Sample size to estimate a correlation

When to use it

Estimating how strongly two continuous measurements are associated, such as agreement between two instruments or a validation study.

Not for

Showing that a correlation differs from zero. Use the powered correlation test under “Detect an effect”.
Agreement between two methods measuring the same quantity, where limits of agreement are more informative than a correlation.

Notation

$n$: Sample size under simple random sampling
$r$: Expected correlation
$E$: Margin of error on the correlation scale
$z(x)$: Fisher’s transformation, ½ ln[(1 + x)/(1 − x)]
$E_z$: The margin carried onto Fisher’s scale
$z_{1-\alpha/2}$: Standard normal quantile for confidence 1 − α (1.960 for 95%)
$N$: Population size
$\mathrm{DE}$: Design effect
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Usable observations per cluster
$k$: Number of clusters
$\mathrm{CV}$: Coefficient of variation of cluster size
$q$: Non-response rate

Formulas

Simple random sampling

E_z = \min\{\,z(r+E) - z(r),\; z(r) - z(r-E)\,\}, \qquad n = \left\lceil \left(\frac{z_{1-\alpha/2}}{E_z}\right)^{2} + 3 \right\rceil

A correlation’s interval is symmetric on Fisher’s z scale, where its standard error is 1/√(n − 3), not on the correlation scale [1, 2]. The margin is carried to that scale on each side and the tighter side is used, so the interval meets the margin on both sides.

Finite population correction

n' = \left\lceil \frac{n}{1 + (n - 1)/N} \right\rceil

Applied when a population size is given [3]. It matters only when the sample is a sizeable share of the population, and is not offered with cluster sampling.

Design effect for a correlation

\mathrm{DE} = 1 + (m - 1)\,\rho^{2}

For a correlation the ICC enters squared, under a bivariate random-intercept model in which both variables share the same ICC and their within- and between-cluster correlations are equal. If those conditions do not fit, use a model-specific calculation. A design effect can also be entered directly.

Unequal cluster sizes

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho^{2}, \qquad \mathrm{CV} = \frac{\operatorname{SD}(s_j)}{\bar{s}}

When each cluster is given its own share s_j of the sample, the CV of those shares enlarges the design effect [5]. With a design effect entered directly the shares only divide the sample, since a measured design effect already reflects its survey’s cluster sizes.

Number of clusters for a chosen cluster size

k = \max\!\left(2,\ \left\lceil \frac{n \cdot \mathrm{DE}}{m} \right\rceil\right), \qquad n_{\text{clustered}} = k\,m

The requirement is rounded up to whole clusters, so the total can exceed n × DE slightly. At least two clusters are needed to estimate between-cluster variation.

Cluster size for a fixed number of clusters

m = \left\lceil \frac{n\,(1-\rho^{2})}{k - n\,\rho^{2}} \right\rceil, \qquad k > n\,\rho^{2}

Solving k·m = n·DE for m. Below that number of clusters no cluster size reaches the margin, because each extra person per cluster adds less and less information; the calculator reports the minimum number of clusters instead.

Few clusters

n \;\leftarrow\; n \left(\frac{t_{k-1,\,1-\alpha/2}}{z_{1-\alpha/2}}\right)^{2}

Optional. With few clusters the interval is built on k − 1 degrees of freedom, so the normal critical value is replaced by t. Because k depends on n, this is iterated until it settles.

Non-response

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

The number to approach so that n respond. Applied last, after any clustering.

Parameters

Confidence level $1-\alpha$

Values: 90%, 95% or 99%

95% is the convention in health research. A higher level widens the interval, so the same margin needs a larger sample.

Margin of error $E$

Values: Above 0 and below 1, on the correlation scale

Half the width of the interval you can accept, for example ±0.1. The expected correlation plus or minus the margin must stay between −1 and 1 on at least one side.

Expected correlation $r$

Values: Between −1 and 1

From comparable studies. Precision depends on it: stronger correlations are estimated more precisely, so assuming too strong a value undersizes the study.

Population size $N$ Optional

Values: A whole number of at least 2

Leave empty for a large population. It matters only when the sample would be more than about 5% of the population. Not available with cluster sampling.

Expected non-response rate $q$ Optional

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

The share of people approached who will not take part; use the response rate of a comparable study. Approaching more people restores the number of respondents, not their representativeness: if non-responders differ, the estimate can still be biased.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

The share of the outcome’s variation that lies between clusters. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [7], but depend on the outcome and the kind of cluster; prefer an estimate from a comparable study.

Design effect $\mathrm{DE}$ Optional

Values: Above 0; usually between 1 and 3

An alternative to the ICC, when a previous study of the same design reports its design effect.

Observations per cluster $m$ Optional

Values: A whole number of at least 1

Usable responses per cluster. Larger clusters are cheaper to reach, but when the ICC is above zero each additional person in a cluster adds less information.

Number of clusters $k$ Optional

Values: A whole number of at least 2

Instead of a cluster size, when the clusters are fixed: the emirates of a country, the schools of a district. The calculator then finds the cluster size, or reports how many clusters the margin needs.

Share of the sample for each cluster $s_j$ Optional

Values: One positive value per cluster, on any scale

When clusters will contribute unequally, for example in proportion to their population. With an ICC, unequal shares enlarge the design effect [5].

Correction for few clusters $t_{k-1}$ Optional

Values: On or off

Uses t on k − 1 degrees of freedom instead of the normal distribution. Worth applying when there are few clusters, roughly fewer than 30.

Assumptions

Participants, or clusters, are selected at random, and respond independently apart from the clustering specified.
The two measurements are roughly bivariate normal and the Pearson correlation is reported. Spearman and Kendall correlations need somewhat larger samples [2].
With clustering, both measurements share the same ICC, and their correlation is the same within and between clusters.
With cluster sampling: one ICC applies to every cluster; clusters are of equal size unless shares are given; the sampled clusters are a small fraction of all clusters; and the analysis accounts for the clustering, for example with survey-design standard errors [4, 6].
Non-response is unrelated to the outcome.

Worked examples

A moderate correlation

Expected r = 0.3, margin ±0.1: upward z(0.4) − z(0.3) = 0.1141, downward z(0.3) − z(0.2) = 0.1068; the tighter is 0.1068, so n = (1.96 / 0.1068)² + 3 = 339.9, rounded up to 340.

Sample size: 340

Protocol statement
To estimate a correlation coefficient with 95% confidence and a margin of error of ±0.1, assuming an expected correlation of 0.3, the required sample size is 340 participants. Calculated with StatWisely (statwisely.com) using: n = (z_(1-α/2) / E_z)² + 3, where E_z is E on Fisher's z scale. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

A stronger correlation

Expected r = 0.6, same margin: the tighter side is z(0.6) − z(0.5) = 0.1438, so n = (1.96 / 0.1438)² + 3 = 188.7, rounded up to 189. Stronger correlations need fewer participants for the same margin.

Sample size: 189

Open this example in the calculator

Cluster sampling with an ICC

ICC 0.1 and 20 per cluster: the ICC enters squared, so DE = 1 + 19 × 0.1² = 1.19. Then 340 × 1.19 = 404.6, so 21 clusters and 420 participants.

Sample size: 420

Protocol statement
To estimate a correlation coefficient with 95% confidence and a margin of error of ±0.1, assuming an expected correlation of 0.3, 340 participants would be required under simple random sampling. Allowing for a design effect of 1.19 (ICC 0.1, 20 per cluster) for cluster sampling, the required sample size is 420 participants, in 21 clusters of 20. This assumes clusters of equal size, with a negligible sampling fraction. Calculated with StatWisely (statwisely.com) using: n = (z_(1-α/2) / E_z)² + 3, where E_z is E on Fisher's z scale; clustered correlation: n x DE, DE = 1 + (m-1)ICC². These are planning estimates; confirm the final design with a qualified statistician.

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 procedure, with and without cluster sampling, every night and before each production release, and the results are compared with the engine’s.
The basic result agrees with the closed form: (1.96 / 0.1068)² + 3 = 339.9, rounded up to 340. In R, ez <- min(atanh(0.4) - atanh(0.3), atanh(0.3) - atanh(0.2)); ceiling((qnorm(0.975) / ez)^2 + 3) returns 340.

How to cite

StatWisely. Sample size to estimate a correlation. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/precision-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.
3.Cochran WG. Sampling Techniques. 3rd ed. New York: John Wiley & Sons; 1977.
4.Kish L. Survey Sampling. New York: John Wiley & Sons; 1965.
5.Eldridge SM, Ashby D, Kerry S. Sample size for cluster randomized trials: effect of coefficient of variation of cluster size and analysis method. International Journal of Epidemiology. 2006;35(5):1292–1300. doi:10.1093/ije/dyl129
6.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
7.Adams G, Gulliford MC, Ukoumunne OC, Eldridge S, Chinn S, Campbell MJ. Patterns of intra-cluster correlation from primary care research to inform study design and analysis. Journal of Clinical Epidemiology. 2004;57(8):785–794. doi:10.1016/j.jclinepi.2003.12.013

When to use it

Not for

Notation

Formulas

Simple random sampling

Finite population correction

Design effect for a correlation

Unequal cluster sizes

Number of clusters for a chosen cluster size

Cluster size for a fixed number of clusters

Few clusters

Non-response

Parameters

Confidence level1−α1-\alpha1−α

Margin of errorEEE

Expected correlationrrr

Population sizeNNNOptional

Expected non-response rateqqqOptional

Intracluster correlation (ICC)ρ\rhoρOptional

Design effectDE\mathrm{DE}DEOptional

Observations per clustermmmOptional

Number of clusterskkkOptional

Share of the sample for each clustersjs_jsj​Optional

Correction for few clusterstk−1t_{k-1}tk−1​Optional

Assumptions

Worked examples

A moderate correlation

A stronger correlation

Cluster sampling with an ICC

Validation

How to cite

References

Confidence level $1-\alpha$

Margin of error $E$

Expected correlation $r$

Population size $N$ Optional

Expected non-response rate $q$ Optional

Intracluster correlation (ICC) $\rho$ Optional

Design effect $\mathrm{DE}$ Optional

Observations per cluster $m$ Optional

Number of clusters $k$ Optional

Share of the sample for each cluster $s_j$ Optional

Correction for few clusters $t_{k-1}$ Optional