About the ANOVA Calculator
This one-way ANOVA calculator tests whether the means of three or more independent groups differ by more than you would expect from random variation. Paste each group’s values on its own line and the tool builds the complete ANOVA table — sums of squares, degrees of freedom, mean squares, the F statistic and its p-value — plus the critical F value and the eta-squared effect size.
Typical uses include comparing test scores across teaching methods, yields across fertiliser treatments, or conversion rates across several page designs. With only two groups, one-way ANOVA gives exactly the same p-value as a pooled two-sample t-test (F = t²).
ANOVA assumes independent observations, roughly normal data within each group and similar variances across groups. A significant result says at least one mean differs, not which one; follow up with a post-hoc test such as Tukey’s HSD to find the specific pairs.
With the default inputs, the f statistic is 2.3575. Change any value above to recalculate instantly.
How to use the anova calculator
- 1Enter each group’s values on a separate line (commas or spaces between numbers).
- 2Optionally add a label before a colon, like "Control: 5, 6, 7".
- 3Choose a significance level — 0.05 is standard.
- 4Read F and the p-value; p below α means at least one mean differs.
- 5Check eta squared for effect size and the chart for which groups stand out.
Formula and method
One-way ANOVA splits the total variation in the data into a between-groups part and a within-groups part. SSB = Σ nᵢ(x̄ᵢ − x̄)² measures how far each group mean sits from the grand mean, weighted by group size. SSW = ΣΣ (xᵢⱼ − x̄ᵢ)² measures the scatter of values around their own group mean. Dividing each by its degrees of freedom gives the mean squares.
If all population means are equal, MSB and MSW both estimate the same variance and F is close to 1. A large F means the group means are further apart than within-group noise can explain. The p-value is the upper tail of the F distribution with (k − 1, N − k) degrees of freedom. Eta squared, SSB ÷ SST, is the proportion of variance explained.
- k
- Number of groups
- N
- Total number of observations
- nᵢ, x̄ᵢ
- Size and mean of group i
- x̄
- Grand mean of all observations
- MSB, MSW
- Mean square between and within groups
Worked examples
Test scores under three teaching methods
The three method means are 83.4, 89.3 and 84.7. SSB = 192.2 on 2 df and SSW = 1100.6 on 27 df give F = 96.1 ÷ 40.76 ≈ 2.36. The p-value of about 0.114 is above 0.05, so the difference in means is not statistically significant.
Plant growth under three fertilisers
Group means are 4.70, 6.08 and 5.26. The between-group spread is large compared with the small within-group scatter, giving F = 2.4087 ÷ 0.0883 ≈ 27.3 and p ≈ 0.00003. About 82% of the variation is explained by fertiliser type.
Frequently asked questions
What does a one-way ANOVA test?+
It tests the null hypothesis that several independent groups all have the same population mean, against the alternative that at least one mean differs. It uses one categorical factor (the grouping) and one numeric outcome.
How do I interpret the F statistic?+
F is the ratio of between-group variance to within-group variance. Values near 1 suggest the group means differ no more than random noise would produce; larger values are evidence of real differences. Compare the p-value with α, or F with the critical F value.
Why not just run several t-tests?+
Each t-test carries its own chance of a false positive, so running many of them inflates the overall error rate — with three groups and three tests at α = 0.05 the chance of at least one false positive is about 14%. ANOVA tests all groups at once at the chosen α.
What is eta squared?+
Eta squared (η² = SSB ÷ SST) is the proportion of total variation explained by the grouping. Common rules of thumb treat 0.01 as small, 0.06 as medium and 0.14 as large. Unlike the p-value, it does not grow simply because the sample is larger.
What should I do after a significant ANOVA?+
Use a post-hoc comparison such as Tukey’s HSD, Bonferroni-adjusted t-tests or Games–Howell (if variances are unequal) to identify which specific pairs of groups differ, and report the effect size alongside the p-value.