Comparing Two Classes on One Test
Two Form 4 classes in Limbe sit the same mathematics test marked out of 50. Class A has 40 students; its mean score is 31.5, its pass rate at the mark of 30 is 70%, and its three highest scores are 48, 46 and 45. Class B has 32 students; its mean score is 34.2, its pass rate is 75%, and its three highest scores are 44, 42 and 41. The head of department must choose one class to represent the school in an inter-school quiz that uses the three best students from the chosen class. She also wants to target extra support at the class whose overall performance is weaker. A colleague claims that Class B is clearly better because both its mean and pass rate are higher. The head of department asks for a numerical comparison before deciding.
1. Calculate the combined mean score and the combined pass rate for the two classes. Show the totals used.
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Total score A = 40 × 31.5 = 1,260. Total score B = 32 × 34.2 = 1,094.4. Combined total = 2,354.4. Combined number = 72. Combined mean = 2,354.4 ÷ 72 = 32.7. Passes A = 70% of 40 = 28. Passes B = 75% of 32 = 24. Total passes = 52. Combined pass rate = 52 ÷ 72 × 100 ≈ 72.2%. The combined mean is pulled toward Class A because Class A has more students.
2. The quiz uses the three best students from one class. Compare the top-three totals and decide which class should be entered. Does the result agree with the combined mean?
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Top three A = 48 + 46 + 45 = 139. Top three B = 44 + 42 + 41 = 127. A's top three total is 12 marks higher, an average of 4 marks per student. For a quiz using the three best students, Class A should be entered because its strongest students scored higher. This does not agree with the combined mean, which favours Class B (34.2 versus 31.5). The mean measures whole-class performance, not the top tail.
3. Which class should receive extra support, and how should the head of department respond to the colleague's claim that Class B is clearly better?
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Class A has the lower mean (31.5 versus 34.2) and the lower pass rate (70% versus 75%), so Class A is the weaker class overall and should receive extra support. The colleague is right about overall performance: B is ahead on both mean and pass rate. But the word 'clearly' is too strong if the decision is about the quiz, because A has the stronger top three. The correct judgement depends on the purpose: B for whole-class average, A for top-student competition. The head should also note the different class sizes (40 versus 32) when comparing totals.