Achieved power for a Pearson correlation

When to use it

Assess the sensitivity of a planned test of one Pearson correlation against zero.

Not for

This method does not compare two correlations or establish a causal relationship.

Notation

$n$: Participants.
$r$: Expected population correlation.
$r_c$: Critical absolute sample correlation under the null.
$z(r)$: Fisher transformation, atanh(r).

Formulas

Null critical correlation

t_c=t_{1-\alpha/\text{tails},n-2}, \qquad r_c=\sqrt{\frac{t_c^2}{t_c^2+n-2}}

The critical t value comes from the exact null distribution of the sample correlation test.

Power on the Fisher scale

m=\operatorname{atanh}(r)+\frac{r}{2(n-1)}, \quad q=\sqrt{n-3}, \quad \text{Power}=\Phi(q[m-z(r_c)])+I_{\text{tails}=2}\Phi(q[-m-z(r_c)])

The alternative is approximated on the Fisher scale with a small-sample bias adjustment. At r = 0, the exact null rejection probability is α [1, 2].

Parameters

Sample size $n$

Values: Integer, at least 2

Enter participants available for analysis.

Expected correlation $r$

Values: Between −1 and 1

Enter the smallest correlation of scientific interest; zero gives power equal to α.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Test direction $s$

Values: 1 or 2

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Assumptions

The null critical value is exact for the Pearson test; power under a nonzero correlation uses a Fisher-scale approximation.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

For 85 participants, an expected correlation of 0.30, α = 0.05, and a two-sided test, use the exact null critical correlation and Fisher-scale alternative approximation.

Sample size: 85

Open this example in the calculator

Validation

The method tests one correlation against zero. It does not use a noncentral-t distribution under the alternative.

How to cite

StatWisely. Achieved power for a Pearson correlation. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-correlation/

Cite the original references for the formulas your design uses as well.

References

1.Fisher RA. On the “probable error” of a coefficient of correlation deduced from a small sample. Metron. 1921;1:3–32.
2.Bonett DG, Wright TA. Sample size requirements for estimating Pearson, Kendall and Spearman correlations. Psychometrika. 2000;65(1):23–28.