One-way ANOVA calculator
Three or more groups, one omnibus test. This compares all the means at once and lays out the full sum-of-squares table behind the F.
A one-way ANOVA tests whether the means of three or more independent groups differ by more than sampling variation would explain, comparing the spread between the group means against the spread within the groups. Paste two or more groups of raw data and this calculator returns the F statistic, both degrees of freedom, the exact p-value, the complete sum-of-squares table, and eta-squared as the effect size.
F statistic
65.414
F(2, 15) = 65.41, p = < .001, a large effect.
A significant F tells you the group means are not all equal, but not which groups differ. Follow up with a post-hoc test such as Tukey's HSD to locate the specific differences.
How the F ratio is built
ANOVA partitions the total variation into a between-groups part and a within-groups part, then turns each into a mean square by dividing by its degrees of freedom. The F statistic is the ratio of the two:
F = MS_between / MS_within
The between-groups sum of squares measures how far each group mean sits from the grand mean, weighted by group size, with k - 1 degrees of freedom for k groups. The within-groups sum of squares measures the scatter of observations around their own group mean, with N - k degrees of freedom. The p-value is the upper-tail area of the F distribution, and eta-squared, the between-groups sum of squares over the total, reports the proportion of variance the grouping explains. The test assumes independent observations, roughly normal residuals, and similar variances across groups.
The guides behind this tool
Frequently asked questions
- What does a one-way ANOVA test?
A one-way ANOVA tests whether the means of three or more independent groups are all equal, against the alternative that at least one differs. It works by comparing the variation between the group means with the variation within the groups, summarised as the F ratio. A significant result says the means are not all the same but does not, by itself, identify which groups differ from which.
- Why not run multiple t-tests instead of an ANOVA?
Running a separate t-test on every pair of groups inflates the chance of a false positive, because each test carries its own 5% error rate and those risks accumulate. With four groups there are six pairwise tests, and the overall error rate climbs well above 5%. A one-way ANOVA holds the error rate at the chosen level with a single test, and post-hoc procedures then compare pairs while controlling that inflation.
- What is a good F value in ANOVA?
There is no single good F value, because the threshold for significance depends on the degrees of freedom and the chosen alpha level. An F near 1 suggests the between-group and within-group variation are similar, consistent with equal means, while an F well above 1 points to real differences. What matters is whether the F exceeds its critical value, reflected in a p-value below your threshold, alongside a meaningful eta-squared.
- What do you do after a significant ANOVA?
A significant ANOVA tells you the group means are not all equal, so the next step is a post-hoc test to find where the differences lie. Tukey's HSD is the common choice when comparing every pair while controlling the family-wise error rate, and Games-Howell is preferred when variances are unequal. You should also report an effect size such as eta-squared so the practical size of the difference is clear, not just its significance.