Sample size for a one-way ANOVA

When to use it

Three or more independent groups compared on a continuous outcome, with the overall F test as the primary analysis.

Not for

Several treatments each compared with one control. Use the multi-arm comparison, which powers the comparisons that matter.
The same people measured several times. Use repeated-measures ANOVA.

Notation

$k$: Number of groups
$N$: Total number of participants
$f$: Cohen’s f: the SD of the group means divided by the within-group SD
$\lambda$: Noncentrality of the F distribution
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$r_b$: Correlation between the outcome and a baseline covariate
$\rho$: Intracluster correlation coefficient (ICC)
$m$: Average number of participants per cluster
$\mathrm{CV}$: Coefficient of variation of cluster size
$C$: Clusters per group
$\ell$: Expected loss of whole clusters
$q$: Expected dropout

Formulas

The smallest N with enough power

\Pr\!\left(F'_{k-1,\,N-k,\,\lambda} > F_{k-1,\,N-k,\,1-\alpha}\right) \ge 1-\beta, \qquad \lambda = f^{2} N

When the means differ, the F statistic follows a noncentral F distribution with k − 1 and N − k degrees of freedom; N is the smallest total whose power reaches the target, then divided equally and rounded up per group [1].

Cohen’s f from expected means

f = \frac{\sigma_m}{\sigma}, \qquad \sigma_m = \sqrt{\frac{1}{k}\sum_{i=1}^{k} (\mu_i - \bar{\mu})^{2}}

From the means you expect in each group and the common within-group SD. For two groups, f = d / 2.

Adjusting for a baseline covariate

n' = \left\lceil n\,(1 - r_b^{2}) \right\rceil + 1

Analysing with a baseline measurement as a covariate (ANCOVA) removes the share r_b² of the outcome’s variance the test would otherwise have to overcome. One participant per group is added back for the degree of freedom the covariate uses [3].

Clustering

\mathrm{DE} = 1 + \left[(\mathrm{CV}^{2} + 1)\,m - 1\right]\rho

When participants are recruited or randomised in clusters, observations within a cluster repeat information, and the sample is multiplied by the design effect. With equal clusters (CV = 0) this is 1 + (m − 1)ρ [4, 5].

Few clusters

F_t = \left(\frac{t_{2(C-1),\,1-\alpha/\tau} + t_{2(C-1),\,1-\beta}}{z_{1-\alpha/\tau} + z_{1-\beta}}\right)^{2}

Optional. A cluster trial is analysed on cluster-level summaries, so its critical values come from t on 2(C − 1) degrees of freedom rather than the normal distribution. With few clusters this costs participants; because C depends on the sample, it is iterated until it settles [6].

All adjustments together

n_{\text{adj}} = \left\lceil n (1 - r_b^{2})\,\mathrm{DE}\;F_t \right\rceil + 1

The adjustments multiply, in this order, and the result is rounded up once. Any adjustment not used contributes a factor of 1.

Dropout

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

Applied last, to the participants actually recruited.

Clusters to recruit

C = \left\lceil \frac{n_{\text{recruit}}}{m} \right\rceil, \qquad C_{\text{recruit}} = \left\lceil \frac{C}{1 - \ell} \right\rceil

Per group. Losing a whole cluster loses every participant in it, so an expected loss of clusters inflates the clusters recruited rather than the participants.

Parameters

Cohen’s f $f$

Values: Above 0

Best computed from the group means you expect, with the formula above. Cohen’s benchmarks of 0.10, 0.25 and 0.40 for small, medium and large are a last resort [1].

Number of groups $k$

Values: A whole number of at least 2

All groups are assumed to be the same size.

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.

Baseline correlation $r_b$ Optional

Values: Between −1 and 1

The correlation between the outcome and its baseline measurement, from previous data. Only use it if the analysis will adjust for the baseline.

Intracluster correlation (ICC) $\rho$ Optional

Values: From 0 to 1

For clustered designs only. Published ICCs for health outcomes in primary-care and community clusters are mostly small, often below 0.05 [7]; prefer an estimate from a comparable trial.

Average cluster size $m$ Optional

Values: At least 1

The average number of participants per cluster. Given with the ICC.

Variation in cluster size $\mathrm{CV}$ Optional

Values: 0 or more; 0 for equal clusters

The SD of cluster sizes divided by their mean. Unequal clusters cost power on their own account [5].

Expected loss of whole clusters $\ell$ Optional

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

Clusters, such as practices or schools, expected to withdraw entirely.

Correction for few clusters $F_t$ Optional

Values: On or off

Worth applying when there are few clusters per group, roughly fewer than 15–20.

Assumptions

The groups are independent and equal in size, and participants are independent apart from any clustering specified.
The outcome is roughly normal with the same standard deviation in every group.
Power is for the overall F test. Pairwise comparisons after it, with a multiplicity correction, need their own sample size and are usually less powerful.
With clustering: clusters are independent, with the same ICC throughout, and the analysis accounts for the clustering, for example with a mixed model or cluster-level summaries [4].
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Three groups, a medium effect

f = 0.25, 3 groups, α = 0.05, 80% power: N = 159, so 53 per group.

53 per group · Total: 159

Protocol statement
To detect an effect size of f = 0.25 with 80% power at a significance level of 0.05, using a one-way ANOVA across 3 groups, the required sample size is 53 per group (159 in total, across 3 groups). Calculated with StatWisely (statwisely.com) using: N: solve the noncentral-F power equation for Cohen's f, k groups, α, and 1-β. These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Four groups

The same effect across 4 groups: 180 in total, 45 per group. Each group needs fewer, but the total grows.

45 per group · Total: 180

Open this example in the calculator

A small effect

f = 0.1 across 3 groups: 969 in total, 323 per group. A small effect needs about six times the sample of a medium one.

323 per group · Total: 969

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.
The results agree with G*Power’s one-way ANOVA: 159 in total for f = 0.25 and three groups, 180 for four groups, and 969 for f = 0.10 and three groups, at α = 0.05 and 80% power [2].

How to cite

StatWisely. Sample size for a one-way ANOVA. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/means-anova/

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

References

1.Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.
2.Faul F, Erdfelder E, Lang AG, Buchner A. G*Power 3: a flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods. 2007;39(2):175–191.
3.Borm GF, Fransen J, Lemmens WAJG. A simple sample size formula for analysis of covariance in randomized clinical trials. Journal of Clinical Epidemiology. 2007;60(12):1234–1238.
4.Donner A, Klar N. Design and Analysis of Cluster Randomization Trials in Health Research. London: Arnold; 2000.
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.Hayes RJ, Moulton LH. Cluster Randomised Trials. 2nd ed. Boca Raton: CRC Press; 2017.
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

The smallest N with enough power

Cohen’s f from expected means

Adjusting for a baseline covariate

Clustering

Few clusters

All adjustments together

Dropout

Clusters to recruit

Parameters

Cohen’s ffff

Number of groupskkk

Significance levelα\alphaα

Power1−β1-\beta1−β

Expected dropoutqqqOptional

Baseline correlationrbr_brb​Optional

Intracluster correlation (ICC)ρ\rhoρOptional

Average cluster sizemmmOptional

Variation in cluster sizeCV\mathrm{CV}CVOptional

Expected loss of whole clustersℓ\ellℓOptional

Correction for few clustersFtF_tFt​Optional

Assumptions

Worked examples

Three groups, a medium effect

Four groups

A small effect

Validation

How to cite

References

Cohen’s f $f$

Number of groups $k$

Significance level $\alpha$

Power $1-\beta$

Expected dropout $q$ Optional

Baseline correlation $r_b$ Optional

Intracluster correlation (ICC) $\rho$ Optional

Average cluster size $m$ Optional

Variation in cluster size $\mathrm{CV}$ Optional

Expected loss of whole clusters $\ell$ Optional

Correction for few clusters $F_t$ Optional