The combined mean score is **7**, because the group of 30 people contributes three times as much to the overall average as the group of 10.

Example

A **weighted mean** is an average in which each value contributes according to its **weight**, meaning how much it counts. When combining group means, the weight is the number of people in each group, so each person counts equally.

Imagine combining the fictional scores of two groups of community volunteers:

- Group A: 10 people with mean score 4 → total score 10 × 4 = 40.
- Group B: 30 people with mean score 8 → total score 30 × 8 = 240.
- Together: 40 people and total score 280 → combined mean 280 ÷ 40 = **7**.

Simply averaging the two group means, (4 + 8) ÷ 2 = 6, would give the smaller group the same influence as the larger group. Counting all 40 people equally gives 7 instead.

Boundary case

If both groups had 10 people, their weights would be equal. Then the combined mean would be (10 × 4 + 10 × 8) ÷ 20 = **6**, the same as the ordinary average of the two group means.

What this does not mean

A mean of 7 does not mean everyone scored 7, or even that anyone scored exactly 7. It does not tell us how spread out the scores are or why the groups differ. The result is exact if the supplied means are exact, the groups do not overlap, and scores use the same scale. If the means were rounded, 7 would be an approximate combined mean. These fictional numbers establish no claim about real volunteers.