Graduate Admissions: When Falling Enrolment Is a Mix Effect
A fictional university's graduate school reports applications, offers and enrolments for four master's programmes. Each programme runs the same three-stage process but sets its own selectivity.
| Programme | 2023 applicants | 2023 offers | 2023 enrolled | 2024 applicants | 2024 offers | 2024 enrolled |
|---|---|---|---|---|---|---|
| Finance | 1000 | 200 | 140 | 900 | 180 | 126 |
| Economics | 800 | 160 | 96 | 1400 | 280 | 168 |
| Data Science | 1200 | 360 | 270 | 1000 | 300 | 225 |
| Management | 500 | 150 | 90 | 400 | 120 | 72 |
The dean says: 'Applications rose 5.7% and offers rose 1.1%, yet enrolments fell slightly. Clearly our yield is collapsing and we need a €150,000 marketing campaign.' The planning office suspects the explanation is compositional rather than a decline in conversion. Tuition is €20,000 per enrolled student in every programme.
1. Compute overall yield (enrolled ÷ offers) for each year and the yield for each programme. Is the dean right that yield is collapsing?
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Overall
- 2023: offers 200+160+360+150 = 870; enrolled 140+96+270+90 = 596 → 596/870 = 68.5%.
- 2024: offers 180+280+300+120 = 880; enrolled 126+168+225+72 = 591 → 591/880 = 67.2%.
By programme
| Programme | 2023 yield | 2024 yield |
|---|---|---|
| Finance | 140/200 = 70% | 126/180 = 70% |
| Economics | 96/160 = 60% | 168/280 = 60% |
| Data Science | 270/360 = 75% | 225/300 = 75% |
| Management | 90/150 = 60% | 72/120 = 60% |
Every programme's conversion rate is unchanged. The overall figure fell only because offers shifted toward lower-yield Economics and away from higher-yield Data Science. 'Yield is collapsing' is false: no programme became worse at converting offers.
2. Decompose the change in enrolment into an offer-volume effect and an offer-mix effect.
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Start from 2023 enrolment of 596 with 870 offers (68.5% overall).
Step 1 - volume: hold the 2023 mix and yield, scale offers to 880: 596 × (880/870) = 602.9, a volume effect of about +6.9 students.
Step 2 - mix: apply the 2024 offer mix at the (unchanged) programme yields: 0.70×180 + 0.60×280 + 0.75×300 + 0.60×120 = 126 + 168 + 225 + 72 = 591. Mix effect = 591 − 602.9 = −11.9 students.
Check: 596 + 6.9 − 11.9 ≈ 591, matching the reported figure.
So the entire decline is compositional: more offers, but a larger share went to Economics (60% yield) and a smaller share to Data Science (75%). No within-programme conversion changed, which means a marketing campaign aimed at 'fixing yield' is aimed at a problem that does not exist in the programme-level data.
3. The dean wants €150,000 to 'restore yield to the 2023 level.' What would actually be required to return to 596 enrolments from 880 offers, and is the campaign the right instrument?
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Target: 596/880 = 67.7% overall yield, up from 67.2%, i.e. 5 extra students.
Because each programme's yield is stable, the lever is the offer mix. Moving one offer from Economics (60%) to Data Science (75%) adds 0.15 enrolled students. To gain 5 students: 5/0.15 ≈ 34 offers reallocated from Economics to Data Science, with no change in total offers or in admissions standards.
Assessment: the €150,000 campaign addresses a misdiagnosis. The 'collapse' is a 1.3-point mix artifact, and the stated target is reachable by reallocating roughly three dozen offers. If the campaign instead raised applications uniformly by 10% at constant offer rates, offers would rise to about 968 and enrolment to roughly 650 - a genuine gain, but that is a growth argument, not a yield-rescue argument. The dean should be asked to justify the campaign on incremental enrolment per euro, not on the false claim that conversion is deteriorating.