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    Method document

    Sample size to detect a Pearson correlation

    How many participants are needed to show that two continuous measurements are correlated, from the smallest correlation worth detecting. Covers dropout.

    Engine version 1.0.0 · Reviewed 1 October 2026

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    https://statwisely.com/methods/correlation-pearson/

    When to use it

    • Testing whether two continuous measurements are linearly associated.

    Not for

    • Measurements that are skewed or ordinal. Use Spearman’s correlation.
    • Estimating the correlation to a stated precision. Use the correlation under “Estimate precisely”.

    Notation

    nn
    Number of participants
    rr
    Correlation to detect
    z(r)z(r)
    Fisher’s transformation, ½ ln[(1 + r)/(1 − r)]
    α\alpha
    Significance level: the chance of a false positive
    1−β1-\beta
    Power: the chance of detecting the effect if it is real
    qq
    Expected dropout

    Formulas

    Fisher’s z approximation

    n=⌈(z1−α/2+z1−βz(r))2+3⌉n = \left\lceil \left(\frac{z_{1-\alpha/2} + z_{1-\beta}}{z(r)}\right)^{2} + 3 \right\rceil

    Fisher’s transformation makes the sampling distribution of a correlation approximately normal with variance 1/(n − 3) [1].

    Dropout

    nrecruit=⌈n1−q⌉n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

    Applied last, to the participants actually recruited.

    Parameters

    Expected correlationrr

    Values: Between −1 and 1, not 0

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

    Significance levelα\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.

    Power1−β1-\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 dropoutqqOptional

    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, and the two measurements are roughly bivariate normal.
    • The test is two-sided, of the correlation being zero.
    • Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

    Worked examples

    A moderate correlation

    r = 0.3, so z(r) = 0.3095; α = 0.05, 80% power: n = (2.8016 / 0.3095)² + 3 = 84.9, rounded up to 85.

    Sample size: 85

    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 Pearson correlation, the required sample size is 85 participants. Calculated with StatWisely (statwisely.com) using: n ≈ [(z_(1-α/tails)+z_(1-β))/atanh(r)]²+3. These are planning estimates; confirm the final design with a qualified statistician.

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    A strong correlation

    r = 0.5: n = 30.

    Sample size: 30

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    Allowing for missing data

    Expecting 10% to have one measurement missing: 85 / 0.9 = 94.4, so 95 are recruited.

    Needed for analysis: 85 · To recruit: 95

    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 Pearson correlation, the required sample size is 85 participants. Allowing for 10% dropout, 95 participants should be recruited. Calculated with StatWisely (statwisely.com) using: n ≈ [(z_(1-α/tails)+z_(1-β))/atanh(r)]²+3. 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 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 Pearson correlation. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/correlation-pearson/

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

    References

    1. 1.Fisher RA. On the “probable error” of a coefficient of correlation deduced from a small sample. Metron. 1921;1:3–32.