One-Way ANOVA Calculator

Compare the means of several independent groups by one-way ANOVA, with the full ANOVA table, F, p-value, effect sizes η² and ω², Levene's test and a summary of each group.

Result

Deterministic calculation

F statistic

F(2, 9) = 35.722

p-value: < 0.0001
Effect size, η² (ω²): 0.888 (0.853)
Total sample size, groups: 12, 3
Levene's test for equal variances: F(2, 9) = 0.098, p = 0.908

Analysis of variance
Source	Sum of squares	df	Mean square	F	p
Between groups	9.7	2	4.85	35.722	< 0.0001
Within groups	1.222	9	0.1358
Total	10.92	11

Groups
Group	n	Mean	SD	SE
Control	4	4.65	0.3873	0.1936
Low dose	5	5.94	0.3362	0.1503
High dose	3	7	0.4	0.2309

Note

The p-value is the probability, if the null hypothesis and every assumption of the test were true, of a result at least as extreme as the one observed. It is not the probability that the hypothesis is true, and it says nothing about how large or important an effect is. Judge the effect size and its interval, the design and the assumptions along with it, and do not treat a p-value on one side of 0.05 as proof of an effect or of its absence.

Note

The F test asks whether the group means are all equal. A small p-value says they are probably not all equal; it does not say which differ. Comparing pairs of groups afterwards needs a procedure that controls the error rate across the comparisons (for example Tukey's HSD), which this tool does not do, and running many unadjusted t-tests on the same data inflates false positives.

Note

Assumptions: independent observations, roughly normal values within each group, and similar variances across groups. Levene's test is shown as a check, but with small groups it has little power, so a large p does not prove the variances are equal. If the variances clearly differ, Welch's ANOVA, which is not included here, is more appropriate.

Significant figuresScientific notation

Calculation

one-way ANOVA: SSbetween = Σ nᵢ (meanᵢ − grand mean)², SSwithin = Σ (x − meanᵢ)², F = MSbetween / MSwithin
η² = SSbetween / SStotal; ω² = (SSbetween − dfbetween MSwithin) / (SStotal + MSwithin), a less biased estimate
Levene's test (median-centred) is an ANOVA on each value's distance from its group median

Formula

F = \dfrac{MS_{\text{between}}}{MS_{\text{within}}}

SS_{\text{between}} = \sum n_i(\bar{x}_i - \bar{\bar{x}})^2;\quad SS_{\text{within}} = \sum (x - \bar{x}_i)^2

\eta^2 = \dfrac{SS_{\text{between}}}{SS_{\text{total}}};\quad \omega^2 = \dfrac{SS_{\text{between}} - df_{\text{between}}\,MS_{\text{within}}}{SS_{\text{total}} + MS_{\text{within}}}

$\mathrm{MS}$: mean square: a sum of squares divided by its degrees of freedom
$\eta^2,\ \omega^2$: the share of the total variance associated with the group, and a less biased estimate of it

How it works

One-way ANOVA compares three or more group means. It splits the total variation into variation between group means and variation within groups. If the group means are equal, the ratio of the two mean squares, F, is expected to be near one; a large F says the means differ by more than the scatter within groups makes likely.

The F test says that not all the means are equal, but not which ones differ. Comparing individual pairs afterwards needs a procedure that controls the error rate across the comparisons, such as Tukey's HSD, which is not included here; running unadjusted t-tests on every pair inflates false positives. Levene's test is shown as a check on the equal-variance assumption.

Worked example

Three groups: control (n = 4, mean 4.65), low dose (n = 5, mean 5.94) and high dose (n = 3, mean 7.00).

SSbetween = 9.70 (2 df) and SSwithin = 1.22 (9 df), so MSbetween = 4.85 and MSwithin = 0.136.
F(2, 9) = 35.72, p < 0.0001; η² = 0.888, ω² = 0.853.
Levene's test: F(2, 9) = 0.098, p = 0.91.

The group means are not all equal, with a large effect (η² = 0.89), and there is no sign of unequal variances.

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

Independent observations, in independent groups.
Roughly normal values within each group.
Similar variances across groups (Levene's test is a rough check only, since it has little power with small groups).

Common mistakes

Reading a small p-value as showing that every group differs.
Running t-tests on all pairs without adjusting for the number of comparisons.
Using it for repeated measures on the same subjects, which are not independent.

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One-Way ANOVA Calculator

Result

Calculation

Formula

How it works

Worked example

Assumptions

Common mistakes

Related tools

Result

Calculation