Sample size to detect a Spearman correlation

When to use it

Skewed or ordinal measurements, or an association expected to be monotonic but not linear.

Not for

Roughly normal measurements, where Pearson’s correlation needs slightly fewer participants.

Notation

$n$: Number of participants
$\rho_s$: Spearman correlation to detect
$z(\rho_s)$: Fisher’s transformation of it
$\alpha$: 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

Fisher’s z with the rank-correlation variance

n = \left\lceil 1.06\left(\frac{z_{1-\alpha/2} + z_{1-\beta}}{z(\rho_s)}\right)^{2} + 3 \right\rceil

A rank correlation’s Fisher-transformed value has about 1.06 times the variance of Pearson’s, so it needs about 6% more participants [2, 1].

Dropout

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

Applied last, to the participants actually recruited.

Parameters

Expected Spearman correlation $\rho_s$

Values: Between −1 and 1, not 0

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

Participants are independent; the approximation is good for moderate correlations and becomes conservative near ±1.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

A moderate correlation

ρₛ = 0.3: n = 1.06 × 81.9 + 3 = 89.8, rounded up to 90, against 85 for Pearson.

Sample size: 90

Protocol statement
To detect a correlation of 0.3 with 80% power at a two-sided significance level of 0.05, using a test of a Spearman correlation, the required sample size is 90 participants. Calculated with StatWisely (statwisely.com) using: n ≈ 1.06[(z_(1-α/tails)+z_(1-β))/atanh(ρ)]²+3. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

A strong correlation

ρₛ = 0.5: n = 31.

Sample size: 31

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 detect a Spearman correlation. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/correlation-spearman/

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

References

1.Fieller EC, Hartley HO, Pearson ES. Tests for rank correlation coefficients. I. Biometrika. 1957;44(3–4):470–481.
2.Fisher RA. On the “probable error” of a coefficient of correlation deduced from a small sample. Metron. 1921;1:3–32.