Sample size to compare two independent correlations

When to use it

Testing whether an association is stronger in one population than another, such as in women and men.

Not for

Two correlations measured on the same people, which are dependent and need a different test.

Notation

$n$: Participants per group
$r_1,\ r_2$: Expected correlations in groups 1 and 2
$z(r)$: Fisher’s transformation
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$q$: Expected dropout

Formulas

Difference of Fisher z values

n = \left\lceil 2\left(\frac{z_{1-\alpha/2} + z_{1-\beta}}{z(r_1) - z(r_2)}\right)^{2} + 3 \right\rceil

Each group’s transformed correlation has variance 1/(n − 3), so their difference has twice that [1, 2].

Dropout

n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

Applied last, to the participants actually recruited.

Parameters

Correlation in group 1 $r$

Values: Between −1 and 1

The smallest correlation worth detecting, from comparable studies. Weak correlations need large samples.

Correlation in group 2 $r_2$

Values: Between −1 and 1, different from r₁

Differences between correlations are hard to detect; expect large samples.

Significance level $\alpha$

Values: Between 0 and 1; usually 0.05

The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

Power $1-\beta$

Values: Between 0 and 1; usually 0.80 or 0.90

The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

Expected dropout $q$ Optional

Values: 0% up to, but not including, 100%

The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

Assumptions

The groups are independent and equal in size; the measurements are roughly bivariate normal in each.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

0.5 against 0.3

r₁ = 0.5 and r₂ = 0.3: z(0.5) − z(0.3) = 0.240, so n = 2 × (2.8016 / 0.240)² + 3 = 276.0, rounded up to 277 per group.

Group 1: 277 · Group 2: 277 · Total: 554

Protocol statement
To detect a difference between correlations of 0.5 and 0.3 with 80% power at a two-sided significance level of 0.05, using a comparison of two independent correlations, the required sample size is 277 per group (554 in total). Calculated with StatWisely (statwisely.com) using: n_per_group = 2[(z_(1-α/tails)+z_(1-β))/(atanh(r₁)-atanh(r₂))]²+3. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Closer correlations

r₁ = 0.4 against 0.3: 1209 per group.

Group 1: 1209 · Group 2: 1209 · Total: 2418

Open this example in the calculator

Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.

How to cite

StatWisely. Sample size to compare two independent correlations. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/correlation-compare/

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.Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.