Mean or median for a class award
The mathematics department of a polytechnic gives the same 100-mark test to nine second-year students who attended a remedial class. Their scores are 45, 52, 58, 61, 62, 65, 70, 74 and 88. A tenth student, who was ill on the test day, takes the test later and scores 96. The department has a rule that a student is eligible for a merit certificate only if the class average on this test is at least 65 marks. The head of department wants to report a single 'typical' score to the principal and also wants to know whether the tenth student's score should be allowed to decide whether the whole class crosses the 65-mark average. A teacher argues that the median is a fairer summary here. The tenth student's answer script is later rechecked and one question is found to have been marked too generously, so the score is reduced to 20.
1. Calculate the mean and median with nine students and then with the tenth student's score of 96, and state the eligibility result in each case.
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Sum of nine scores = 575, mean = 575/9 = 63.9; median is the 5th value = 62. With the tenth score of 96: sum = 671, mean = 67.1; the ordered list is 45, 52, 58, 61, 62, 65, 70, 74, 88, 96, median = (62+65)/2 = 63.5. The 65-mark average rule gives 'not eligible' on nine students (63.9 < 65) but 'eligible' with the tenth (67.1 ≥ 65), so one score flips the decision.
2. Which measure should the department use, and should the tenth student's score be allowed to decide the award? Justify your judgement.
Read the worked answer
The mean is sensitive to the single extreme score of 96; the median moves only from 62 to 63.5. If the aim is a 'typical' score, the median is the fairer summary. But the written rule says 'class average', so formally the mean must be used and the tenth student's score counts - the department cannot ignore a legitimate score just because it changes the result. The real problem is that the rule was not written with outliers in mind; a better policy would be fixed before the test, for example using the median or requiring a minimum number of students.
3. Recompute the mean and median with the corrected score of 20, and compare how far each measure moved.
Read the worked answer
With 20 instead of 96: sum = 575 + 20 = 595, mean = 59.5. Ordered: 20, 45, 52, 58, 61, 62, 65, 70, 74, 88; median = (61+62)/2 = 61.5. Changing the tenth score from 96 to 20 moves the mean by 7.6 marks (67.1 → 59.5) but the median by only 2.0 marks (63.5 → 61.5). This confirms the median's resistance to a single extreme value, while the mean changes enough to reverse the eligibility decision again.