Whether any group differs — which is the only question this answers.
Comparing three or more groups
An optional name, a colon, then the values. F is large. Which pair differs? ANOVA does not say.
F(2, 12) over 3 groups and 15 observations
32.000000
p = 1.550e-5. At least one group mean differs from the others — and the test does not say which.
Between-group MS
80.00000
2 df
Within-group MS
2.50000
12 df
η² (eta squared)
0.842105
share of variance between groups
ω² (omega squared)
0.805195
bias-corrected
Control
7.00000
n = 5, SD 1.5811
Drug A
11.00000
n = 5, SD 1.5811
Drug B
15.00000
n = 5, SD 1.5811
The F test says something differs. It does not say what. The pairwise gaps are Control vs Drug B: -8.0000, Control vs Drug A: -4.0000, Drug A vs Drug B: -4.0000 — but the F statistic pooled every group into one number and cannot be read backwards to rank them. Identifying which pairs differ needs a post-hoc test (Tukey’s HSD is the usual choice), and eyeballing the largest gap is exactly the informal multiple comparison that the correction exists to prevent.
Comparing 3 groups with pairwise t-tests instead would mean 3 tests and a 14.26% chance of at least one false positive. That is the number ANOVA exists to control: one test at 5% instead of 3 tests each at 5%. If you do run them, the Bonferroni threshold for this many comparisons is 0.016667 rather than 0.05 — conservative, but honest about how many chances you took.
η² says 84.2% of the total variation is between groups; ω² says 80.5%. The gap is bias. Eta squared is computed from the sample and includes the between-group variation that appears by chance, so it always overstates — badly with small groups. Omega squared subtracts the expected chance component and is the one to report. Neither is affected by the sample size the way the p-value is, which is the same distinction the effect size calculator makes for two groups.
What this tool shows
A significant F says something differs and never says what. So this prints every pairwise gap alongside, and the family-wise error rate you would incur by testing them instead — 14.26% for three groups, 90.06% for ten.
The full F table: sums of squares, degrees of freedom, mean squares
Eta squared and omega squared, which differ by a real bias
Every group’s mean, n and standard deviation
All pairwise gaps, since F cannot rank them
The family-wise error rate for your number of groups
The Bonferroni threshold, if you run them anyway
Full F table Post-hoc gap shown ω² beside η² F = t² at k = 2
Ten groups compared pairwise: a 90% chance of at least one false positive.
Updated 8 September 2026 · Works in any browser, no installation
ANOVA compares the variation between group means with the variation inside the groups. If the first is large relative to the second, at least one group differs. That is the entire claim — it does not identify which group, and it is not a refinement of the output but a property of the statistic.
At a glance
Formula shown
F = MS_between ÷ MS_within, where MS_between = SS_between/(k − 1) and MS_within = SS_within/(N − k). Under the null both estimate the same variance, so F is near 1. Eta squared = SS_between/SS_total overstates the effect; omega squared = (SS_between − df_between·MS_within)/(SS_total + MS_within) corrects it.
Scenario support
Comparing three or more treatments, groups or conditions on one measurement; testing whether a categorical factor matters; screening before post-hoc comparisons; any case where running several t-tests would be the alternative.
Educational estimate
Planning support from the values you enter — not professional advice.
It tells you something differs, and refuses to say what
This is not a shortcoming of the software. It is what the statistic is.
F pools every group into a single number. The between-group sum of squares adds up how far each group mean sits from the grand mean, across all of them at once. Three groups at 5, 10 and 15 produce the same between-group variation as three at 15, 10 and 5, and F cannot distinguish those arrangements because it never held them separately.
So a significant F is the beginning of the analysis. The tool prints every pairwise gap so you can see the candidates, and states plainly that the F test did not rank them.
Picking the largest gap by eye is the trap. That is an informal multiple comparison — you looked at all of them and chose the most extreme — and it carries exactly the inflated error rate that a post-hoc correction exists to control. The maximum of several differences is larger than any one of them by construction.
Tukey’s HSD is the standard next step when every pair matters, because it controls the error rate across all comparisons simultaneously. Dunnett’s test is tighter when you only care about comparisons against a control, since it makes fewer of them.
A caution that surprises people: a significant F does not guarantee that any pair is significantly different under a post-hoc test, and a non-significant F does not guarantee that no pair is. The tests control different error rates over different families of hypotheses, so they can disagree. That is not a contradiction; it is what “family-wise” means.
Why not just run t-tests
The obvious alternative — compare every pair with a t-test — is what ANOVA exists to prevent, and the cost is quantifiable.
Each test at α = 0.05 has a 5% false-positive rate. Several tests compound. Three groups means three pairwise tests, and the chance of at least one false positive is 1 − 0.95³ = 14.26%. Five groups is ten tests and 40.13%. Ten groups is 45 tests and 90.06% — a near-certainty of finding something that is not there.
ANOVA replaces all of them with one test at 5%. That is the whole point: the family-wise error rate is controlled by construction rather than by correction.
If you do run pairwise tests, Bonferroni is the simplest fix — divide α by the number of comparisons. With three groups that is 0.0167. It is conservative, and it is honest about how many chances you took. The tool prints the threshold for your k.
The same arithmetic explains a broader problem. Any analysis that tries many hypotheses and reports the significant ones faces this, whether or not the tests are formally pairwise. Testing 20 outcomes at 5% gives a 64% chance of at least one false positive, which is why pre-registering the primary outcome matters more than any correction applied afterwards.
With two groups, F is exactly t squared
The tool keeps the two-group case enabled deliberately, because the identity is the clearest way to see what ANOVA generalises.
Load the “Two groups” preset. F = 16.000000, and the t statistic is 4.000000. Exactly, not approximately — and the two tests return identical p-values, because the F distribution with 1 numerator degree of freedom is the square of the t distribution.
So ANOVA is not an alternative to the t-test; it is the same comparison extended. The question “should I use a t-test or ANOVA?” has no content at two groups. It has content at three, where a t-test cannot be applied without the multiplicity problem above.
The identity also explains the assumptions. ANOVA inherits the t-test’s requirements: independent observations, roughly normal residuals, and similar variances across groups. The first is the one that actually matters — repeated measures on the same subjects break it entirely, and that needs a repeated-measures design rather than this test.
Unequal variances are the common violation, and ANOVA is reasonably robust to them when the groups are the same size. When they are not, the test can be badly off, and Welch’s ANOVA is the right substitute — the same adjustment the Welch t-test makes for two groups.
For non-normal data, the Kruskal-Wallis test is the rank-based counterpart, standing to ANOVA as the Mann-Whitney test stands to the t-test — and carrying the same caveat about what a rank test actually tests.
Eta squared flatters; omega squared does not
F and its p-value depend on the sample size, so neither says how large the difference is. Two effect sizes are standard, and one of them is biased.
Eta squared is the share of total variation that sits between groups — SS_between / SS_total. Simple, and systematically too large: the between-group sum of squares picks up variation that arises by chance even when the groups are identical, and eta squared counts all of it as effect.
Omega squared subtracts the expected chance component. It is the estimate to report, and on small datasets the difference is substantial — the tool prints both so the bias is visible rather than assumed. With large samples they converge.
Neither is affected by sample size the way p is. That is the same argument the effect size calculator makes for two groups: with enough data every trivial difference becomes significant, and only the effect size distinguishes trivial from important.
One further caution about interpreting either. They describe how much of the variance the grouping accounts for in this sample, with these group sizes and this range of conditions. A designed experiment with extreme groups will show a much larger eta squared than an observational study of the same effect, for the same reason a restricted range shrinks R². They are not comparable across designs.
Three assumptions, and only one of them usually matters
ANOVA inherits the t-test's requirements, and they are not equally fragile. Treating them as a checklist of three equal items is how the important one gets missed.
Normality is the least important. The test concerns the sampling distribution of group means, not the data, so the central limit theorem does most of the work once the groups reach moderate size. A significant normality test on large groups detects a deviation too small to affect the result.
Equal variances matter when the groups are unequal in size. With balanced groups ANOVA tolerates a fair amount of heterogeneity. With unbalanced groups it does not: a large variance paired with a small group inflates the false-positive rate, and a large variance with a large group makes the test conservative. Welch's ANOVA drops the assumption entirely and costs almost nothing when it holds, which makes it a reasonable default rather than a fallback.
Independence is the one that actually breaks, and no sample size repairs it. Repeated measures on the same subjects, students within classrooms, measurements taken in time order, several readings from one machine — correlated observations carry less information than their count suggests, so the within-group mean square underestimates the real variability and F comes out too large. The result is confidently wrong.
The diagnostic is the design, not a test. No statistic run on the data will reveal that two rows came from the same person; only knowing how the data was collected will. If observations cluster, the analysis needs a repeated-measures or mixed model rather than this one.
And for genuinely non-normal small samples, the Kruskal-Wallis test is the rank-based counterpart — standing to ANOVA as the Mann-Whitney test stands to the t-test, and carrying the same caveat about what a rank test actually compares.
Method. The sums of squares are accumulated from the group means directly rather than through the computational shortcut, which loses precision on data far from zero for the same reason the correlation engine avoids it. The F tail probability comes from the regularised incomplete beta function rather than a table, so it is exact at any degrees of freedom. Omega squared is computed from the mean squares rather than approximated from eta squared, so the two are independent estimates and their gap is real. The suite asserts that F equals t² exactly at two groups across a grid of datasets, that the family-wise error rate is 14.26% at three groups and 90.06% at ten, that eta squared always exceeds omega squared on the same data, and that the sums of squares partition exactly — SS_between plus SS_within equals SS_total to machine precision. That engine is verified on every change against 115 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
t-testOne-sample, two-sample and paired t-tests defaulting to Welch, with Student's pooled version printed beside it — and a warning when the two disagree on the verdict.
Effect SizeCohen d, Hedges g and the overlap between groups, with a sample-size control that moves the p-value while leaving the effect size fixed — the same d gives t = 1.29 at n=30 and 23.57 at n=10,000.
p-valueA p-value from a t or z statistic, one- or two-tailed — with a panel that holds an effect fixed and grows the sample, so you can watch significance appear from nothing but n.
Mann-Whitney UMann-Whitney U with the tie correction and both medians shown, because it is not a test of medians — two samples with identical medians of 4 are separated at p = 0.01.
Critical ValueCritical values for z, t, chi-square and F at any alpha and any degrees of freedom — with one- and two-tailed values shown together, because reading the wrong column of a printed table is the classic error.
VarianceSample and population variance from your data, with a live simulation that shows exactly how much the wrong divisor costs — 20% low at n = 5, closing as the sample grows.
An educational tool. ANOVA assumes independent observations, roughly normal residuals and similar variances across groups; repeated measures on the same subjects break the first assumption entirely and need a different design.
Published a one-way ANOVA calculator that prints every pairwise gap and states plainly that the F statistic cannot rank them — a significant F says at least one group differs and identifies none, which is a property of the statistic rather than a gap in the output.
Quantifies the alternative: comparing groups with pairwise t-tests instead means a 14.26% chance of at least one false positive at three groups and 90.06% at ten. The Bonferroni threshold for the current number of comparisons is shown beside it.
Keeps the two-group case enabled deliberately, because F is exactly the square of the t statistic there — 16.000000 and 4.000000 on the preset — which is the clearest way to see that ANOVA generalises the t-test rather than replacing it.
Reports omega squared beside eta squared, since eta squared includes the between-group variation that arises by chance and always overstates the effect.
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