One-Way ANOVA Calculator
Run a one-way ANOVA test on up to four groups.
Calculate the F-statistic and p-value to determine if group means differ significantly.
One-Way ANOVA (Analysis of Variance)
One-way ANOVA tests whether the means of three or more independent groups are equal. It answers: “Is at least one group mean significantly different from the others?”
The null hypothesis (H₀): All group means are equal. The alternative hypothesis (H₁): At least one group mean differs.
How it works — partitioning variance:
| Source | Sum of Squares | df | Mean Square | F |
|---|---|---|---|---|
| Between groups | SSB | k − 1 | MSB = SSB/dfB | F = MSB/MSW |
| Within groups | SSW | N − k | MSW = SSW/dfW | |
| Total | SST | N − 1 |
Where k = number of groups, N = total observations.
Formulas:
Grand mean x̄ = (sum of all values) / N
SSB = Σᵢ nᵢ × (x̄ᵢ − x̄)² (between-group variance)
SSW = Σᵢ Σⱼ (xᵢⱼ − x̄ᵢ)² (within-group variance)
F = MSB / MSW
Interpreting the result:
A large F-statistic (and small p-value) suggests at least one group mean is different. The conventional threshold is p < 0.05 for statistical significance. If significant, use a post-hoc test (Tukey, Bonferroni) to find which specific groups differ.
Assumptions:
- Observations are independent
- Residuals are approximately normally distributed
- Groups have similar variances (homoscedasticity)
- If variances differ, use Welch’s ANOVA instead
Enter data as comma-separated numbers per group. Example: Group 1 → 23, 25, 28, 22, 27