Combining two group means by simple averaging is only valid when the groups are the same size. The combined mean is (n₁m₁ + n₂m₂) ÷ (n₁ + n₂) — group sizes are the weights.
The simple average of two numbers is automatic; weighting requires noticing that group sizes exist and matter. When a question shows two tidy means, the urge to split the difference is strong.
When two means meet, ask "how many in each?" before touching them. If sizes differ, go back to totals: rebuild each group's sum, add, divide by combined count. Sanity check: the combined mean must sit closer to the larger group's mean.
| Group | Size | Mean |
|---|---|---|
| A | 8 | 40 |
| B | 9 | 86 |
Two groups are combined. What is the mean of the combined group?
Combined mean = (n₁ × mean₁ + n₂ × mean₂) ÷ (n₁ + n₂) — weight by group size, never average the two means. The correct answer is 64.40. Traps to avoid: 61.60 comes from the "swapped weights" error; 126 comes from the "summed means" error; 86 comes from the "took larger group mean" error; 63.0 comes from the "unweighted mean" error.
This trap appears in 25 of our questions, across: Weighted average · Expected value · Combined mean · Mixtures.
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