How correlation power analysis works
This calculator uses Fisher's z transformation (Cohen, 1988) — the standard method behind most correlation power calculators, including R's pwr.r.test. The correlation r is transformed to zr = arctanh(r), which has an approximately normal sampling distribution with standard error 1/√(N − 3); power and sample size follow directly from the normal distribution rather than requiring simulation.
Cohen's (1988) benchmarks are r = .10 (small), .30 (medium), and .50 (large). Because correlations near ±1 have a compressed sampling distribution, the Fisher transformation is what makes the normal approximation accurate across the full range — a plain normal approximation on r itself would be noticeably off for larger correlations. This calculator assumes a single bivariate correlation test, not a multiple-correlation or partial-correlation design, which would need a different effect size and degrees of freedom.
Worked example
Detecting a medium correlation (r = .30) at two-tailed α = .05 with target power of .80 calls for 85 paired observations — the standard textbook benchmark.
Required N = 85, achieved power = 80.0%, Fisher z critical = 1.960.