0 and 1. Cohen's conventions: 0.1 small, 0.3 medium, 0.5 large — but a pilot estimate beats a convention.

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