One-way ANOVA, comparing multiple means
SS_between = Σ nᵢ (x̄ᵢ − x̄)²
Enter at least two groups of measurements and an alpha threshold: the tool calculates the one-way analysis of variance (between-group sum of squares, within-group sum of squares, total sum of squares, degrees of freedom, mean squares, F-statistic and p-value) then displays the complete ANOVA table and a conclusion that distinguishes absence of evidence from evidence of equal means.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasSS_between = Σ nᵢ (x̄ᵢ − x̄)²
SS_between = Σ nᵢ (x̄ᵢ − x̄)²
SS_within = Σ Σ (xᵢⱼ − x̄ᵢ)²
SS_total = SS_between + SS_within
F = MS_between / MS_within
One-way ANOVA decomposes the total variability of the data into a component explained by group membership (between-groups) and a residual component (within-groups). The ratio of their mean squares, the F-statistic, departs from 1 when groups differ more from each other than within themselves.
- 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 truly equal.
- Alpha (α)
- · significance level chosen before the test, typically 0.05: the p-value is compared to it to decide whether or not to reject the null hypothesis.
Validity domainOne-way ANOVA assumes independent groups, a distribution close to normal within each group, and comparable variances across groups.
One-way ANOVA assumes independent groups, a distribution close to normal within each group, and comparable variances across groups. With small sample sizes or very unequal variances, the p-value becomes less reliable; a Welch test or data transformation may then be preferable. Each group must contain at least two numerical values: with a single value, the within-group variance of that group is undefined.
Interpreting the resultA p-value below alpha signals a difference between group means too large to be attributed to chance: the equality hypothesis is rejected, but the test does not say which groups differ (a post-hoc test would be needed for that).
A p-value below alpha signals a difference between group means too large to be attributed to chance: the equality hypothesis is rejected, but the test does not say which groups differ (a post-hoc test would be needed for that). A p-value above or equal to alpha does not prove equal means: it only means that the data do not provide sufficient evidence against it, which may also reflect too small a sample size.