The correct method for finding the grand mean in an unequal n summary data ANOVA is obtained by taking each treatment mean, multiplying it by the number of observations (n) in that treatment, summing all of these products, and then dividing by the total number of observations across all treatments (N). This calculation provides a weighted average that accounts for the differing sample sizes of each treatment group.
This approach is essential because it ensures that treatments with larger sample sizes contribute more to the overall mean than those with smaller sample sizes, reflecting the data's structure more accurately. It effectively combines the information from all groups, allowing for a proper representation of the overall data set's central tendency without being skewed by differences in group sizes.
In contrast, averaging treatment variances, taking the maximum treatment mean, or calculating the variance of all treatment means do not yield the grand mean as they do not account for the combinations of means and their respective sample sizes.