One-Way ANOVA Calculator

Compare 2 to 10 group means with a full ANOVA table, F and p-value, eta and omega squared, Welch's ANOVA, Levene's test and Holm or Bonferroni pairwise tests.

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About this tool

What is the One-Way ANOVA Calculator?

One-way analysis of variance asks whether three or more groups share the same mean. Paste the values for each group, or type the n, mean and standard deviation of each, and the One-Way ANOVA Calculator builds the full ANOVA table (sums of squares, degrees of freedom, mean squares, F and p), reports eta squared and omega squared as effect sizes, and draws the group means with 95% intervals and the F distribution with your statistic marked.

It also runs the checks and follow-ups a careful analysis needs. Welch's ANOVA gives a result that does not assume equal variances, Levene's test (median-centred, the Brown-Forsythe form) checks that assumption when you supply raw data, and a pairwise table compares every pair of groups with Holm and Bonferroni adjustments so you can see which means differ. It handles 2 to 10 groups of unequal sizes, and the data stay in your browser.

  • Full ANOVA table with SS, df, MS, F and p for 2 to 10 groups
  • Raw data or summary statistics, unequal group sizes allowed
  • Welch's ANOVA and Levene's (Brown-Forsythe) test
  • Pairwise comparisons with Holm and Bonferroni adjustments
  • Eta squared, omega squared, means plot and F curve
  • Matches R aov, oneway.test and pairwise.t.test; runs in your browser
Read the guide: One-Way ANOVA Explained: F, Effect Sizes and Post Hoc Tests

How to use the One-Way ANOVA Calculator

  1. Choose "Raw data" to paste each group's values, or "n, mean, SD" if you only have summaries.
  2. Name each group and enter its data. Use "Add group" for up to 10 groups and the bin icon to remove one; at least 2 are needed.
  3. Pick the significance level.
  4. Read F, the p-value and the effect sizes, then check Welch's F and Levene's test. If Levene's p is small or the SDs differ by more than about two to one, rely on Welch's result.
  5. If the overall test is significant, read the pairwise table to see which groups differ after the Holm adjustment, and copy or share the result.

Worked examples

PlantGrowth (the default data)

Input

ctrl, trt1, trt2, 10 plants each; means 5.032, 4.661, 5.526

Output

SS between 3.766, SS within 10.492, F(2, 27) = 4.846, p = 0.01591, η² = 0.264, ω² = 0.204

MS between = 3.766 / 2 = 1.883 and MS within = 10.492 / 27 = 0.3886, so F = 1.883 / 0.3886 = 4.846. About a quarter of the variation in weight lies between the groups.

Which groups differ?

Input

Same data, pairwise comparisons

Output

trt1 vs trt2: difference -0.865, p = 0.0134 (Holm and Bonferroni); ctrl vs trt1 and ctrl vs trt2 not significant

The overall effect comes from the two treatments pulling in opposite directions; neither differs clearly from the control on its own.

Checking the equal-variance assumption

Input

Same data

Output

Levene F(2, 27) = 1.119, p = 0.341; Welch F(2, 17.13) = 5.181, p = 0.01739

Levene's test finds no strong sign of unequal spread, and Welch's ANOVA agrees with the classic result, so the conclusion does not depend on the assumption.

From summary statistics

Input

Method A n 12, mean 72.4, SD 8.1; Method B n 12, mean 78.9, SD 7.6; Method C n 11, mean 75.2, SD 9.0

Output

F(2, 32) = 1.883, p = 0.1686; Welch F(2, 20.98) = 2.013, p = 0.1586

A 6.5-point spread between means is not enough to rule out chance with groups this size and this much scatter.

Background and insight

R. A. Fisher developed analysis of variance at the Rothamsted agricultural station in the early 1920s and set it out in Statistical Methods for Research Workers (1925). The idea is to split the total variation into two parts: variation between the group means and variation within the groups. If the groups share one true mean, the between-group mean square and the within-group mean square both estimate the same error variance, and their ratio F should be near 1. A large F means the group means are further apart than the scatter inside the groups can explain. George Snedecor named the F distribution in Fisher's honour.

Running one ANOVA instead of every pairwise t-test controls the error rate: with five groups there are ten pairs, and ten tests at the 5% level give roughly a 40% chance of at least one false positive if all means are equal. When the overall F is significant, follow-up comparisons find which groups differ. This tool uses t-tests on the pooled within-group variance with Holm's (1979) step-down adjustment, which is uniformly more powerful than Bonferroni and makes no extra assumptions. Tukey's HSD is a common alternative that this tool does not compute.

Classic ANOVA assumes equal variances. Welch (1951) proposed a weighted F test that drops the assumption, and simulation studies have shown it holds its error rate far better when variances and group sizes differ. The default data are R's PlantGrowth set (Dobson, 1983: dried plant weights under a control and two treatments), and the test suite reproduces R's aov (F = 4.846, p = 0.01591), oneway.test (Welch F = 5.181, p = 0.01739), pairwise.t.test and car::leveneTest results for it.

Common use cases

  • Experiments with several treatments

    Compare fertilisers, drug doses, teaching methods or machine settings in one test instead of many t-tests.

  • Product and UX research

    Compare task times or satisfaction scores across three or more designs or customer segments.

  • Manufacturing

    Check whether output differs between shifts, lines or suppliers.

  • Reports and coursework

    Produce the standard ANOVA table with effect sizes and post hoc comparisons, and check software output.

Tips and limitations

  • ANOVA tells you that some means differ, not which. Use the pairwise table, and only after a significant overall test if you did not plan specific comparisons in advance.
  • Watch the spreads. If the largest SD is more than about twice the smallest, or Levene's test is significant, report Welch's ANOVA.
  • Observations must be independent. Repeated measurements on the same subjects need repeated-measures ANOVA, which this tool does not perform.
  • ANOVA is fairly tolerant of non-normal data with similar, moderate group sizes. For small, skewed groups consider the Kruskal-Wallis test.
  • Report an effect size. η² is the share of variation between groups in this sample; ω² corrects its upward bias and is the better estimate for the population.
  • With two groups, one-way ANOVA is the same as the pooled t-test: F equals t².

Frequently asked questions

What does a one-way ANOVA test?

It tests the null hypothesis that all group means are equal against the alternative that at least one differs. "One-way" means there is a single grouping factor, such as treatment or region.

How is the F statistic calculated?

F = MS between / MS within. MS between is the sum over groups of n × (group mean - grand mean)², divided by k - 1. MS within is the sum of (n - 1) × SD² over groups, divided by N - k. The calculator shows every intermediate number.

Can I do an ANOVA with only means and standard deviations?

Yes. Switch to "n, mean, SD" and enter the summaries for each group. The ANOVA table, Welch's test and the pairwise comparisons are exact from summaries; only Levene's test needs the raw values.

What should I do if the variances are not equal?

Use Welch's ANOVA, which the calculator reports alongside the classic F. It does not assume equal variances and keeps its error rate close to α when group spreads and sizes differ.

Which post hoc test does this calculator use?

It runs t-tests on every pair of groups using the pooled within-group variance, with p-values adjusted by Holm's method and by Bonferroni. These match R's pairwise.t.test. Tukey's HSD is not included.

What is the difference between eta squared and omega squared?

Eta squared is SS between / SS total, the proportion of variation explained in this sample. It overstates the population effect, especially with small samples. Omega squared adjusts for that and is usually a little smaller.

Why not just run several t-tests?

Each test has its own chance of a false positive, and they add up. With 4 groups there are 6 pairs, and six tests at 5% give about a 26% chance of at least one false alarm when nothing is going on. ANOVA tests everything at once at the chosen level.

Is there an in-depth guide to the One-Way ANOVA Calculator?

Yes. “One-Way ANOVA Explained: F, Effect Sizes and Post Hoc Tests” covers the background, how it works, common pitfalls and worked examples. Read the full guide.

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