groupmean

A group mean is the average value of a variable calculated within predefined groups of observations. For each group, the group mean is obtained by summing the observations in that group and dividing by the number of observations in the group. Collecting the group means provides a concise summary of how the variable behaves across different groups, in contrast to the grand or overall mean, which pools all observations regardless of group.

Computation and relation to the overall mean: if a dataset is divided into groups, each group has its own mean. The overall mean across all observations can be expressed as a weighted average of these group means, with weights equal to the group sizes. If groups are unweighted, the simple average of the group means may differ from the grand mean, especially when group sizes vary.

Applications: group means are central to descriptive statistics and are widely used in analysis of variance (ANOVA) to compare groups. They help illustrate group differences and inform modeling decisions, such as whether to include a group effect in a linear model. In regression, one may use group-mean centering, where each observation is adjusted by its group’s mean, to separate within-group variation from between-group variation.

Variants and modeling approaches: in hierarchical or multilevel models, group means may be treated as random effects with a common distribution, allowing partial pooling of information across groups. This can stabilize estimates for groups with small sample sizes. Group means are also used in data visualization and exploratory data analysis to identify patterns and potential outliers within groups.

Limitations: relying solely on group means ignores within-group variability and can be misleading for small groups or skewed distributions. Outliers within groups can disproportionately affect a group mean.