Reading a Two-Centre Placement Comparison Correctly
A district skill-mission review compares two training centres, A and B, for 2023. Each centre runs a Long-term course and a Short-term course, and each centre trained exactly 500 candidates that year. Neither centre is at capacity, and course-wise placement rates have been stable for three years. The review records only these figures:
| Centre | Long-term trainees | Long-term placed | Short-term trainees | Short-term placed |
|---|---|---|---|---|
| A | 100 | 60% | 400 | 30% |
| B | 400 | 55% | 100 | 25% |
The mission director tells the press that "Centre B is the better centre because it placed more of its candidates overall". Centre A's principal objects. The mission has funds to add 100 seats in 2024 at one centre in one course, and wants the largest increase in the number of placements per 100 extra seats, assuming current rates continue.
1. Check the director's claim with numbers. Is it factually right, and does it support the conclusion drawn?
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Convert rates to counts.
- Centre A: Long-term 60% of 100 = 60 placed; Short-term 30% of 400 = 120 placed. Total 180 of 500 = 36%.
- Centre B: Long-term 55% of 400 = 220 placed; Short-term 25% of 100 = 25 placed. Total 245 of 500 = 49%.
So the factual half of the claim is correct: B placed 245 candidates against A's 180, and 49% against 36%. The conclusion does not follow.
Look at the course-wise comparison, which is the like-for-like one: A placed 60% against B's 55% in the Long-term course, and 30% against 25% in the Short-term course. A is ahead in both courses, yet behind overall. The overall figure is therefore being driven by the mix of courses, not by teaching quality, and the director has compared two numbers that are not measuring the same thing. The claim "B is better" is unsupported; the defensible statement is "B placed more candidates in 2023", with the reason stated.
2. Explain why the overall rates move in the opposite direction to the course-wise rates, and state what a fair comparison would look like.
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This is a weighting effect, often called Simpson's paradox. Centre B put 400 of its 500 candidates (80%) into the Long-term course, where placement is high, and only 100 into the Short-term course. Centre A did the reverse: 80% of its candidates took the low-placement Short-term course. Each centre's overall rate is a weighted average of its two course rates, weighted by its own intake mix, so B's average is pulled up by the favourable mix even though it is worse in both cells.
The fair comparison is to apply the same mix to both centres. Using a 50:50 standard:
- A: 0.5 × 60 + 0.5 × 30 = 45%
- B: 0.5 × 55 + 0.5 × 25 = 40%
A is ahead by 5 percentage points, and stays ahead under any common weighting, because it wins in both course types. At district level the same correction should be applied to every centre before ranking. The mission should also publish counts alongside rates, since percentages over 100 trainees hide the fact that B's Short-term result rests on a very small group.
3. Where should the 100 new seats go, and what is the expected gain? Give one important caveat.
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Take the highest current placement rate, since the question asks for placements per 100 extra seats with current rates continuing. The rates are A Long-term 60%, B Long-term 55%, A Short-term 30%, B Short-term 25%. The best option is therefore 100 additional Long-term seats at Centre A, expected to add about 60 placements (100 × 0.60). The next best, B Long-term, would add about 55.
Two observations sharpen the answer. First, this is not the same question as "which centre is better": A wins the marginal-seat calculation and also wins the standardised comparison, so the two criteria happen to agree here. Second, the expected gain of 60 assumes the 60% rate survives expansion. It may not. A 60% rate achieved on 100 trainees selected over three years may depend on the calibre of applicants or on employer relationships that do not scale, and the Short-term course at A has four times the volume with half the success. A defensible decision funds 60–70 Long-term seats at A, watches the rate for one cycle, and releases the rest only if it holds.