Two-way ANOVA calculator
One-way ANOVA tells you whether any group differs. A two-way design tells you which factor matters — and whether the two interact, meaning the effect of one depends on the level of the other. That interaction term is usually the reason you run the second factor at all.
| Source | SS | df | F | p |
|---|---|---|---|---|
| Factor A | 8.00 | 1 | 4.00 | 0.116 |
| Factor B | 32.00 | 1 | 16.00 | 0.016 |
| A × B | 0.00 | 1 | 0.00 | 1.000 |
| Error | 8.00 | 4 | — | — |
A significant interaction means the effect of A depends on the level of B, so main effects alone mislead.
y = μ + α + β + αβ + ε · Two factors with interaction.
MS_source / MS_error · One test per source.
cell − α − β + μ · The part not explained by the main effects.
A clear path from inputs to results.
One-way ANOVA tells you whether any group differs. A two-way design tells you which factor matters — and whether the two interact, meaning the effect of one depends on the level of the other. That interaction term is usually the reason you run the second factor at all.
How to use
- Enter two observations in each of the four cells.
- Read the F ratios for A, B and the interaction.
- Check which p-values are below 0.05.
- Look at the cell means to interpret an interaction.
The formula
Reference: Two-way analysis of variance with interaction, balanced design.
Worked example
2 × 2 design, two replicates per cell
- A low, B low: 10, 12.
- A low, B high: 14, 16.
- A high, B low: 12, 14.
- A high, B high: 16, 18.
- 1
Grand mean 14; cell means 11, 15, 13, 17.
- 2
SS_A = 8, SS_B = 32, SS_AB = 0, SS_E = 8.
- 3
F_A = 4.0 (p = 0.116); F_B = 16.0 (p = 0.016).
- 4
F_AB = 0, so no interaction: the effects are additive.
- 5
Factor B matters; factor A does not, at this sample size.
F_B 16.0 (p 0.016) · no interaction.
Interpreting the result
With only two replicates the design has four error degrees of freedom, so it can only detect large effects. That is the trade: a 2×2 with replicates is cheap, but the interaction test is weak. If the interaction is the question, replicate more.
Assumptions & limitations
- Balanced design with equal replicates.
- Independent, normally distributed errors with equal variance.
- Two levels per factor.
- No blocking or covariates.
Frequently asked questions
What does a significant interaction mean?
The effect of A depends on the level of B, so a single main effect is misleading.
Can I add more levels?
Yes, but the arithmetic is the same; this form is fixed at two levels each.
Why is F_AB zero here?
The cell means are perfectly additive, so there is no interaction to detect.
Methodology last reviewed 2026-10-21. Results are informational and not financial or engineering advice — see our disclaimer.