Checking Equal Variances With Residual Plots

Why Look at Residual Spread?

One-way ANOVA compares group means by measuring differences between groups relative to variation within groups. The usual ANOVA model assumes that the responses in every group have the same variance. If some groups are much more variable than others, that assumption may be questionable, so the usual ANOVA analysis may not behave as expected. Residual plots offer a visual way to look for such differences.

A residual plot does not prove that variances are equal or unequal. It helps identify patterns that merit attention: a gradual increase in spread across groups, or one group whose residuals are much more dispersed than the others. The comparison must respect the fact that one-way ANOVA has categorical groups, not a continuous explanatory variable.

Model and Residuals

Suppose there are \(k\) groups, with \(n_i\) observations in group \(i\). Write observation \(j\) in group \(i\) as \(y_{ij}\), and let \(\bar{x}_i\) denote the sample mean of the responses in group \(i\). In a one-way ANOVA, the fitted value for every observation in that group is \(\bar{x}_i\); its residual is \(e_{ij}=y_{ij}-\bar{x}_i\).

A useful model for describing the assumption is \(y_{ij}=\mu_i+\varepsilon_{ij}\), where \(\mu_i\) is the mean response in group \(i\) and \(\varepsilon_{ij}\) is the random error for that observation. For the usual one-way ANOVA assumptions, errors are independent, have mean zero, and have a common variance: \(\operatorname{Var}(\varepsilon_{ij})=\sigma^2\) for every group. Normality is a separate assumption considered in residual normality assessment; this tutorial focuses on variance.

When the common-variance assumption is in question, compare the residuals across groups. Residuals are measured in the original response units, and the zero line is a useful reference: values above it are observations above their group mean, and values below it are observations below their group mean. The spread around zero—not the position of the group mean itself—is the feature of interest.

Reading Residual Plots

A residual-versus-fitted plot places fitted values on the horizontal axis and residuals on the vertical axis. Since all observations in a group share the same fitted value, a one-way ANOVA plot typically has one vertical strip of points per group. Compare the vertical spread of those strips. Similar spreads are broadly consistent with equal variances; noticeably different spreads raise concern.

A groupwise residual plot places residuals against group labels, or shows a separate residual distribution for each group. It makes the group comparison explicit. Keep a common vertical scale across groups: changing the scale from group to group can hide meaningful differences in spread.

A funnel shape is a pattern in which the vertical spread tends to widen or narrow as fitted values increase. In a one-way ANOVA, fitted values are group means, so the plot has separate group strips rather than a continuous range of fitted values. If group means are placed in numerical order and the strips widen from left to right, that can suggest that variability changes across groups. But the horizontal spacing of categorical groups has no intrinsic meaning: a visual “funnel” is a clue to compare group spreads, not evidence of a continuous trend.

Groupwise plots can also show unequal variances without a funnel. For example, two groups may have similar residual spreads while a third is much more dispersed. The key question is whether the within-group variability looks comparable, not whether the points follow one particular shape.

A Short Justification for Comparing Residual Spread

The residuals are deviations from the group means, so their spread reflects within-group variation. In fact, for group \(i\), the sum of squared residuals is exactly the sum of squared deviations used to calculate that group’s sample variance:

This gives a direct reason to compare residual spreads by group: larger squared residuals within a group produce a larger group sample variance. The residuals are centered within each group, since subtracting the group mean gives \(\sum_j e_{ij}=0\). They are therefore not independent points; the plot is a diagnostic display, not a collection of independent measurements for a visual hypothesis test.

There is also a small-sample detail behind the display. Under independent errors with common variance \(\sigma^2\), the variance of a residual within a group is

To see the last equality, \(\operatorname{Var}(\varepsilon_{ij})=\sigma^2\), \(\operatorname{Var}(\bar{\varepsilon}_i)=\sigma^2/n_i\), and \(\operatorname{Cov}(\varepsilon_{ij},\bar{\varepsilon}_i)=\sigma^2/n_i\). Subtracting twice that covariance gives \(\sigma^2+\sigma^2/n_i-2\sigma^2/n_i=\sigma^2(1-1/n_i)\). Thus, residuals in a small group have slightly reduced variance because the same observations were used to estimate their group mean. This is one reason not to treat minor differences in plotted spread as decisive, especially when group sizes differ.

What a Plot Can and Cannot Tell You

Residual plots are valuable because they show the pattern of variation directly and in context. They can help identify whether spread appears stable across groups, increases across ordered fitted values, or is unusually large in a particular group. They can also reveal when the visual evidence is weak or ambiguous. A plot cannot establish equality of population variances: samples vary, and a small sample may make important differences hard to see.

This assumption matters because one-way ANOVA pools within-group variation when calculating its test statistic. The usual ANOVA result relies on its model conditions, including a common variance across groups. A marked spread difference is therefore a reason to describe the diagnostic concern and consider the analysis carefully, rather than to report the plot as a formal test.

Group sizes matter when judging plots. If one group has many more observations, it will contribute more points and may make its spread easier to assess. Small groups can look deceptively similar simply because they contain few observations. Compare spreads on the same scale, note the group sample sizes, and avoid interpreting the number of plotted points itself as a variance difference.

Common Mistake

A common mistake is to say, “The residuals are centered around zero in every group, so the variances must be equal.” Centering is built into the residual calculation: each group’s residuals sum to zero because they are deviations from that group’s sample mean. It says nothing by itself about how far the residuals lie from zero. To assess equal variances, compare the vertical spread around zero across groups, while keeping sample sizes and the limits of a visual diagnostic in mind.