Two Campuses, One Stagnant Pass Rate
A university publishes first-year pass rates for two campuses. In 2023, Campus A enrolled 400 first-year students and 80% passed; Campus B enrolled 100 and 60% passed. In 2024, Campus A enrolled 200 first-year students and 85% passed, while Campus B enrolled 300 and 70% passed. A newspaper headline reads: "Pass rate stuck at 76% - nothing has changed in two years." The dean of Campus B objects: "Our pass rate rose by ten percentage points, the biggest improvement in the institution, yet we are blamed for the stagnation." A senate member responds by proposing that the university shift more students to Campus A, since Campus A's pass rate is higher in both years.
1. Verify the 76% in the headline for both years, showing the number of students passing at each campus.
Read the worked answer
2023
- Campus A: 0,80 × 400 = 320 passes
- Campus B: 0,60 × 100 = 60 passes
- Total: 320 + 60 = 380 of 500 → 380 ÷ 500 = 0,76 = 76% ✓
2024
- Campus A: 0,85 × 200 = 170 passes
- Campus B: 0,70 × 300 = 210 passes
- Total: 170 + 210 = 380 of 500 → 380 ÷ 500 = 0,76 = 76% ✓
The headline figure is arithmetically correct. What it hides: the total number of passes is identical (380), so the overall rate could not move - even though every campus's own rate rose.
2. Both campuses improved, yet the overall rate did not change. Explain how this is possible and name the mechanism.
Read the worked answer
The mechanism: a shift in the composition (the weights), not in the rates.
The overall rate is a weighted average of the two campus rates, weighted by enrolment shares:
| 2023 | 2024 | |
|---|---|---|
| Campus A share | 400/500 = 80% | 200/500 = 40% |
| Campus B share | 100/500 = 20% | 300/500 = 60% |
| A rate | 80% | 85% |
| B rate | 60% | 70% |
Showing the arithmetic both ways
- 2023: 0,80(80%) + 0,60(20%) = 64 + 12 = 76%
- 2024: 0,85(40%) + 0,70(60%) = 34 + 42 = 76%
The university shifted its intake towards the campus with the lower pass rate. Each component rose, but the weight moved towards the weaker component by exactly enough to cancel the gains.
Consequence for the headline: "nothing has changed" is false. Per-campus teaching outcomes improved at both sites; what did not change is the enrolment mix, and it is the mix that pinned the aggregate at 76%. Reporting only the aggregate makes a genuine improvement invisible.
3. Using the 2024 figures, work out what the overall pass rate would be if 100 students were moved from Campus B to Campus A, assuming pass rates stay unchanged. Then explain why the senate member's proposal does not follow from that calculation.
Read the worked answer
The arithmetic Move 100 students from B to A: Campus A = 300, Campus B = 200 (total still 500).
- A: 0,85 × 300 = 255 passes
- B: 0,70 × 200 = 140 passes
- Total = 395 / 500 = 79%
So on the assumption of fixed rates, the aggregate would rise from 76% to 79%.
Why the proposal does not follow
- Campus rates are not causal effects. They describe cohorts that enrolled at each campus, not what would happen to a given student if relocated. Entry requirements, programme mix, language of instruction and student background all differ. The 85% may reflect whom Campus A admits rather than what it does.
- The rates would not stay fixed. Adding 100 students to Campus A is a 50% increase in its intake; class sizes, staff ratios and support capacity would change, and the 85% is precisely the number most likely to move.
- The counterfactual is untested. To justify shifting students you need evidence that the same students do better at A than at B - a comparison of matched or randomly assigned students, not two aggregate rates.
- It optimises the indicator, not the outcome. The proposal raises the published number without any claim that more students actually pass; the total number of passes (395 vs 380) rises only because the same 500 students are re-weighted.
The senate member has confused a weighted average with a lever.