Mock test scores: mean, median and claims
A Form 4 mathematics teacher records the marks of 40 learners in a mock test marked out of 60. The marks are grouped as follows: 0–9, 3 learners; 10–19, 6 learners; 20–29, 10 learners; 30–39, 11 learners; 40–49, 7 learners; 50–59, 3 learners. The pass mark is 30. The teacher tells the class that the average mark is about 35 and that three-quarters of the class passed. A parent challenges these claims at a meeting. The teacher also plans to remove the 9 learners who scored below 20 for a remediation class and wants to know the mean of the remaining 31 learners. The school needs an accurate summary, not a comforting one.
1. Estimate the mean and median from the grouped data. Comment on the teacher's claim that the average is about 35.
Read the worked answer
Use midpoints 4.5, 14.5, 24.5, 34.5, 44.5, 54.5.
Sum fx = 3×4.5 + 6×14.5 + 10×24.5 + 11×34.5 + 7×44.5 + 3×54.5
= 13.5 + 87 + 245 + 379.5 + 311.5 + 163.5 = 1,200.
Estimated mean = 1,200 ÷ 40 = 30.0.
Cumulative frequencies: 3, 9, 19, 30, 37, 40. The median position is the 20.5th value, which lies in the 30–39 class.
Median = 29.5 + ((20.5 − 19) ÷ 11) × 10 = 29.5 + 1.36 = 30.86 (2 d.p.).
The teacher's claim of about 35 is not supported: both the mean and median are close to 30, well below 35.
2. Estimate the probability that a learner selected at random passed, and the probability that a learner selected at random scored at least 40.
Read the worked answer
Pass means 30 or more: 11 + 7 + 3 = 21 learners.
P(pass) = 21/40 = 0.525, or 52.5%, not three-quarters.
At least 40: 7 + 3 = 10 learners.
P(at least 40) = 10/40 = 0.25, or 25%.
3. The teacher removes the 9 learners who scored below 20. Calculate the mean of the remaining 31 learners. Does this support the teacher's original claim?
Read the worked answer
Total fx for the 9 learners below 20 = 3×4.5 + 6×14.5 = 13.5 + 87 = 100.5.
Remaining total = 1,200 − 100.5 = 1,099.5 over 31 learners.
Mean = 1,099.5 ÷ 31 = 35.47 (2 d.p.), which is above 35.
But this is the mean of a selected group after removing the lowest scorers. It does not show that the whole class average is 35. The selection has raised the mean, so the teacher's original claim about all 40 learners remains misleading.