Reporting a class result after a late test
A Year 5 teacher has marked 15 of the 16 students in a mental-maths quiz scored out of 20. The scores are 12, 15, 9, 18, 14, 11, 16, 13, 10, 15, 17, 8, 14, 12 and 16. The sixteenth student, absent on the day, sits the quiz a week later and scores 7. The teacher must write a one-line summary for the school newsletter describing how the class 'typically' performed, and the principal has already told a parent group that the class average was 'above 65%'.
1. Calculate the mean and median of all 16 scores, and express the mean as a percentage of 20.
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Sum of the 15 original scores = 200, so with the late score the total is 200 + 7 = 207.
- Mean: 207 ÷ 16 = 12.9375 ≈ 12.94 out of 20
- As a percentage: 12.9375 ÷ 20 × 100 = 64.7%
- Median: ordered scores are 7, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 17, 18. With 16 values, average the 8th and 9th: (13 + 14) ÷ 2 = 13.5 out of 20 (67.5%)
2. The principal's 'above 65%' claim was made before the late test was marked. Advise the teacher on what can now be reported, and which average best fits the word 'typically'.
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The claim no longer holds on the usual meaning of 'average'. The mean has fallen from 13.33 (66.7%) to 12.94, i.e. 64.7%, which is below 65%. The teacher cannot repeat the claim without correction.
On 'typically':
- The median of 13.5 (67.5%) describes the middle student and is unaffected by how low the single score of 7 is.
- The mean is dragged down by that one low score, because 16 is a small group where one result carries weight (each score is worth about 6 percentage points of the mean).
A defensible newsletter line: 'The typical student scored 13.5 out of 20; the class mean was 12.9 out of 20.' Quoting the median alone would be true but would look evasive if the principal's figure is queried, so both should appear.
3. One student is chosen at random from the 16. What is the probability that this student scored above the class mean? Explain what the answer shows about using the mean to describe a class.
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- Mean = 12.9375, so 'above the mean' means a score of 13 or more.
- Scores of 13 or more: 13, 14, 14, 15, 15, 16, 16, 17, 18 = 9 students.
- Probability = 9/16 = 0.5625 (56.25%)
Interpretation: 9 of 16 students beat the mean, while only 7 fell below it, so the class is not split evenly around its average - several low scores (7, 8, 9, 10) pull the mean down. That is exactly why the mean alone can mislead in a small class, and why the median is a safer word for 'typical'.