One Way ANOVA

Comparing group means

One way analysis of variance tests whether independent groups have equal population means. Each line represents a group and group sizes may differ. The calculation starts with group means and the overall mean of all observations.

Sums of squares and example

The between groups sum of squares weights each squared difference between a group mean and the overall mean by the group size. The within groups sum of squares uses deviations of observations from their own group mean. Divide each sum by its degrees of freedom to obtain mean squares, then take their ratio for F. For groups 1,3 and 3,5 both sums of squares are 4, F is 2 and p is about 0.292893.

Limits

This classical method assumes equal variances and independent observations. A small p value indicates that at least one mean differs, but does not identify a particular pair. Repeated measures and multiple comparison procedures are not included. Check model assumptions and practical effect sizes separately.

Follow the sums through to the F ratio

Enter groups on separate lines and use the same measurement unit throughout. Rescaling one group's values but leaving others unchanged creates an artificial difference in means. Distinguish the group count from the observation count. Degrees of freedom combine these counts differently, and each group's own size remains part of the calculation.

Choose rows 1 2 3, 4 5 6 and 7 8 9. Group means are 2, 5 and 8, with grand mean 5. Between group sum of squares is 3 × (9 + 0 + 9) = 54. Each group's squared deviations from its own mean sum to 2, so the within group total is 6. Degrees of freedom are respectively 2 and 6. Mean squares are 27 and 1, giving F = 27. With numerator degrees of freedom 2, the upper tail is (d₂/(d₂ + 2F))^(d₂/2). Here d₂ = 6 and F = 27 give (6/60)³ = 0.001.

The result does not say that every pair of groups differs. The F ratio assesses between group variation relative to within group variation. A large ratio can provide evidence against a common mean when assumptions hold. The direction and magnitude of differences, and which comparisons are needed, require separate examination.

Frequently asked questions

How do you calculate one way ANOVA?

For example, dividing between group mean square by within group mean square gives F = 1 and p = 0.4648 for groups [0 2], [1 3] and [2 4]. Classical one way ANOVA assumes independent observations, equal population variances and normally distributed errors, and does not identify a differing pair or perform multiple comparisons.

Calculate with these values