Quantitative - Ranking schools by mean score
Four secondary schools entered candidates for a national aptitude test. School W entered 40 candidates with a mean score of 78. School X entered 120 candidates with a mean of 72. School Y entered 60 candidates with a mean of 81. School Z entered 80 candidates with a mean of 75. A newspaper ranks the schools by mean score and prints the headline "School Y is the strongest school in the district." The district office replies that the ranking is misleading. Both sides accept the raw numbers and disagree only about what should be computed and compared. A parent whose child is an average student in the district asks which school she should prefer.
1. Compute the district-wide mean and the mean of the four school means. Explain why they differ.
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District-wide (weighted) mean: total score ÷ total candidates.
- W: 40 × 78 = 3,120
- X: 120 × 72 = 8,640
- Y: 60 × 81 = 4,860
- Z: 80 × 75 = 6,000
- Sum = 22,620; candidates = 40 + 120 + 60 + 80 = 300
- 22,620 ÷ 300 = 75.4
Mean of the school means (unweighted): (78 + 72 + 81 + 75) ÷ 4 = 306 ÷ 4 = 76.5
The gap of 1.1 exists because the unweighted figure gives a 60-candidate school the same influence as a 120-candidate school. X is both the largest school and the lowest-scoring, so under-weighting it pushes the average up. Neither number is wrong; they answer different questions. "What does a candidate in this district typically score?" needs 75.4. "How do the four schools' averages sit relative to each other?" is closer to 76.5 - and neither addresses which school teaches best.
2. Is the headline supported? Describe a legitimate question for which School Y's mean of 81 would be the right statistic.
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The headline is not supported as a claim about school quality. A mean score mixes at least three things: what the school teaches, whom it admits, and how many it admits. Y entered only 60 candidates, the smallest group, which is consistent with selective entry; a school that admits stronger pupils will post a higher mean with no better teaching. The headline converts a description ("Y's candidates scored highest on average") into a judgement ("strongest school") without evidence for the step in between.
When 81 is the right number: if the question is descriptive - "if a pupil is already enrolled at Y, what score should the family expect?" - the school mean is exactly the statistic wanted. It is also the right figure for a counsellor comparing typical outcomes for admitted pupils.
What the headline would need instead: prior-attainment data for the same pupils, so the schools are compared on progress rather than on intake. Without that, the ranking is a ranking of admissions, dressed as a ranking of teaching.
3. Suppose School Y doubled its intake to 120 candidates while keeping the same mean of 81. Recompute the district mean and comment on what this shows about rankings.
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Recalculate: Y's contribution becomes 120 × 81 = 9,720, and the candidate total becomes 360.
Sum = 3,120 + 8,640 + 9,720 + 6,000 = 27,480. District mean = 27,480 ÷ 360 = 76.3, up from 75.4.
The teaching at Y did not change at all - same mean, same programme - yet the district figure rose by nearly a full point, purely because a high-scoring school took a larger share of candidates. This is the same arithmetic that made the unweighted average (76.5) exceed the weighted one (75.4), run in reverse.
The lesson for rankings: any statistic that is a ratio of two totals moves when the mix changes, so a school's position can shift without a single pupil learning more. That is why comparing means across schools of different size, and different intake, cannot support a word like "strongest".
For the parent: the school mean says little about her child, who is average for the district. She should ask about the spread of scores, the school's admission criteria, and how much pupils improve between entry and the test - a school can have a modest mean and still add more value than one with a high mean.