Averaging averages without weights
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.
Why your brain does this
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.
The fix
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.
See the trap in a real question
| Holding £m | Return % |
|---|---|
| 20 | 7% |
| 10 | 4% |
| 10 | 2% |
What was the fund's overall return, weighted by holding size?
Weight each group's value by its size: Σ(size × value) ÷ Σ(size). A simple mean of the group values is the classic error. The correct answer is 5.0%. Traps to avoid: 13.0% comes from the "summed returns" error; 4.3% comes from the "unweighted mean" error; 6.0% comes from the "dropped a group" error; 4.0% comes from the "median not mean" error.
Where it strikes
This trap appears in 25 of our questions, across: Weighted average · Expected value · Combined mean · Mixtures.
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