Reporting an Honest Average Mark at an Admission Cell
A government polytechnic is preparing its results note. The first-semester mathematics paper was written by 65 first-year students across two branches: Mechanical (40 students) and Civil (25 students). The clerk has only branch-wise figures and reports the institute's "average mark" as the simple mean of the two branch averages, Mechanic 58 and Civil 66. The figure 58 for Mechanical includes three students who were marked absent and entered as 0; the Mechanical total of 40 × 58 = 2320 marks is unaffected by whether those three are counted, since each contributed nothing. The principal must publish one defensible institute average, must award a merit certificate to the top 10 % of the 65 students, and must answer a newspaper that has already printed 62 as the institute average.
1. Compute the correct combined mean mathematics mark and explain why the clerk's 62 is wrong. Show how the size of the error depends on branch sizes.
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The combined mean is a weighted mean, weighted by the number of students:
(40 × 58 + 25 × 66) / 65 = (2320 + 1650) / 65 = 3970 / 65 = 61.08 ≈ 61.1
The clerk computed (58 + 66) / 2 = 62, which is the mean of the two branch means - a legitimate statistic only when the branches have equal strength. Mechanical is the larger branch and also the lower-scoring one, so it pulls the institute average down; averaging the averages silently gives Civil's 25 students the same weight as Mechanical's 40.
The error here is a modest 0.9 marks. Its size depends entirely on the imbalance: if Civil had only 5 students, the true mean would be (2320 + 330) / 45 = 58.9 against the clerk's 62, an error of over 3 marks. Small error in this case, wrong method in principle - and the newspaper should be corrected to 61.1.
2. Recompute the Mechanical mean excluding the three absentees and decide which figure the institute should publish.
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The three absentees were entered as 0, so the branch total is 2320 whether they are counted or not:
- As recorded, over 40 students: 2320 / 40 = 58.0
- Excluding the absentees, over 37 students: 2320 / 37 = 62.7
The difference is 4.7 marks on the branch average, and that difference is manufactured entirely by the convention of writing 0 for absence. A zero conflates "did not attempt the paper" with "attempted and scored nothing", which is a false statement about the student and a distortion of the branch. It drags the branch down by roughly 3 × 62.7 ≈ 188 marks in total.
The publishable figure is 62.7 with the denominators printed alongside - "37 appeared, 3 absent, mean of those who appeared 62.7" - and the institute average recomputed on the same basis, over 62 students: (2320 + 1650) / 62 = 64.0. Publishing 58 without the note is the one option that cannot be defended, because the reader will assume 40 students averaged 58.
3. The principal must award merit certificates to the top 10 % of 65 students. Decide how many certificates, whether ranking should be branch-wise or combined, and flag any procedural risk.
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Count: 10 % of 65 = 6.5, so 7 certificates - you cannot award half a certificate, so round up. But note how the denominator changes the answer: if the absentees are excluded, 10 % of 62 = 6.2, i.e. 6 certificates. The principal must fix the base (students on roll, or students who appeared) before announcing the number, because the choice alone moves the count by one.
Ranking method: the branches wrote the same paper under the same marking scheme, so rank all candidates together on total marks and set one cut-off at rank 7. Proportional branch quotas (Mechanical 4, Civil 3) are defensible only if the branches were taught or assessed differently; otherwise a quota would reward a weaker branch and penalise a strong one.
Risks to flag: (i) absentees cannot be ranked at all, and their zeros must not be treated as scores; (ii) if two candidates tie at rank 7, the tie-break (higher marks in mathematics, then date of birth) must be published before the list, not improvised after; (iii) the newspaper's 62 must be corrected with the method shown, since the institute's credibility is now attached to that number.