Analysis of variance (ANOVA) answers a simple question: are the averages of several groups different, or could the gaps between them be ordinary sampling noise? One-way ANOVA handles one grouping factor with any number of levels, such as three teaching methods, four fertilizers or five suppliers. Paste one group per line and the calculator produces the complete ANOVA table, the F statistic and p-value, two effect sizes, and a summary of each group.
How to use the ANOVA calculator
- Type or paste each group on its own line. Start a line with a label and a colon (for example
Supplier B: 4.1, 3.9, 4.4) to name it; otherwise groups are numbered. - Groups may have different sizes, but each needs at least one value, and at least one group needs two or more.
- Set the significance level α.
- Read F and the p-value on the tape, then the ANOVA table, the group summary and the bar comparison of group means.
ANOVA formulas
With k groups, N observations in total, group sizes nᵢ, group means x̄ᵢ and grand mean x̄:
Each sum of squares is divided by its degrees of freedom to get a mean square:
The p-value is the right-tail area of the F distribution with (k − 1, N − k) degrees of freedom. Effect sizes: η² = SSB ÷ SST and ω² = (SSB − (k − 1)MSW) ÷ (SST + MSW).
Worked example
Eighteen students are split evenly among three teaching methods, and their exam scores are:
- Method A: 84, 90, 88, 81, 92, 86 (mean 86.83)
- Method B: 78, 74, 80, 72, 79, 76 (mean 76.50)
- Method C: 91, 94, 89, 96, 90, 93 (mean 92.17)
The grand mean is 1,533 ÷ 18 = 85.17.
- SSB = 6(86.83 − 85.17)² + 6(76.50 − 85.17)² + 6(92.17 − 85.17)² = 761.33, with 2 df.
- SSW = 5 × (16.17 + 9.50 + 6.97) = 163.17, with 15 df.
- MSB = 761.33 ÷ 2 = 380.67 and MSW = 163.17 ÷ 15 = 10.88.
- F = 380.67 ÷ 10.88 = 34.99. The critical value at α = 0.05 is 3.68, and the p-value is about 0.0000022.
- η² = 761.33 ÷ 924.5 = 0.82: about 82% of the variation in scores lines up with teaching method.
The methods clearly differ on average. The means suggest Method C is highest and Method B lowest, but a post-hoc test is needed before claiming that each pair differs.
Assumptions and checks
Independence
Each observation should come from a different subject, and groups should not overlap. Measuring the same people under every condition calls for repeated-measures ANOVA instead.
Normality within groups
ANOVA assumes the values in each group come from a roughly normal population. With moderate group sizes the test is robust to mild skew; with small groups, look for outliers first using the descriptive statistics calculator.
Equal variances
The pooled MSW assumes all groups share one variance. A quick rule: if the largest group standard deviation is less than twice the smallest, ANOVA is fine. The calculator warns you when that ratio is exceeded. In the example the SDs are 4.02, 3.08 and 2.64, comfortably within the rule.
ANOVA and the t-test
With exactly two groups, one-way ANOVA and the pooled two-sample t-test are the same test: F equals t², and the p-values match. For two groups with unequal spreads, use Welch’s version in the t-test calculator. To check any F statistic against its distribution directly, use the p-value calculator.
Frequently asked questions
Why not just run t-tests between every pair of groups?
Each test carries its own chance of a false positive. With three groups there are three pairwise tests, and the chance that at least one is falsely significant at α = 0.05 rises to about 14%. ANOVA asks one overall question at a single α, then post-hoc tests examine pairs with a proper correction.
What does the F statistic mean?
F compares the spread between group means with the spread inside the groups. If the groups truly share one mean, both mean squares estimate the same variance and F hovers around 1. A large F means the group means are further apart than the within-group noise can explain.
My ANOVA is significant. Which groups differ?
The F-test does not say. Follow up with a post-hoc procedure such as Tukey's HSD, which compares every pair while controlling the overall error rate, or with planned contrasts if you specified comparisons in advance. Looking at the group means and their confidence intervals is a good first step.
Can the groups have different sizes?
Yes. One-way ANOVA handles unequal group sizes, and this calculator weights each group by its size. Very unbalanced designs combined with unequal variances can distort the test, so in that case consider Welch's ANOVA.
What is the difference between eta squared and omega squared?
Both estimate the share of total variation explained by group membership. Eta squared (SSB ÷ SST) describes the sample and tends to overstate the population effect, especially with small samples. Omega squared adjusts for that bias and is usually a little smaller.