Expected Mean Squares in One-Way ANOVA

Why Expected Mean Squares Matter

A one-way ANOVA separates variation within groups from variation among group means. The corresponding mean squares are random quantities: their values would change if the study were repeated. Their expectations explain what each mean square measures under the model. In particular, the expected within-group mean square reflects the common error variance, while differences among fixed group means add a nonnegative component to the expected between-group mean square.

This distinction is useful when interpreting ANOVA calculations: a large observed between-group mean square may arise from ordinary sampling variation, from genuinely different group means, or from both. The results here concern the expectations under the model; they do not by themselves determine a test decision or the distribution of an F statistic.

Model and Notation

Consider \(k\) fixed groups. Group \(i\) contains \(n_i\) independent observations, and \(N=\sum_{i=1}^k n_i\) is the total sample size. Write the observation from unit \(j\) in group \(i\) as \(Y_{ij}\). The one-way fixed-effects model is

Here \(\mu_i\) is the fixed mean for group \(i\), and the errors are independent with \(E[\varepsilon_{ij}]=0\) and \(\operatorname{Var}(\varepsilon_{ij})=\sigma^2\) for every observation. Thus all groups share a common error variance. Normal errors are not needed to derive the expected mean squares; independence, zero error means, and a common finite variance are sufficient.

For the sample, let \(\bar{Y}_{i\cdot}\) be the mean of group \(i\), and let \(\bar{Y}_{\cdot\cdot}\) be the overall sample mean, weighted by the group sizes. The within-group sum of squares is \(SS_W=\sum_i\sum_j(Y_{ij}-\bar{Y}_{i\cdot})^2\); the between-group sum of squares is \(SS_B=\sum_i n_i(\bar{Y}_{i\cdot}-\bar{Y}_{\cdot\cdot})^2\). Their mean squares are \(MS_W=SS_W/(N-k)\) and \(MS_B=SS_B/(k-1)\), respectively, when the indicated degrees of freedom are positive.

The Expected Mean Squares

The between-group expression has two parts. The first, \(\sigma^2\), is the contribution from sampling error, which remains even when all fixed group means are equal. The second, \(\lambda/(k-1)\), is the additional contribution from differences among the fixed means. The group sizes matter: the grand mean is weighted, and a group mean farther from that weighted center contributes in proportion to its size.

Why These Expectations Hold

The within-group result follows by removing each group’s sample mean. Within group \(i\), write \(\bar{\varepsilon}_{i\cdot}=n_i^{-1}\sum_j\varepsilon_{ij}\). Since the fixed mean \(\mu_i\) cancels when centering observations within a group,

The expected first term is \(n_i\sigma^2\). Independence gives \(\operatorname{Var}(\bar{\varepsilon}_{i\cdot})=\sigma^2/n_i\), and the error mean is zero, so \(E[n_i\bar{\varepsilon}_{i\cdot}^{\,2}]=\sigma^2\). Therefore group \(i\) contributes \((n_i-1)\sigma^2\) in expectation. Summing over groups gives \(E[SS_W]=(N-k)\sigma^2\); dividing by \(N-k\) gives \(E[MS_W]=\sigma^2\).

For the between-group result, write \(\bar{Y}_{i\cdot}=\mu_i+\bar{\varepsilon}_{i\cdot}\) and \(\bar{\varepsilon}_{\cdot\cdot}=N^{-1}\sum_i n_i\bar{\varepsilon}_{i\cdot}\). Then \(\bar{Y}_{\cdot\cdot}=\bar{\mu}_w+\bar{\varepsilon}_{\cdot\cdot}\). Substitution into \(SS_B\) separates each centered group mean into its fixed part and its random error part. The cross-product has expectation zero because each error mean is zero.

The first term is \(\lambda\). For the last term, use \(\sum_i n_i(\bar{\varepsilon}_{i\cdot}-\bar{\varepsilon}_{\cdot\cdot})^2=\sum_i n_i\bar{\varepsilon}_{i\cdot}^{\,2}-N\bar{\varepsilon}_{\cdot\cdot}^{\,2}\). The expectation of the first sum is \(k\sigma^2\), since each group mean error has variance \(\sigma^2/n_i\). The overall mean error has variance \(\sigma^2/N\), so the expectation of the second term is \(\sigma^2\). Hence the random error part contributes \((k-1)\sigma^2\), and \(E[SS_B]=\lambda+(k-1)\sigma^2\). Dividing by \(k-1\) proves the stated formula for \(E[MS_B]\).

Worked Examples

Common Mistakes and Interpretation

Expected mean squares are model-based averages over repeated samples, not predictions of exact observed values. In particular, \(E[MS_W]=\sigma^2\) does not imply that every within-group mean square equals \(\sigma^2\). Similarly, the formula for \(E[MS_B]\) includes the error-variance component as well as the fixed-effects component.

For example, in the unequal-size study above, omitting the \(\sigma^2=4\) term would give \(45/2=22.5\), not the correct expected between-group mean square \(26.5\). Using an unweighted center instead of the weighted grand mean is another error: it changes the deviations used to calculate \(\lambda\), and can therefore change the result.

• Do not confuse fixed means with sample means. The \(\mu_i\) values are model parameters; \(\bar{Y}_{i\cdot}\) values vary from sample to sample.
• Use the appropriate degrees of freedom. Divide \(SS_B\) by \(k-1\) and \(SS_W\) by \(N-k\) when defining the two mean squares.
• Keep the common variance component. Even when group means differ, \(E[MS_B]\) is \(\sigma^2+\lambda/(k-1)\), not just \(\lambda/(k-1)\).
• Use the weighted grand mean. With unequal group sizes, \(\bar{\mu}_w\) is not generally the simple average of the \(k\) group means.