Achieved power for a one-way ANOVA

When to use it

Assess the sensitivity of a comparison of three or more independent group means at the group sizes available.

Not for

The omnibus test says whether any means differ, not which. Power for a specific pairwise contrast is lower and needs its own calculation.

Notation

$n$: Participants in each group.
$k$: Number of groups.
$N$: Total participants, kn.
$f$: Cohen’s f: the SD of the group means divided by the common within-group SD.
$\lambda$: Noncentrality, f²N.

Formulas

Degrees of freedom and noncentrality

N = kn, \qquad \nu_1 = k-1, \qquad \nu_2 = N-k, \qquad \lambda = f^2 N

Cohen’s benchmarks are f = 0.10, 0.25, and 0.40 for small, medium, and large effects [1].

Power of the F test

\text{Power}=P\left(F_{\nu_1,\nu_2,\lambda} > F_{1-\alpha;\,\nu_1,\nu_2}\right)

The alternative is a noncentral F distribution [1, 2].

Parameters

Sample size per group $n$

Values: Integer, at least 2

Enter the participants available in each group.

Number of groups $k$

Values: Integer, at least 2

The number of independent groups compared.

Effect size (Cohen’s f) $f$

Values: Positive

From expected group means μᵢ and a common SD σ: f = √(Σ(μᵢ − μ̄)²/k)/σ.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Assumptions

Groups are independent and equally sized, the outcome is approximately normal within each group, and the groups share one standard deviation.
F and chi-square tests reject only in the upper tail, so there is no one- or two-sided choice.

Worked examples

A worked input set

For 3 groups of 20 participants (N = 60) and a medium effect, f = 0.25, at α = 0.05: ν₁ = 2, ν₂ = 57, and λ = 0.25² × 60 = 3.75.

Needed for analysis: 20 · To approach: 60

Open this example in the calculator

Validation

Power is the exact noncentral-F tail probability. It agrees with pwr::pwr.anova.test in R and with G*Power, and at the per-group size the ANOVA calculator returns it reaches that calculator’s target power.

How to cite

StatWisely. Achieved power for a one-way ANOVA. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-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.