Comparing Two Class Test Results
Two classes take the same 50-mark mathematics test. Class X has 20 students with a mean of 34 and a standard deviation of 6. Class Y has 25 students with a mean of 30 and a standard deviation of 11. A student in Class X scored 44; a student in Class Y scored 41. The head of department wants to know which performance was stronger relative to the class, and whether the classes differ in consistency. A parent notes that Class Y's top score was 49 while Class X's top score was 46, and concludes Class Y must be the stronger group.
1. Compute the z-scores of the two named students and decide whose performance was stronger relative to their class.
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z = (score − mean)/standard deviation.
Class X student: z = (44 − 34)/6 = 10/6 ≈ 1.67. Class Y student: z = (41 − 30)/11 = 11/11 = 1.00.
The Class X student is about 1.67 standard deviations above the class mean, versus 1.00 for the Class Y student. Relative to their classmates, the Class X student performed more strongly. The raw scores (44 vs 41) point the same way, but the z-scores show the gap is larger than the raw difference suggests, because Class X is a tighter group.
2. Compare the consistency of the two classes and explain what the standard deviations imply for interpreting individual scores.
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Class X has standard deviation 6 on a 50-mark test; Class Y has 11, nearly double. Class Y's scores are much more spread out, so its mean of 30 is a less reliable summary of a typical student.
Implications: in Class X, most scores lie roughly within 34 ± 6, i.e. about 28 to 40, so a score of 44 is unusual. In Class Y, scores spread roughly 30 ± 11, i.e. about 19 to 41, so a 41 is near the top of the typical range and a 49 is exceptional but not astonishing. Comparing means alone would hide this: the classes differ as much in spread as in average. Consistency is better in Class X, which may reflect more uniform teaching or a narrower intake.
3. Evaluate the parent's conclusion that Class Y is the stronger group because its top score is higher.
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The parent is comparing single extreme values, which is a weak basis for judging a group. A top score depends on one student and is highly variable; it says nothing about the group as a whole.
On the group measures, Class X has the higher mean (34 vs 30) and the lower standard deviation (6 vs 11), so Class X is stronger on average and more consistent. Class Y does have the highest individual score (49), and converting it: z = (49 − 30)/11 ≈ 1.73, which is only slightly above the Class X student's 1.67. So even Class Y's best performance is not dramatically more exceptional relative to its own class.
The sound conclusion: Class X performed better overall; Class Y has greater spread and one outstanding individual. Judgements about group strength should rest on means and spreads, not on the maximum.