Mr Banda's test marks: which average to report, and a false claim about the average
Mr Banda teaches Standard 8 mathematics. He gives a test marked out of 50 to his class of 10 pupils. The marks are 34, 41, 28, 45, 41, 37, 22, 41, 39 and 32. One pupil, Chisomo, was away at a district athletics competition and sits the same test a week later, scoring 48. Mr Banda must write a short report for the head teacher, who wants one sentence describing how the class performed. The head teacher also asks whether it is true that 'more than half of the class scored below the class average'. Mr Banda's colleague says he should simply report the mean, because 'the mean is the average and the average is what a report needs'. Two pupils scored below the pass mark of 30.
1. Calculate the mean, median, mode and range for the 10 pupils who wrote the test first. Which single measure should Mr Banda put in the report, and why?
Read the worked answer
Sum: 34 + 41 = 75; + 28 = 103; + 45 = 148; + 41 = 189; + 37 = 226; + 22 = 248; + 41 = 289; + 39 = 328; + 32 = 360.
Mean = 360 ÷ 10 = 36.
Median: ordered marks are 22, 28, 32, 34, 37, 39, 41, 41, 41, 45. With 10 values the median is the mean of the 5th and 6th: (37 + 39) ÷ 2 = 38.
Mode = 41 (three pupils). Range = 45 − 22 = 23.
Which to report: the mean of 36 is pulled down by the single mark of 22; the mode of 41 is just one cluster of three pupils and overstates the class. The median of 38 says half the class scored 38 or more, which is the fairest one-line summary for a class of 10, and the wide range of 23 should be quoted with it. Reporting only '36' hides both the strong group on 41 and the two pupils below the pass mark of 30.
2. Chisomo's mark of 48 is added. Recalculate the mean, median and range, and explain how the two averages respond differently to one high score.
Read the worked answer
New sum = 360 + 48 = 408, with 11 pupils: mean = 408 ÷ 11 = 37.09…, i.e. 37.1 to 1 decimal place (11 × 37 = 407, so plain 37 would be wrong).
Ordered list: 22, 28, 32, 34, 37, 39, 41, 41, 41, 45, 48. With 11 values the median is the 6th value: 39.
Range = 48 − 22 = 26.
Comparison: one pupil scoring 48 raises the mean by about 1.1 marks (36 → 37.1) but moves the median by only 1 mark (38 → 39). The mean uses the size of every mark, so a single unusual score shifts it; the median only uses position in the ordered list. That is why the median is called resistant to outliers, and why a report on a small class should not rely on the mean alone.
3. Is the head teacher's claim - 'more than half the class scored below the class average' - true? Test it for both the 10-pupil and the 11-pupil class, and explain what would make the statement true.
Read the worked answer
10 pupils, mean 36: marks below 36 are 22, 28, 32, 34 - that is 4 pupils, which is less than half of 10. Claim false.
11 pupils, mean 37.1: marks below 37.1 are 22, 28, 32, 34, 37 - that is 5 pupils out of 11, and half of 11 is 5.5, so again not more than half. Claim still false.
Why: the mean is dragged towards extreme marks (here the low 22 and 28), so it does not split a class into two equal halves unless the marks are symmetric around it. The measure that always splits the class in half is the median: for the 10-pupil class exactly half scored 38 or above.
Correct statement: 'The median mark is 38, so half of the class scored 38 or above.' The head teacher's sentence would only be true if the mean happened to fall above the median, which is not the case here.