This ANOVA calculator runs a one-way analysis of variance from raw group data — testing whether several group means differ more than within-group noise can explain — with F, degrees of freedom, p-value and η² effect size.
A significant F says “the means are not all equal” — not which pairs differ. Post-hoc comparisons (Tukey HSD etc.) with multiplicity control are the required next step before naming winners; running raw t-tests between every pair inflates false positives exactly as running this omnibus test was designed to avoid.
F = Between-Group Variance ÷ Within-Group Variance
ANOVA partitions total variation into the part explained by group membership (means scattered around the grand mean) and the leftover noise (points scattered around their own group means). If group labels were meaningless, both would estimate the same variance and F ≈ 1. Large F says the means spread more than noise allows. Two groups only? F = t² — the same test as our T-Test Calculator wearing its multi-group generalization. Part of our Statistics Calculators collection.
Worked Example (Verified)
- Groups {5,6,7}, {8,9,10}, {3,4,5} (defaults): group means 6/9/4, grand mean 6.33 → SSB = 38, SSW = 6 → MSB 19, MSW 1 → F = 19.00, df (2, 6), p = 0.0025, η² = 0.864 — an enormous effect: 86% of variance sits between groups (calibration: F(2,6) = 5.14 is the 5% critical value, which this engine reproduces at p = 0.050)
- The noise-only world: identical means with the same within-spread gives SSB ≈ 0 → F ≈ 0, p ≈ 1 — ANOVA correctly unimpressed
- η² benchmarks (rough): 0.01 small, 0.06 medium, 0.14 large — the defaults' 0.86 is a textbook sledgehammer chosen for clean arithmetic
- Sample-size lever: the same means with n=30 per group would shrink MSW's relative weight and drive p orders of magnitude smaller — F blends effect and evidence
Assumptions & Next Steps
- Approximately normal residuals and roughly equal within-group variances (Levene's test checks the second; Welch-ANOVA and Kruskal–Wallis are the fallbacks).
- Independent observations — repeated measures need repeated-measures ANOVA, not this page.
- Post-hoc discipline: Tukey HSD for all-pairs, Dunnett vs a control, planned contrasts when hypotheses predate data.
- Report η² (or ω², less biased) beside p — with big N, meaningless separations turn "significant."
Related Tools
- T-Test Calculator — the two-group case
- Chi-Square Calculator — the categorical counterpart
- Statistics Calculators — the full collection
Frequently Asked Questions
How do you calculate one-way ANOVA?
Compute the grand mean; SSB = Σ nᵢ(meanᵢ − grand)² with df = groups − 1; SSW = Σ within-group squared deviations with df = N − groups; F = (SSB/dfB) ÷ (SSW/dfW), and p comes from the F-distribution. The defaults ({5,6,7}/{8,9,10}/{3,4,5}) give F = 19.0, p = 0.0025.
What does a significant ANOVA tell you?
Only that the group means are not all equal — somewhere a difference exists. It never says which groups differ or by how much. Post-hoc tests (Tukey, Dunnett) with error-rate control, plus the effect size η², complete the inference.
Why not just run t-tests between every pair?
Each t-test carries its own false-positive risk; six pairs at α = 0.05 compound to ~26% chance of at least one spurious "discovery." ANOVA's omnibus gate plus adjusted post-hocs exist precisely to hold the family-wise rate at 5%.
What is η² (eta squared)?
The proportion of total variance attributable to group membership: SSB ÷ (SSB + SSW). Rough benchmarks: 0.01 small, 0.06 medium, 0.14 large. It's slightly optimistic in small samples (ω² corrects this), but it's the standard first report alongside F and p.
What if my variances are very unequal or data are skewed?
Classical ANOVA assumes both away. Practical ladder: transform the data (log often fixes both), use Welch's ANOVA for unequal variances, or Kruskal–Wallis (rank-based) when normality fails badly. Mild violations with balanced, decent-sized groups are tolerated — robustness is ANOVA's quiet virtue.
Is this ANOVA calculator free?
Yes — free, instant, no sign-up.