One-way ANOVA calculator with post hoc tests

Name: One-way ANOVA calculator with post hoc
Author: Richard Injante

Compare the means of three or more groups: ANOVA, equal-variance check, Tukey or Games-Howell, effect size, tables and a box plot.

Content reviewed by Richard Injante Published by SolucionesAhora.com Updated on 2026-10-08 How we verify

Calculator inputs

Enter your figures, or press “See example” to watch it work.
If your case is unusual, open “Advanced options”.
Read the result and, below it, what it means and how it was worked out.

Data for each group

Enter one group per line, with values separated by spaces or commas and a decimal point. You can put the group name before a colon (“Control: 4.2 5.1 3.9”); otherwise they are called Group 1, Group 2… You need at least two groups with two values each, and up to twelve groups.

One group per line

Significance level (alpha) (optional)

The threshold used to decide whether a difference is significant. 0.05 is the habit; 0.01 is stricter.

Advanced options

Pairwise comparisons (post hoc) (optional)

“Automatic” uses Tukey if variances are homogeneous (Brown-Forsythe test) and Games-Howell if they are not. Bonferroni and Holm correct simple t tests using the pooled variance: they are more conservative than Tukey.

See the result

Results

F statistic80.529

p valuep < 0.001

Between-groups df2

Error df15

Eta squared (η²)0.915

Omega squared (ω²)0.898

Equal variances (Brown-Forsythe)p = 0.125

Welch’s F74.057

Welch df9.46

Welch p valuep < 0.001

Post hoc appliedTukey

Descriptives by group
Group	n	Mean	SD
Control	6	4.583	0.471
Dosis baja	6	5.933	0.273
Dosis alta	6	7.083	0.232

Pairwise comparisons
Comparison	Difference	Adjusted p	Cohen’s d	Significant?
Control − Dosis baja	-1.35	0	-3.952	Yes
Control − Dosis alta	-2.5	0	-7.319	Yes
Dosis baja − Dosis alta	-1.15	0.0001	-3.367	Yes

See the working step by step

Number of groups = 3; total participants = 18; grand mean = 5.87
F(2, 15) = 80.529, p < 0.001
η² = 0.915; ω² = 0.898
Brown-Forsythe: p = 0.125
Welch’s ANOVA: F(2; 9.46) = 74.057, p < 0.001
Post hoc Tukey: 3 significant out of 3 comparisons

In short

What it calculates: Compare the means of three or more groups: ANOVA, equal-variance check, Tukey or Games-Howell, effect size, tables and a box plot.
Formula used: Between-groups sum of squares: SSB = Σ nj × (mean_j − grand mean)²
Example: Example 1. A significance level of 0.01 with three groups.

One-way ANOVA calculator with post hoc

With three or more groups, the one-way ANOVA is the standard way to ask whether their means differ. But a significant ANOVA only says that some group is different; to find out which, you need pairwise comparisons.

Paste each group’s data, one per line. The calculator runs the ANOVA, checks whether variances are equal, computes the effect size and picks the right post hoc test. Everything runs in your browser: the data is never sent to a server.

How it works

ANOVA compares the variation between group means with the variation within each group. If the first is much larger than the second, the F statistic is large and the p value small.

It then checks the equal-variance assumption with the Brown-Forsythe test. If it holds, pairwise comparisons use Tukey; if not, Games-Howell, which needs neither equal variances nor equal sizes. Welch’s ANOVA is also shown as an alternative to the classic F.

Formula

Between-groups sum of squares: SSB = Σ nj × (mean_j − grand mean)²
Within-groups sum of squares: SSW = Σ (nj − 1) × s_j²
F = (SSB ÷ (k − 1)) ÷ (SSW ÷ (N − k))

Effect size:
η² = SSB ÷ (SSB + SSW)
ω² = (SSB − (k − 1) × MSW) ÷ (SSB + SSW + MSW)

Brown-Forsythe: ANOVA on |value − group median|
Welch’s ANOVA: weights w_j = nj ÷ s_j²

Tukey: q = |diff| ÷ √(MSW ÷ 2 × (1 ÷ ni + 1 ÷ nj)); p = 1 − F_q(q; k, N − k)
Games-Howell: q = √2 × |diff| ÷ √(s_i²÷ni + s_j²÷nj); with Welch’s degrees of freedom

Worked example

Example 1. A significance level of 0.01 with three groups.
Five people per group. F(2, 12) = 16.52, p < 0.001, η² = 0.734. The variances are equal (Brown-Forsythe p = 0.824), so Tukey is used. At alpha 0.01 only 1 of the 3 pairwise comparisons is significant, even though the ANOVA is very significant.

Example 2. Variances that differ a lot.
Three groups of six with very different spreads. Classic ANOVA gives F(2, 15) = 5.51, p = 0.016, but Brown-Forsythe gives p = 0.009. Welch’s ANOVA, F(2, 6.81) = 3.43, p = 0.093, is not significant, and Games-Howell finds no significant pair. The classic ANOVA was too optimistic.

Explanation

What ANOVA does and does not do

One-way ANOVA tests the hypothesis that all means are equal. If it is significant, it says at least one mean differs, not which. To find out, run pairwise comparisons with a correction that controls false positives: three groups mean three pairs, five groups ten.

ANOVA assumptions

Independent observations within and between groups.
Roughly normal residuals: with groups of 15–20 or more, ANOVA tolerates moderate departures well.
Equal variances across groups. This is the assumption most often broken and the one that does most harm when group sizes differ. Brown-Forsythe tests it using the median, so it resists skewed data.

When variances are unequal, the alternative is Welch’s ANOVA for the global test and Games-Howell for the pairs.

Which post hoc test to choose

Tukey: the standard with equal variances; it controls error across all pairs at once. With unequal sizes it is called Tukey-Kramer.
Games-Howell: needs neither equal variances nor equal sizes. The safe choice when Brown-Forsythe is significant.
Bonferroni: multiplies each p value by the number of comparisons. Simple and conservative.
Holm: like Bonferroni but stepwise; same error control and more powerful. Almost always preferable.

How to report it

A typical sentence: “ANOVA showed significant differences between groups, F(2, 15) = 80.53, p < .001, η² = .91. Tukey comparisons showed that...”. Always report the effect size: the p value only says whether there is a difference, not how big.

Common mistakes

Running several t tests instead of one ANOVA: it inflates false positives.
Reading a significant ANOVA as meaning every group differs from every other.
Applying Tukey with very unequal variances and group sizes.
Leaving out the effect size.
Confusing η² with ω²: η² overestimates in small samples; ω² corrects the bias.

Limits of this calculator

It is a one-way ANOVA for independent groups. It does not do repeated measures, two-way ANOVA, ANCOVA or mixed models.
It does not check residual normality or flag outliers: look at the box plot.
Each pair’s Cohen’s d uses the pooled standard deviation of all groups.
With very small samples the p values may be unreliable.

Notice

This is an estimate. For publication, confirm the analysis with a statistics package and justify the assumptions you checked.

Frequently asked questions

What is a one-way ANOVA?

A test that compares the means of three or more independent groups at once. It tests whether all are equal; if significant, at least one differs.

Why not run several t tests?

Because each test carries its own false-positive risk and they add up: with ten comparisons, the chance of an error somewhere exceeds 40%. ANOVA and post hoc tests control that risk.

When do I use Tukey and when Games-Howell?

Tukey when variances are equal; Games-Howell when they are not, or group sizes differ a lot. The automatic option decides using the Brown-Forsythe test.

What is eta squared?

The proportion of total variation explained by group. Commonly read as 0.01 small, 0.06 medium and 0.14 large. Omega squared is a less biased version.

What if variances are unequal?

Use Welch’s ANOVA for the global test and Games-Howell for the pairs. This calculator shows both results.

How many groups can I compare?

Up to twelve groups, with at least two values each. With two groups, ANOVA is equivalent to Student’s t test.

Is my data sent to a server?

No. The calculation runs in your browser and nothing leaves your device.

How do I report the result?

Give F with its degrees of freedom, the p value, the effect size (η² or ω²) and the post hoc test used, for example: F(2, 15) = 80.53, p < .001, η² = .91.

Sources and references

The formula behind this calculator is not ours. These are the sources it comes from, in APA 7 format, in case you want to go to the original or cite them in your own work.

Tukey, J. W. (1949). Comparing individual means in the analysis of variance. Biometrics, 5(2), 99–114. https://doi.org/10.2307/3001913 https://doi.org/10.2307/3001913 The source of Tukey’s test for comparing all pairs of means.
Games, P. A., & Howell, J. F. (1976). Pairwise multiple comparison procedures with unequal n’s and/or variances: A Monte Carlo study. Journal of Educational Statistics, 1(2), 113–125. https://doi.org/10.3102/10769986001002113 https://doi.org/10.3102/10769986001002113 The post hoc for groups with unequal sizes or variances.
Welch, B. L. (1951). On the comparison of several mean values: An alternative approach. Biometrika, 38(3–4), 330–336. https://doi.org/10.2307/2332579 https://doi.org/10.2307/2332579 Welch’s ANOVA, which does not require equal variances.
Lakens, D. (2013). Calculating and reporting effect sizes to facilitate cumulative science: A practical primer for t-tests and ANOVAs. Frontiers in Psychology, 4, Artículo 863. https://doi.org/10.3389/fpsyg.2013.00863 https://doi.org/10.3389/fpsyg.2013.00863 How to calculate and report eta squared and omega squared.

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