Analysis of variance (ANOVA) answers one question: are the means of several groups all equal, or does at least one differ? Instead of running a t-test for every pair of groups – which inflates the false-positive rate – ANOVA splits the total variation in the data into two parts. The between-group part measures how far the group means sit from the overall mean; the within-group part measures the ordinary scatter of values around their own group mean.
If the null hypothesis of equal means is true, both parts estimate the same underlying variance and their ratio, the F statistic, is close to 1. A large F means the group means are further apart than random scatter would explain. The p-value comes from the F distribution with k − 1 and N − k degrees of freedom, where k is the number of groups and N the total number of observations.
Typical uses include comparing crop yields under several fertilisers, test scores under different teaching methods or reaction times across age bands. ANOVA assumes independent observations, roughly normal data within each group and similar variances; it is fairly robust to mild departures when the groups are of similar size.
How to use the One-Way ANOVA Calculator
- Groups – one group per line (optional “Name:” before the values)
- Significance level (α)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Type each group on its own line (an optional “Name:” prefix labels it). The calculator finds each group mean and the grand mean, adds up the between- and within-group sums of squares, divides by their degrees of freedom to get the mean squares and takes their ratio F. The p-value is the upper-tail area of the F distribution beyond the observed F.
Worked example
Eighteen students are taught by three methods and their test scores are: A = 6, 8, 4, 5, 3, 4 (mean 5); B = 8, 12, 9, 11, 6, 8 (mean 9); C = 13, 9, 11, 8, 7, 12 (mean 10). The grand mean is 8.
SSB = 6 × (5 − 8)² + 6 × (9 − 8)² + 6 × (10 − 8)² = 54 + 6 + 24 = 84 with 2 degrees of freedom, so MSB = 42. Adding the squared deviations inside each group gives SSW = 16 + 24 + 28 = 68 with 15 degrees of freedom, so MSW = 4.533.
F = 42 ÷ 4.533 = 9.265. The critical F(2, 15) at α = 0.05 is 3.682 and the p-value is 0.0024, so we reject H₀: the teaching methods do not all give the same mean score. η² = 84 ÷ 152 = 0.553.
How to read the ANOVA table
Each row splits the variation: “Between groups” is the signal, “Within groups” the noise and “Total” their sum. Mean squares are sums of squares divided by degrees of freedom, and F is the ratio of the two mean squares. If p ≤ α, report the result as F(df₁, df₂) = value, p = value – for the example, F(2, 15) = 9.26, p = 0.002 – together with η² so readers see how big the difference is, not just that it exists.
Formula
SSB = Σ nᵢ(x̄ᵢ − x̄)² · SSW = Σ Σ (xᵢⱼ − x̄ᵢ)² · MSB = SSB ÷ (k − 1) · MSW = SSW ÷ (N − k) · F = MSB ÷ MSW · η² = SSB ÷ SST
Worked example
Inputs — Groups – one group per line (optional "Name:" before the values): Method A: 6, 8, 4, 5, 3, 4
Method B: 8, 12, 9, 11, 6, 8
Method C: 13, 9, 11, 8, 7, 12; Significance level (α): 0.05.
Results — p-value = 2.40e-3; F statistic = 9.2647; Decision = Reject H₀ at α = 0.05 – p = 0.0024 ≤ 0.05 and F = 9.2647 lies in the rejection region.; Critical F (right-tailed) = 3.6823; Degrees of freedom between groups (k − 1) = 2; Degrees of freedom within groups (N − k) = 15.
Key terms
- Between-group variation (SSB)
- How far the group means lie from the grand mean, weighted by group size.
- Within-group variation (SSW)
- Scatter of observations around their own group mean; also called error or residual sum of squares.
- Mean square
- A sum of squares divided by its degrees of freedom – an estimate of variance.
- F statistic
- The ratio MSB ÷ MSW; values well above 1 point to differences between means.
- Eta squared (η²)
- The share of total variation explained by group membership.
Frequently asked questions
What does a significant ANOVA tell me?
Only that at least one group mean differs from the others. To find which pairs differ, follow up with a post-hoc test such as Tukey's HSD or Bonferroni-corrected t-tests.
Why not run several t-tests instead?
With three groups you would run three t-tests, and the chance of at least one false positive rises from 5% to about 14%. ANOVA tests all groups at once at the chosen α.
Can I use ANOVA with only two groups?
Yes. With two groups the F statistic equals the square of the pooled two-sample t statistic, and the p-values are identical.
Do the groups need the same size?
No. Unequal group sizes are allowed, but ANOVA is less robust to unequal variances when the sizes differ a lot – then consider Welch's ANOVA or a Kruskal–Wallis test.
What is a large effect size?
Common rules of thumb for η² are 0.01 small, 0.06 medium and 0.14 large. In the default example η² = 0.553, a very large effect.
What is the critical value of F for 2 and 15 degrees of freedom?
3.682 at α = 0.05 and 6.359 at α = 0.01. An observed F above the critical value is significant.
What if my data are not normal?
With moderate group sizes ANOVA copes well with skewness. For small, clearly non-normal samples or ordinal data use the Kruskal–Wallis test.