Nine Scores and an 18 Per Cent Claim
In a community secondary school, nine students sat a mathematics test marked out of 100 at the end of the first term and again after eight weeks of remedial classes. The first-term scores are 42, 45, 38, 50, 47, 44, 41, 39 and 95. The second-term scores, in the same student order, are 55, 58, 52, 60, 57, 61, 54, 56 and 70. At the assembly the principal announces that the class mean rose from 49 to 58, an 18 per cent jump, and that the remedial classes have clearly worked. A mathematics teacher objects that one student dropped from 95 to 70, so the picture is not that simple. The school must now write a short paragraph for the school report and for the parents' meeting, and the head teacher asks the class to decide what figure to quote and what claim, if any, the data can actually support.
1. Compute the mean and the median for both terms, then decide which average better describes a typical student in this class, giving your reasoning.
Read the worked answer
First term: 42+45+38+50+47+44+41+39+95 = 441, so mean = 441 ÷ 9 = 49.0. In order: 38, 39, 41, 42, 44, 45, 47, 50, 95, so median = 44.
Second term: 55+58+52+60+57+61+54+56+70 = 523, so mean = 523 ÷ 9 ≈ 58.1. In order: 52, 54, 55, 56, 57, 58, 60, 61, 70, so median = 57.
With only nine scores, the single 95 lifts the first-term mean about five marks above the median, so the mean is describing that one student as much as the class. The median says the middle student moved 44 → 57, a gain of 13; the mean says 49 → 58, a gain of 9. Both move the same way, so the improvement is not created by the choice of average - but the size of the gain depends on the average quoted, and a report that quotes only the mean hides the fact that its baseline was inflated.
2. The principal claims an 18 per cent jump proves the remedial classes worked. The teacher points to the 95 → 70 fall. Evaluate both positions.
Read the worked answer
The teacher's objection is about causation, not about the total. Eight of the nine students rose, and they rose by fairly similar amounts (between 10 and 17 marks), which is the pattern you expect when a whole class receives the same extra teaching. The ninth student fell by 25 marks. That single fall does not cancel the other eight, but it does two things: it shows the classes were not uniformly effective, and it removes the explanation that the second paper was simply easier for everyone.
The principal's percentage is technically correct but rhetorically loose. '18 per cent' compares two means, one of which was pushed up by an unusually high score in a class of nine; as a description of a typical student it overstates and understates at the same time. Also, no class that did not receive remedial teaching was measured, so 'the classes worked' is a correlation dressed as a cause - it might be revision, test familiarity, or the second paper's difficulty.
3. Draft what the school should report, and name the extra evidence that would turn the claim into a defensible one.
Read the worked answer
Reportable wording: 'Of nine students, eight improved between the first and second term. The median score rose from 44 to 57 and the mean from 49 to 58. One student's score fell from 95 to 70.' This quotes both averages, gives the direction and size honestly, and lets the reader see the exception rather than hiding it. It makes no causal claim.
Evidence that would strengthen a causal claim:
- attendance records for the remedial classes, so that the eight who improved can be matched against who actually attended;
- a comparison group, for example another class at the same level that did not receive the classes and sat the same two tests;
- item-wise marks, to see whether gains cluster in the topics the classes covered - a targeted gain is far better evidence than an overall rise;
- the difficulty level of the two papers, or a common item included in both. Until at least the attendance data exist, the school should report the change and resist crediting it entirely to the classes.