One-Way Analysis of Variance (ANOVA)#

The one-way ANOVA (Analysis of Variance) tests whether the means of three or more independent groups differ significantly. It is the extension of the independent samples t-test to more than two groups.

When to Use#

Use the one-way ANOVA when you want to:

Compare three or more independent groups on a metric variable
The dependent variable is metric (continuous)
The data in all groups are approximately normally distributed
The variances across groups are approximately equal (homoscedasticity)

Important: The ANOVA only tests whether any difference exists between the groups (omnibus test). It does not indicate which groups differ. Post-hoc tests are required for this (e.g., Tukey HSD, Bonferroni).

Assumptions#

Independence of observations (between and within groups)
Metric scale of the dependent variable
Normal distribution in each group (Shapiro-Wilk test per group)
Homogeneity of variance (Levene's test) – if violated: Welch's ANOVA

Note: The ANOVA is relatively robust against mild violations of the normality assumption, especially with large and equal sample sizes. For severe violations, the Kruskal-Wallis test is the appropriate nonparametric alternative.

Formula#

The test statistic is based on the ratio of variance between groups to variance within groups:

F = \frac{MS_{between}}{MS_{within}}

The mean squares are calculated from the sums of squares:

MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{\sum_{j=1}^{k} n_j (\bar{X}_j - \bar{X})^2}{k - 1}

MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{\sum_{j=1}^{k} \sum_{i=1}^{n_j} (X_{ij} - \bar{X}_j)^2}{N - k}

where:

$k$ is the number of groups
$n_j$ is the size of the j-th group
$N$ is the total sample size
$\bar{X}_j$ is the mean of the j-th group
$\bar{X}$ is the grand mean

The test statistic follows an F-distribution with $df_1 = k - 1$ and $df_2 = N - k$ degrees of freedom.

Example#

Practical Example: Comparing Teaching Methods

An education researcher wants to compare three different teaching methods. They randomly assign 90 students to three groups:

Group 1 (n=30): Traditional lecture
Group 2 (n=30): Problem-based learning
Group 3 (n=30): E-learning

At the end of the semester, all students take the same exam. The one-way ANOVA tests whether the mean exam scores of the three groups differ significantly.

If the result is significant ( $p < .05$ ), post-hoc tests (e.g., Tukey HSD) follow to determine which specific groups differ from each other.

Effect Size#

Eta-squared ( $\eta^2$ ) as a measure of effect size:

\eta^2 = \frac{SS_{between}}{SS_{total}}

Partial eta-squared is commonly reported in practice:

\eta_p^2 = \frac{SS_{between}}{SS_{between} + SS_{within}}

Effect Size	η²
Small	0.01
Medium	0.06
Large	0.14

Tip: Alternatively, omega-squared ( $\omega^2$ ) can be reported, which provides a less biased estimate of the population effect size.