Mock Examination Scores and the Principal's Adjustment
A school enters 180 candidates for WASSCE. In the mock mathematics examination the mean score is 42 and the standard deviation is 11. The school's internal pass mark is 50. Only 38 candidates currently score 50 or above. The principal proposes adding 8 marks to every candidate's score so that the pass rate will double, before results are shown to parents and used to predict WASSCE grades. The mathematics head says the scores are roughly normal but slightly skewed to the right, and that item analysis shows weak performance in three topics: mensuration, probability, and linear inequalities. Teachers have four weeks before the external examination. The school has no external benchmark data for the mock, and the principal has already told the parent-teacher association that results will be released next week.
1. Estimate the current pass rate using the normal approximation and compare it with the actual 38 candidates; explain the difference.
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Normal approximation: z = (50 − 42) / 11 = 0.727. The proportion above z = 0.727 is about 0.233, so about 23.3% of 180 = 42 candidates would be expected to pass.
Actual: 38 candidates, or 21.1%.
The estimate is close but slightly high. The head teacher says the distribution is slightly skewed to the right, meaning a few high scores pull the mean up while many candidates sit below it. In that case the normal approximation overestimates the pass rate, which fits the actual count. The difference is small enough that the normal model is useful for a rough check, but it should not be used as a precise prediction. The school should also note that the pass mark of 50 is an internal judgement, not a fixed natural boundary.
2. Evaluate the principal's proposal: what adding 8 marks does to the mean, standard deviation, pass rate, rank order, and the use of scores for predicting WASSCE.
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Adding 8 marks to every candidate:
- Mean: rises from 42 to 50.
- Standard deviation: stays 11, because adding a constant does not change spread.
- Pass rate: if the pass mark stays at 50, the new z = 0, so about 50% pass under the normal model, roughly 90 candidates. That is more than double the actual 38, so the principal's double claim is arithmetically plausible but not a real improvement.
- Rank order: unchanged. Every candidate moves by the same amount, so positions and gaps are identical.
- Prediction of WASSCE grades: weakened. The mock scores no longer reflect the original scale, and parents may be misled. The external examination is marked independently, so the added marks do not travel with the candidates.
Judgement: The proposal is cosmetic and ethically questionable if presented as raising achievement. It does not show that learning improved. If the school wants a fair internal comparison, it can use standardised scores for reporting, clearly labelled as such, but it should not change the recorded raw marks without explanation.
3. Recommend how the school should use the mock results in the four weeks, including one statistical and one pedagogical action.
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Statistical action: Use item analysis and standardised scores, not a flat addition. Identify the three weak topics, calculate the facility index for each item, and compare the school's mean with a relevant reference if one is available. Report scores as raw marks plus a clear explanation of the internal pass mark.
Pedagogical action: Teach to the diagnosed weaknesses in the four weeks: mensuration (formula selection and units), probability (sample space and complementary events), and linear inequalities (number-line representation and testing points). Use short weekly quizzes with items similar in style to WASSCE, and reteach any topic where fewer than 50% of candidates succeed.
What not to do: Do not add marks and tell parents the pass rate doubled. Do not use the mock to predict individual grades with false precision. Set a realistic target, such as moving 10–15 candidates across the pass mark through targeted support, and track whether that happens.