One-way Analysis of Variance

Tests in one go whether the means of three groups differ somewhere. It divides the spread between the groups by the spread within them to give an F value. Unlike running three separate t-tests, it does not inflate the chance of a false positive.

Do three fertilisers produce different yields? Comparing three group means is not a job for three separate t-tests. Allowing a 5% error rate on each test lets the chance of erring somewhere across the three grow to $1 - 0.95^3$ , about 14.3%. Analysis of variance handles all the groups at once.

The idea is to split the total variation in two: the variation between the group means, and the variation within each group. If the groups genuinely differ, the former will be large relative to the latter.

F = \dfrac{\dfrac{SS_b}{k - 1}}{\dfrac{SS_w}{N - k}}

$SS_b$ is the between-groups variation: each group mean's squared distance from the grand mean, weighted by group size, summed
$SS_w$ is the within-groups variation: each value's squared distance from its own group mean, summed
$k$ is the number of groups and $N$ the total number of values

Each is divided by its degrees of freedom before the ratio is taken. In effect the within-group variation becomes a ruler, and F measures how many times larger the between-group variation is.

Example

Take the defaults: group A as 12, 15, 18, 20, group B as 22, 25, 27, 24, and group C as 17, 19, 16, 21.

The group means are 16.25, 24.5 and 18.25, with a grand mean of about 19.6667. The between-groups variation is about 148.1667 and the within-groups variation is 64.5.

The degrees of freedom are $3 - 1 = 2$ between and $12 - 3 = 9$ within. F is $(148.1667 \div 2) \div (64.5 \div 9)$ , about 10.3372.

The p-value is about 0.004660 against a critical value of about 4.2565. F clears it comfortably, so at the 5% level the three groups differ somewhere.

Points to watch

Analysis of variance says only that a difference exists somewhere. It does not identify whether A differs from B or B from C. Finding that out requires a follow-up multiple comparison procedure such as Tukey's method.

This calculator is fixed at three groups; the count cannot be changed.

It assumes each group is normally distributed and that the groups share a common spread. When the spreads differ substantially, an alternative such as Welch's analysis of variance is needed.

With only two groups, analysis of variance and the t-test always agree. In fact the F value for two groups is exactly the square of the t value.