What this calculator does
One-way ANOVA tests whether three or more group means differ, by comparing the variation between groups against the variation within them. Three groups of five observations give an F-statistic of 15.272 on 2 and 12 degrees of freedom.
The logic is a ratio. If the groups really are the same, the spread between their means should be about the same size as the spread inside each group, and F should be near 1. When F is much larger than 1, the group means are further apart than within-group noise can explain. Identical groups give exactly zero.
The formula
The between-group sum of squares measures how far each group mean sits from the overall mean, weighted by group size. The within-group sum of squares measures variation inside the groups. Each is divided by its degrees of freedom to give a mean square, and F is the ratio of between to within. The test assumes roughly normal data with similar variances across groups.
| Term | Meaning |
|---|---|
| F-statistic | The ratio of between-group to within-group mean square. Near 1 under the null. |
| SS between | Variation explained by group membership. |
| SS within | Variation remaining inside the groups, the residual noise. |
| Post hoc test | A follow-up needed to identify which specific groups differ, since ANOVA only says that some do. |
The inputs explained
| Field | What to enter |
|---|---|
| Groups, separated by ; (values comma separated within each) | Groups separated by semicolons, with values comma separated inside each group. Groups need not be the same size. |
When to use it
Comparing three or more treatments
Running separate t-tests on every pair inflates the false positive rate, which is the problem ANOVA solves.
Testing across categories
Whether a continuous outcome differs across several groups, such as regions, suppliers or conditions.
Screening before detailed analysis
A non-significant ANOVA suggests no pairwise comparison is worth pursuing.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does group separation affect F?
Groups ranging from identical to clearly separated.
| Groups | F-statistic | SS between, SS within | Degrees of freedom (between, within) |
|---|---|---|---|
| 20,21,22,23,24; 20,21,22,23,24; 20,21,22… | 0 | 0, 30.000 | 2, 12 |
| 21,22,23,24,25; 22,23,24,25,26; 23,24,25… | 2.000 | 10.000, 30.000 | 2, 12 |
| 23,25,29,34,28; 19,20,22,24,21; 30,33,31… | 15.272 | 276.933, 108.800 | 2, 12 |
Questions
Why not just run multiple t-tests?
Because each test carries its own false positive risk, and running several compounds it. Three groups means three pairwise tests and roughly a 14% chance of at least one false positive at the 5% level. ANOVA tests everything at once at the stated level.
What does a significant ANOVA tell me?
Only that at least one group differs from at least one other. It does not say which, or how many, or in what direction. Identifying the specific differences needs a post hoc test such as Tukey HSD, which handles the multiple comparison problem properly.
What assumptions does ANOVA make?
Independent observations, roughly normal data within each group, and similar variances across groups. It is reasonably robust to mild departures, especially with equal group sizes. For badly unequal variances, Welch ANOVA is the safer choice.
What is a large F value?
Anything meaningfully above 1, judged against the critical value for your degrees of freedom. At (2, 12) the 5% threshold is 3.89. An F below 1 means the groups differ less than within-group noise alone would produce, which is entirely consistent with no real difference.
For comparing exactly two means, see the t-statistic calculator. For comparing two variances, see the F-test calculator.