Why do group sizes matter when combining means?

The arithmetic mean is the total of all values divided by how many values there are. A group’s mean summarizes its values, but its count remains important when combining groups. A group of six values contributes six individual observations rather than one observation simply because it has one displayed mean.

Group A initially has two values with mean four units, so its total is eight units. Group B has six values with mean eight units, giving total forty-eight. Together there are eight values and a total of fifty-six units. Dividing fifty-six by eight gives a combined mean of seven units.

This is a weighted mean of the group means. The weights are two and six, the group sizes: (2 × 4 + 6 × 8)/(2 + 6). Each original value receives the same influence. B’s mean receives more group-level weight because B contains more individual values.

Averaging just four and eight would give six, assigning equal influence to the two groups regardless of size. That happens to agree with the combined mean when their sizes are equal. The important choice is what should receive equal weight: individual values in this lesson, rather than the groups as two named units.