Skip to content
CALX
Appearance

Appearance

Light or dark

Colour
Accessibility

CategoriesSearch

One-way ANOVA calculator, comparing multiple means

Comparison of means across several groups with the complete ANOVA table: sums of squares, F, p-value and cautious conclusion.

Parameters

Separate values with spaces, line breaks or semicolons. A comma is read as a thousands separator (1,234) when unambiguous, and rejected otherwise.

A value strictly between 0 and 1, for example 0.05.

Result

Awaiting the quantities required for the calculation.

Calculation, assumptions and sources

OverviewCompares the means of at least two groups by one-way ANOVA: sums of squares, degrees of freedom, mean squares, F statistic and p-value.

Compares the means of at least two groups by one-way ANOVA: sums of squares, degrees of freedom, mean squares, F statistic and p-value.

Method & formulasSS_between = Σ nᵢ (x̄ᵢ − x̄)²
SSbetween = Σ nᵢ (x̄ᵢ − x̄)²SSwithin = Σ Σ (xᵢⱼ − x̄ᵢ)²SStotal = SSbetween + SSwithinF = MSbetween / MSwithin

ANOVA splits the variability into a between-group and a within-group component. The F statistic is the ratio of the corresponding mean squares.

k
number of groups being compared.
N
total sample size, all groups combined.
df between
between-group degrees of freedom, k − 1.
df within
within-group degrees of freedom, N − k.
MS
mean square: sum of squares divided by its degrees of freedom.
P-value
probability of observing an F-statistic at least as large if the population means were equal.
Alpha (α)
significance level set before the test, typically 0.05.
Validity domainThe test assumes independent groups, a distribution close to normal within each group and comparable variances.

The test assumes independent groups, a distribution close to normal within each group and comparable variances. With small samples or very unequal variances, the p-value is less reliable; a Welch test or a data transformation may be preferable. Each group needs at least 2 values, otherwise its variance is undefined.

Interpreting the resultIf p < α, the hypothesis of equal means is rejected.

If p < α, the hypothesis of equal means is rejected. ANOVA does not show which groups differ; a post-hoc test does. A small sample can hide a difference between the means.