Pearson Correlation Coefficient Sample Size Calculator

Methodology

This calculator sizes a one-sample Pearson correlation test under the assumption that the paired observations arise from a bivariate normal population. The null hypothesis is that the population correlation is zero, while the alternative hypothesis specifies a nonzero correlation of the magnitude entered above.

Fisher z transformation

Because the sampling distribution of the Pearson correlation is bounded between −1 and 1 and is not well approximated by a normal distribution on the raw correlation scale, the anticipated correlation is transformed using Fisher's z transformation:

For a test of H₀: ρ = 0, the standard large-sample planning equation is:

For a two-sided test, α* = α. For a one-sided test, the corresponding critical normal quantile uses α rather than α/2. The result is rounded upward because a fractional participant is not possible.

Interpretation

The result is the number of complete, analyzable pairs of observations required. If loss to follow-up, missing measurements, or other exclusions are expected, the enrollment target should be increased accordingly.

Worked validation example

A two-sided test with ρ = 0.40, α = 0.05, and 80% power gives z(ρ) = 0.42365. Substitution into the Fisher z sample-size equation gives a required sample size of 47 paired observations after rounding up. This agrees with the published nQuery example in which a two-sided Fisher z test with 5% significance was designed to detect a Pearson correlation of 0.40 with 80% power using n = 47.

References

• Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10, 507–521.
• Fisher, R. A. (1921). On the “probable error” of a coefficient of correlation deduced from a small sample. Metron, 1, 3–32.
• William C. Guenther (1977). Desk Calculation of Probabilities for the Distribution of the Sample Correlation Coefficient. The American Statistician, 31(1), 45–48.
• NCSS/PASS. Pearson's Correlation Tests, PASS Sample Size Software. PASS documents sample-size and power calculations for testing a population correlation and provides published validation examples.
• nQuery sample-size methodology: published applications document the two-sided Fisher z test for a Pearson correlation, including an example with ρ = 0.40, α = 0.05, 80% power, and n = 47.