Means: Many Groups (ANOVA)
Determine the required sample size per group for a one-way ANOVA F-test using the exact noncentral F distribution. Enter the anticipated group means, common within-group standard deviation, significance level, and desired power. The calculation searches for the smallest equal group size that achieves the requested power.
This calculator implements the one-way analysis of variance F-test assuming normally distributed observations, a common within-group variance, independent groups, and equal group sample sizes. The null hypothesis is that all group means are equal; the alternative is that at least one mean differs.
For G groups with equal sample size n, the total sample size is N = Gn. The numerator degrees of freedom are G − 1 and the denominator degrees of freedom are N − G.
Let the hypothesized group means be μ1, …, μG, and let σ be the common within-group standard deviation. With equal allocation, the weighted mean of the group means is their arithmetic mean:
The noncentrality parameter is then
The critical value is the upper α quantile of the central F distribution:
Power is the probability that a noncentral F random variable with numerator degrees of freedom G − 1, denominator degrees of freedom N − G, and noncentrality parameter λ exceeds this critical value. The calculator evaluates this probability directly and increments the equal group size until the requested power is reached.
The omnibus ANOVA tests whether there is evidence that at least one population mean differs. It does not specify which pair of means differs. If the study also requires adequately powered pairwise comparisons or multiplicity-adjusted comparisons, those should be planned separately.
this method provides a validation example from Fleiss (1986): four equal-sized groups with hypothesized means 9.775, 12, 12, and 14.225, common standard deviation 3, α = 0.05, and desired power 0.80. The published Validation 11 subjects per group, for a total sample size of 44, with actual power 0.80273. The calculator reproduces this result.
this method also reports that its calculation uses the exact noncentral F distribution rather than the normal approximation used in the original Fleiss calculation.
the software/this method. One-Way Analysis of Variance Assuming Equal Variances (F-Tests). this method Sample Size Software, Chapter 543. The documentation defines the noncentrality parameter, power calculation, assumptions, examples, and validation against Fleiss (1986) and Desu (1990).
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Lawrence Erlbaum Associates, Hillsdale, NJ. Note that Cohen as a reference for the one-way ANOVA effect-size framework.
Fleiss, J. L. (1986). The Design and Analysis of Clinical Experiments. John Wiley & Sons. This calculator uses the four-group example from page 374 as a validation example and obtains N = 44.
the relevant methodological literature: Sample Size for Multiple Means in this method. Note that separate analytical one-way ANOVA procedures, including the equal-variance F-test and an effect-size formulation.