SigmaCalc

ANOVA Calculator

Test whether 3+ group means differ significantly, without running pairwise t-tests that inflate your error rate.

Groups (semicolon-separated, values comma-separated)

Example: 12,15,14; 20,22,19; 25,28,24

df between: 2
df within: 9
F-statistic: 33.756

What you entered

Groups: 3

1
Sum of squares between groups
Σ nᵢ(x̄ᵢ − x̄grand)²= 253.1667
2
Sum of squares within groups
Σ Σ (xᵢⱼ − x̄ᵢ)²= 33.75
3
Mean square between
253.1667 ÷ 2 df= 126.5833
4
Mean square within
33.75 ÷ 9 df= 3.75
5
F-statistic
126.5833 ÷ 3.75= 33.7556

Result

F-statistic: 33.756

An F-statistic of 33.756 with (2, 9) degrees of freedom — compare it against an F-distribution critical value for your chosen significance level to decide whether the group means differ significantly.

One-way ANOVA F-statistic for worked groups

ANOVA compares variance between groups against variance within them. A large F means the group means differ by more than the internal scatter explains.

Groups	F-statistic	df between	df within
[1,2,3] [4,5,6] [7,8,9]	27.000	2	6
[10,12,11] [14,15,13] [20,19,21]	63.000	2	6
[5,6,7,8] [6,7,8,9]	1.200	1	6

The third row is the instructive one: the two groups differ by exactly 1 at every position, yet F is only 1.200 because that shift is small relative to the spread within each group. The first two rows have identical group sizes and degrees of freedom but very different F values, driven entirely by how far apart the means sit relative to internal variation. Compare F against the critical value for your degrees of freedom - at 0.05 with 2 and 6 df it is 5.14, so the first two rows are significant. A significant F tells you at least one group differs but not which, which is what post-hoc tests are for.

Why not just run multiple t-tests?

Running a t-test for every pair of groups multiplies your chance of a false positive with each additional comparison. ANOVA tests all groups at once with a single F-statistic, controlling that error rate properly.

ANOVA tells you 'if', not 'which'

A significant F-statistic tells you at least one group mean differs from the others — it doesn't say which pair. Follow up with a post-hoc test (like Tukey's HSD) to pinpoint exactly which groups differ.

Frequently asked questions

Three fertilizers were tested on plant growth (cm): A=[20,22,19], B=[25,28,24], C=[21,23,20]. Do they differ?

Group means: A=20.3, B=25.7, C=21.3. Grand mean = 22.4. ANOVA compares between-group variance to within-group variance. If F exceeds the critical value for df=(2,6), the fertilizers produce significantly different growth. Here F ≈ 11.3, p ≈ 0.009 — yes, they differ significantly.

How is ANOVA different from a t-test?

A t-test compares exactly 2 groups. ANOVA compares 3 or more groups simultaneously. Running multiple t-tests inflates the false positive rate (3 groups = 3 comparisons = ~14% chance of a false positive instead of 5%). ANOVA controls this with a single omnibus test.

What do I do after ANOVA shows significance?

ANOVA only tells you 'at least one group differs.' To find WHICH pairs differ, run a post-hoc test (Tukey's HSD is most common). This adjusts for multiple comparisons while identifying specific group differences.