ANOVA Calculator
Compare the means of three or more groups. Paste each group's values and Calvo builds the full one-way ANOVA table, F-value, p-value, effect size and pairwise comparisons.
One-way ANOVA calculator
What one-way ANOVA tests
One-way ANOVA (analysis of variance) checks whether three or more group means are all equal. H₀: μ₁ = μ₂ = … = μk. H₁: at least one mean is different. It compares the variation between the group means with the variation within the groups; if the between-group variation is much larger than expected from chance, F is large and p is small.
Formulas
| Quantity | Formula |
|---|---|
| Grand mean | x̄̃ = Σx / N |
| Between-group SS | SSB = Σ ni(x̄i − x̄̃)² |
| Within-group SS | SSW = ΣΣ (xij − x̄i)² |
| Degrees of freedom | dfB = k − 1, dfW = N − k |
| Mean squares | MSB = SSB / dfB, MSW = SSW / dfW |
| F statistic | F = MSB / MSW |
| Effect size | η² = SSB / SST |
Worked example
Three teaching methods A, B, C give these test scores: A = 6, 8, 4, 5, 3, 4; B = 8, 12, 9, 11, 6, 8; C = 13, 9, 11, 8, 7, 12. The group means are 5, 9 and 10, the grand mean is 8 and N = 18.
SSB = 6(5−8)² + 6(9−8)² + 6(10−8)² = 84, SSW = 68, df = (2, 15). MSB = 42, MSW = 4.533, so F = 9.265. The 5% critical value is F(2, 15) = 3.682 and p = 0.0024. Since p < 0.05 we reject H₀: at least one teaching method has a different mean.
After a significant ANOVA
A significant F only says some mean differs. Calvo also lists pairwise comparisons using the pooled MSW with a Bonferroni adjustment, so you can see which pairs differ without inflating the false-positive rate.
Assumptions
- Independence: observations are independent of each other.
- Normality: each group is roughly normal (less important with larger groups).
- Equal variances: group SDs are similar. Calvo shows each SD; a largest-to-smallest SD ratio above about 2 is a warning sign.
FAQ
What does the F-value in ANOVA mean?
F is the ratio of between-group variance (MSB) to within-group variance (MSW). A large F means the group means are more spread out than random variation within groups would explain.
When do I use ANOVA instead of a t-test?
Use a t-test for two groups. For three or more groups, ANOVA tests all means at once; running many t-tests instead would inflate the chance of a false positive.
What if the ANOVA is significant?
Follow up with pairwise comparisons to find which groups differ. Calvo shows Bonferroni-adjusted pairwise t-tests using the pooled within-group variance.
What is eta squared?
Eta squared is SSB divided by SST. It is the proportion of the total variation explained by the group differences. Around 0.01 is small, 0.06 medium and 0.14 large.
Can groups have different sizes?
Yes. One-way ANOVA works with unequal group sizes. Each group needs at least one value, and at least one group needs two or more so the within-group variance can be estimated.
Does Calvo do two-way ANOVA?
Not yet. This calculator handles one-way ANOVA, which covers most statistics course problems with a single factor.