The Class Average That Looked Fine
A Year 9 class of 25 students sits a maths quiz out of 20. The teacher calculates the mean score as 14.2 and tells the class they did well. One student, Priya, scored 4 because she was absent for most of the topic. Another student, Tom, scored 20. The teacher later realises that two students' scores were entered twice by mistake: a score of 18 and a score of 9 were each counted twice. The teacher must decide whether the class 'did well' and how to report the results fairly to parents.
1. The reported mean of 14.2 was based on 27 entries (25 students plus 2 duplicates). Work out the correct mean for the 25 students.
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Total of the 27 entries = 14.2 × 27 = 383.4.
Remove the two duplicate scores (18 and 9): 383.4 − 18 − 9 = 356.4.
Correct mean = 356.4 ÷ 25 = 14.256, about 14.3.
The mean barely changes (from 14.2 to about 14.3). So the duplicate-entry error did not meaningfully affect the average. The teacher's conclusion that the class 'did well' cannot be overturned just by fixing the duplicates - the average is essentially the same.
A strong answer notes that the duplicates happened to be close to the mean, so they pulled the average only slightly. If the duplicates had been extreme scores, the effect would have been larger.
2. Priya's score of 4 pulls the mean down. The teacher wants to describe the class fairly. Compare the mean with the median and explain which is more useful here.
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We do not know every score, but we can reason about the shape. Priya's 4 is an outlier - much lower than the rest. An outlier pulls the mean toward it.
The median is the middle score when the 25 scores are ordered. Because 25 is odd, the median is the 13th score. If most students scored around 15–17 and only Priya scored very low, the median would be higher than the mean (roughly 15 or 16).
The median is more useful for describing a typical student because it is not affected by Priya's unusually low score. The mean of 14.3 makes the class look weaker than it is.
A fair report to parents would give both: 'The median score was about 15, and the mean was about 14.3, lowered by one student who missed most of the topic.' This is honest and avoids misleading either way.
3. The teacher wants to decide whether to count Priya's score. What factors should influence this decision, and what is the fairest approach?
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Factors to weigh:
- Reason for absence - Priya missed most of the topic, so the quiz does not measure what she has learned.
- Consistency - whatever rule is applied to Priya should apply to any student in the same situation.
- Effect on the class - excluding her raises the mean slightly, but the class result should not depend on one student.
- Fairness to Priya - reporting a 4 to parents without context could be misleading.
Fairest approach: report Priya's result separately with a note that she was absent for most of the topic, and do not use her score in the class average. Alternatively, allow her to sit the quiz again after catch-up teaching. A strong answer recognises that the decision is about what the number is for - measuring the class's learning versus measuring Priya's - and that the same rule must be applied to everyone.