Enter one group per box — they may be different sizes — to get the F statistic, the p value and the complete sum-of-squares partition. Use "Add group" for a fourth and beyond.
Separate values with commas, spaces, tabs or new lines — paste a column straight from a spreadsheet. Groups may be different sizes.
Show the working, step by step
The idea in one line
F = variation between the group means ÷ variation within the groups
Think of it as a signal-to-noise ratio. The numerator is the signal: how far apart the group means sit. The denominator is the noise: how much individual observations scatter around their own group's mean. If the signal is not clearly bigger than the noise, the apparent differences between groups are just what random sampling produces.
The partition
ANOVA's central trick is that the total variation splits cleanly into two pieces that add back up exactly:
SS_total = SS_between + SS_within
| Source | Sum of squares | df | Mean square |
|---|---|---|---|
| Between groups | Σ nᵢ(x̄ᵢ − x̄)² | k − 1 | SS ÷ df |
| Within groups | Σ Σ (x − x̄ᵢ)² | N − k | SS ÷ df |
| Total | Σ (x − x̄)² | N − 1 |
k is the number of groups and N the total number of observations.
Dividing each sum of squares by its degrees of freedom gives a mean square — which is simply a
variance. F is the ratio of the two.
Why F sits near 1 under the null
If all the groups really share a mean, then both mean squares are unbiased estimates of the same population variance. Two estimates of the same quantity have a ratio near 1, so that is where F lands.
When the groups differ, only the numerator inflates — the between-group sum of squares picks up the real separation, while the within-group figure keeps measuring ordinary scatter. F climbs, and the F distribution with (k − 1, N − k) degrees of freedom says how unusual that climb is.
Note that F cannot go below zero, and only its upper tail is evidence. That is why the test is one-tailed by construction, unlike a two-tailed t-test.
What ANOVA assumes
- Independent observations, both within and between groups.
- Roughly normal data within each group. Reasonably forgiving at moderate sample sizes.
- Similar variances across groups — homogeneity of variance. This is the one that bites. As a rough check, if the largest group standard deviation is more than about twice the smallest, the F test is on shaky ground and Welch's ANOVA or a Kruskal–Wallis test is the safer route.
- Balanced designs help. ANOVA tolerates unequal group sizes, but it is most robust to violated assumptions when the groups are of similar size.
After a significant result
A significant F is the beginning of the analysis, not the end. The natural next steps are:
- A post-hoc test — Tukey's HSD compares every pair while holding the overall error rate at 5%.
- Planned contrasts, if you decided before collecting data which comparisons mattered. These are more powerful than post-hoc tests precisely because you committed in advance.
- An effect size. η² is reported above; report it alongside the p value, not instead of it.
Common questions
What does one-way ANOVA test?
Whether three or more group means are all equal. The null hypothesis is that every group is drawn from a population with the same mean; a small p value is evidence that at least one of them is not.
Note the phrasing. ANOVA tells you that something differs. It does not tell you which group, or how many — that needs a follow-up comparison.
Why is it called analysis of variance if it compares means?
Because it answers a question about means by comparing two variances. If every group really shares a mean, then the spread between the group means and the spread within the groups are both estimating the same underlying variance, and their ratio should sit near 1.
When the groups genuinely differ, the between-group figure inflates while the within-group figure does not, and the ratio climbs. That ratio is F.
Why not just run several t-tests instead?
Because each test carries its own 5% false-positive risk, and they accumulate. Comparing four groups pairwise takes six tests, and the chance of at least one false positive rises to about 26% — you are five times more likely to be fooled than the 5% you thought you had signed up for.
ANOVA asks the whole question in a single test at a single 5% risk. That is the entire reason it exists.
The result is significant — which groups differ?
ANOVA cannot say. A significant F establishes only that the groups are not all alike. To find out which pairs differ you need a post-hoc test that corrects for multiple comparisons — Tukey's HSD is the usual choice, or Bonferroni-corrected t-tests if you have a small, pre-planned set of comparisons.
The group means and standard errors shown on the chart above are a reasonable first look, but they are not a formal test.
What is eta squared?
The proportion of the total variation explained by group membership — the ANOVA equivalent of r². An η² of 0.30 means 30% of the variation in the data is attributable to which group an observation is in, and 70% is variation within groups.
Like every effect size, it answers the question the p value ignores: not whether the difference is real, but whether it is large.
What if I only have two groups?
Then ANOVA and the pooled two-sample t-test are the same test, and F is exactly t squared. Either gives an identical p value. Use the t-test for two groups — it also gives you a confidence interval for the difference, which ANOVA does not.
Related calculators
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T-test calculator
One or two groups, with a confidence interval.
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Pooled standard deviation
The within-group spread, on its own.
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P-value calculator
Convert an F statistic into a p value.
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Variance calculator
The quantity ANOVA partitions.