ANOVA Calculator
Run a one-way ANOVA across two or more groups — get the F statistic, between- and within-group degrees of freedom, the p-value and a verdict at α = 0.05.
How it works
Grand mean x̄ = Σx ÷ N · SS_between = Σ nⱼ(x̄ⱼ − x̄)² · SS_within = Σ Σ (x − x̄ⱼ)² · df_between = k − 1, df_within = N − k · MS = SS ÷ df · F = MS_between ÷ MS_within · p = P(F > statistic)
One-way ANOVA tests whether three or more group means differ by more than chance, using a single F test instead of many pairwise t-tests (which would inflate the false-positive rate). It splits the total variation in the data into two parts: the between-groups sum of squares, which grows when group means are far apart, and the within-groups sum of squares, which reflects the natural scatter inside each group. Dividing each by its degrees of freedom gives mean squares, and their ratio F = MS_between ÷ MS_within is large when the groups really differ. The upper-tail p-value, computed here from the regularized incomplete beta function, is the chance of an F this large if every group shared the same true mean; a significant result tells you at least one group differs but not which one.
Worked example
For the three groups 5, 7, 6, 8 · 10, 12, 9, 11 · 15, 14, 16, 13 (N = 12), the group means are 6.5, 10.5 and 14.5 and the grand mean is 10.5. SS_between = 4·(6.5−10.5)² + 4·(10.5−10.5)² + 4·(14.5−10.5)² = 64 + 0 + 64 = 128, and SS_within = 5 + 5 + 5 = 15. With df_between = 2 and df_within = 9, MS_between = 64 and MS_within ≈ 1.667, so F = 64 ÷ 1.667 ≈ 38.4 with a p-value below 0.0001 — the group means differ significantly.
Frequently asked questions
Why use ANOVA instead of several t-tests?
Running a separate t-test for every pair of groups multiplies the chance of a false positive: with several comparisons at α = 0.05 the overall error rate climbs well above 5%. ANOVA answers the single question "do any of these means differ?" with one test at one significance level, keeping the family-wise error rate under control.
What does a significant F tell me?
A significant F (small p-value) says that at least one group mean differs from the others by more than sampling noise would explain. It does not say which groups differ — for that you follow up with a post-hoc test such as Tukey’s HSD, which compares pairs while adjusting for the multiple comparisons.
What assumptions does one-way ANOVA make?
It assumes the observations are independent, the values within each group are roughly normally distributed, and the groups have similar variances (homogeneity of variance). Moderate violations are often tolerable with balanced group sizes, but strong departures — especially very unequal variances — call for a robust alternative such as Welch’s ANOVA.
Related calculators
For education — check the test assumptions (independence, sample size, distribution) before drawing conclusions from a p-value.