Track 10 · Comparing Groups and Categorical Data · Lesson 4 of 11

One-way ANOVA

Test whether several group means differ with one-way ANOVA, explain the F ratio as between- versus within-group variation, check its assumptions and report η² and ω².

40 minIntermediate8 graded exercisesMembers

A retail chain wants to know whether weekly sales differ between its small, medium and large stores — and, more subtly, between its four regions. Comparing three or four means at once needs a method that looks at all of them together, and an honest check that the observations really are independent.

One-way analysis of variance (ANOVA) compares the means of $k \ge 2$ independent groups defined by one categorical factor.

$H_0: \mu_1 = \mu_2 = \cdots = \mu_k$ (all population means equal).
$H_1$ : at least one mean differs. (Not “all means differ” — ANOVA does not say which ones; that is the job of post-hoc comparisons.)

Despite its name, ANOVA tests means. It does so by comparing two kinds of variation.

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Explain the F statistic as between-group variation relative to within-group variation
Compute an ANOVA table from data or from group summaries
Check independence from the design, and normality and equal variances from the data (Brown–Forsythe, Welch's ANOVA)
Report η² or ω² with F and p, and recognise pseudo-replication

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Formula sheet

Sums of squares, mean squares and the F statistic
$SS_B = \sum_{i=1}^{k} n_i(\bar{x}_i - \bar{x})^2, \quad SS_W = \sum_{i=1}^{k}\sum_{j}(x_{ij} - \bar{x}_i)^2 = \sum_{i=1}^{k}(n_i - 1)s_i^2, \quad F = \frac{SS_B/(k-1)}{SS_W/(N-k)} = \frac{MS_B}{MS_W}$
Effect sizes for one-way ANOVA
$\eta^2 = \frac{SS_B}{SS_{total}}, \qquad \omega^2 = \frac{SS_B - (k-1)\,MS_W}{SS_{total} + MS_W}$

Reference

F distribution
The F distribution is the distribution of a ratio of two independent chi-square variables, each divided by its degrees of freedom; it is used for comparing variances and in ANOVA and regression.
One-way ANOVA
Tests whether k ≥ 2 group means are all equal by comparing variation between group means with variation within groups.
Welch's ANOVA
A one-way ANOVA that does not assume equal group variances, weighting each group by n/s².
Levene's and Brown–Forsythe tests for equal variances
Test whether several groups have equal population variances by running a one-way ANOVA on absolute deviations from each group's mean (Levene) or median (Brown–Forsythe).

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.