Final answer:
When k population means are indeed different, it usually results in a decrease in the average error deviation, as the variance is better explained by actual differences in populations rather than within-group variations.
Step-by-step explanation:
When the k population means are truly different from each other, the likelihood is that the average error deviation decreases. This is because the differences between the group means are not due to random error or variation within the groups, but due to actual differences between the populations themselves. As a result, the within-group variations (errors) are relatively lower compared to the between-group differences (true differences in means).
The concept is related to the analysis of variance (ANOVA), which looks at the variances within groups compared to the variances between groups to determine if population means are different. When the k population means are indeed different, it suggests a greater variance between groups than expected by chance, which usually translates to a smaller average error deviation within each of the groups.