ANOVA Calculator
One-way analysis of variance with the full F-table and p-value.
df = (2, 9) · p = 0.0004
| Source | SS | df | MS |
|---|---|---|---|
| Between | 72 | 2 | 36 |
| Within | 15 | 9 | 1.667 |
| Total | 87 | 11 | — |
One-way ANOVA compares three or more group means with F = MS_between ÷ MS_within. For three groups with means 2, 5, and 8, MS_between = 27 and MS_within = 1, so F = 27 on (2, 6) df — far above the 5.14 critical value, so at least one mean differs significantly.
Comparing three or more groups
A t-test compares two means; one-way ANOVA generalizes it to any number of groups in a single test, avoiding the inflated error rate of running many pairwise t-tests. It splits the total variation into a between-groups part and a within-groups part, then asks whether the spread between group means is large relative to the spread within the groups.
MS = SS ÷ df; df_between = k − 1, df_within = N − k
Worked example
Three groups of three — A: 1, 2, 3 (mean 2); B: 4, 5, 6 (mean 5); C: 7, 8, 9 (mean 8). Grand mean 5, N = 9, k = 3.
- 1 Between-groups mean square. SS_between = 3(2−5)² + 3(5−5)² + 3(8−5)² = 27 + 0 + 27 = 54; df = k − 1 = 2, so MS_between = 54 ÷ 2 = 27.
- 2 Within-groups mean square. Each group’s deviations (−1, 0, +1) sum to 2; SS_within = 2 + 2 + 2 = 6; df = N − k = 6, so MS_within = 6 ÷ 6 = 1.
- 3 Form the F-ratio. F = 27 ÷ 1 = 27 on (2, 6) df. The 0.05 critical F is 5.14, so 27 ≫ 5.14 — clearly significant.
Critical F-values (α = 0.05)
Numerator df across the top, denominator df down the side; reject when F exceeds the value.
| df_within ↓ / df_between → | 1 | 2 | 3 |
|---|---|---|---|
| 6 | 5.99 | 5.14 | 4.76 |
| 10 | 4.96 | 4.10 | 3.71 |
| 20 | 4.35 | 3.49 | 3.10 |
| 30 | 4.17 | 3.32 | 2.92 |
Reading the table and the assumptions
What a significant F means. A large F with a small p-value says at least one group mean differs from the rest — but not which one. Follow up with post-hoc comparisons (e.g. Tukey’s HSD) to locate the difference.
Why not many t-tests. Running a t-test on every pair inflates the chance of a false positive. ANOVA tests all groups at once at your chosen α, keeping the overall error rate in check.
The assumptions. One-way ANOVA assumes independent observations, roughly normal groups, and similar variances across groups (homogeneity of variance). It is fairly robust when group sizes are equal.
Entering data. Put one group per line, with values separated by commas or spaces. The tool reports the full F-table — sums of squares, degrees of freedom, mean squares, F, and the p-value.