Averages, Weighted Averages and Combined Groups
Average questions in EPSO papers are almost never "add these and divide". They are weighted-average, combined-group or missing-value problems, and the fastest route is nearly always to work with sums rather than averages.
Core rules
- Mean = sum / count, therefore sum = mean × count. This identity solves most missing-value items in a single line.
- Weighted mean = (sum of weight × value) / (sum of weights), where the weights are counts, not percentages to be averaged naively.
- The combined mean always lies between the two group means, closer to the mean of the larger group.
- Balance method: the combined mean divides the gap between the group means in the inverse ratio of the group sizes. With sizes 3:2 it sits 3/5 of the way from the smaller group's mean towards the larger one.
- The median is the middle value of an ordered list; it is not the mean and is unaffected by outliers.
Worked example
30 staff in Unit A earn an average of €42,000; 20 staff in Unit B earn an average of €47,000. What is the combined average?
Total A = 30 × 42,000 = 1,260,000. Total B = 20 × 47,000 = 940,000. Combined total = 2,200,000 across 50 staff = €44,000.
Balance check: the gap is 5,000 and the group sizes are 3:2, so the combined mean lies 3/5 of the gap above 42,000, i.e. 42,000 + 3,000 = 44,000. Note that 44,000 is not the midpoint 44,500 - pooling the two averages is the classic error.
Adding or removing a value
The mean of 12 items is 25, so their sum is 300. If one item is removed and the remaining 11 have a mean of 24, their sum is 264, so the removed item is 36.
Traps
- Averaging two group means without weighting by group size.
- Forgetting that the count changes when a value is added or removed.
- Using a percentage as a weight when the underlying counts differ.
- Returning the median when the mean was requested, particularly in a skewed distribution.