Two Schools, Two Pass Rates, One Unsupported Claim
A regional newspaper reports on Grade 12 mathematics results at two schools. School A entered 60 candidates; 45 passed, and the average mark across all 60 candidates was 54%. School B entered 20 candidates; 12 passed, and the average mark across all 20 was 58%. School A's principal says the results prove her school 'teaches mathematics better than School B'. School B's principal replies that his school 'outperforms School A on average'. A parent is quoted as saying that 'School A's pass rate is inflated because it enters weak candidates in an easier paper', but offers no evidence. Three further facts are on the record: School A admits only learners who scored above 55% in Grade 10 mathematics; School B is an open-enrolment school that absorbed 30 learners in January after another school closed; 12 of School B's 20 candidates were repeating the subject; and School A's mathematics teacher was on sick leave for six weeks in Term 2.
1. Calculate the pass rates, then evaluate both principals' claims. Which claim, if any, does the data support?
Read the worked answer
Pass rates. School A: 45 ÷ 60 = 75%. School B: 12 ÷ 20 = 60%. Mean marks: A = 54%, B = 58%.
School A's principal: 'teaches better'. This is a causal claim and the data cannot carry it. The two intakes are not comparable: A admits only learners above 55% in Grade 10, while B is open-enrolment and absorbed 30 displaced learners in January. A 15-point pass-rate gap between a selected group and an open group is expected before any teaching happens. The claim also contradicts the one number that favours B - the higher mean mark.
School B's principal: 'outperforms on average'. Literally true of the mean (58 > 54), but it is a comparison of intakes as much as of teaching, and B's mean is computed over 20 candidates, so a single learner moves it by 1.4 marks.
The small-number trap. With 20 candidates, one learner is 5 percentage points of pass rate; the 15-point gap is only three learners. A claim about a school's teaching built on three learners is fragile.
Verdict. Neither claim is supported as stated. The defensible statement is descriptive: A had the higher pass rate, B had the higher mean mark, and the two schools started from different points. Choosing between those two facts is not a statistical question - it is a choice about which measure answers the question being asked, and neither answers 'which teaches better'.
2. Which single additional piece of information would most change the picture, and why is it more important than, for example, the teacher's six-week absence?
Read the worked answer
The most valuable addition is prior attainment per candidate - each learner's Grade 10 (or Grade 11) mathematics mark, matched to that learner's Grade 12 result, so that a value-added figure can be computed: how much each candidate gained, not where each ended up.
Why it dominates. The current figures compare two groups that were already different before Grade 12 began. A matched comparison removes that difference from the answer. Without it, the 75% versus 60% gap measures intake selection as much as teaching.
Why the teacher's absence is secondary. The six-week absence in Term 2 is a plausible explanation for a dip, but it is only a hypothesis until marks are compared: a school can lose six weeks and still gain more per learner than a school with full staffing. It is one factor among several that the passage already lists - open enrolment, 30 January arrivals, 12 repeaters - and each competes to explain the same gap. Prior attainment is the only one that lets you subtract the starting point.
What would not change much. A larger sample from the same design would sharpen the percentages but leave the comparison of unlike groups intact. Precision is not the same as validity: a very precise measurement of the wrong comparison is still the wrong comparison.
Practical consequence. If the matched data showed B's learners gaining more marks on average, the pass-rate story would reverse; if it showed A's learners gaining more, A's claim would finally have support it does not now have.
3. Write two sentences suitable for the newspaper that report these results accurately without implying cause, and explain the choices you made.
Read the worked answer
Model sentences. 'At School A, 45 of 60 mathematics candidates passed (75%), compared with 12 of 20 candidates (60%) at School B. The average mark was 54% at School A and 58% at School B; the two schools admit learners with different prior attainment, so the figures describe their results rather than rank their teaching.'
Why these choices.
- Counts with denominators, not bare percentages: '12 of 20' tells the reader how thin B's evidence is; '60%' alone hides it.
- Both measures reported. Putting the pass rate and the mean mark side by side makes the contradiction visible instead of letting each principal quote the number that suits them.
- No causal verbs. 'Teaches better', 'proves' and 'outperforms' all assert cause; 'passed', 'was' and 'compared with' describe.
- The confounder is stated, not editorialised. One clause naming prior attainment is fair reporting; a paragraph attacking either principal is not.
- The parent's claim is omitted. It has no evidence, so printing it would give an untested accusation the same standing as the figures. If it must appear, it should be attributed and labelled as unsupported in the same sentence.