Two Sections, One Outlier
Two sections of the same college algebra course took a common final exam.
- Section A (10 students): 70, 72, 74, 76, 78, 80, 82, 84, 86, 88
- Section B (10 students): 88, 89, 90, 90, 91, 92, 92, 93, 93, 10
The department chair reports that Section A's mean is 79 and Section B's mean is 82.8, and concludes that Section B was taught more effectively, though he calls the 3.8-point gap "modest." A dean points out that Section B's lowest score belongs to a student who sat the exam the morning after being discharged from the hospital.
1. Compute the median of each section and explain why the mean and median tell different stories about Section B.
Read the worked answer
Medians:
- Section A is already ordered; the middle two scores are 78 and 80 → median = 79.
- Section B ordered: 10, 88, 89, 90, 90, 91, 92, 92, 93, 93. Middle two are 90 and 91 → median = 90.5.
Why the two statistics diverge for B:
- In Section A the distribution is nearly symmetric, so mean and median coincide at 79.
- In Section B, nine scores sit between 88 and 93, but the single 10 pulls the mean down about 7.7 points. The mean is doing bookkeeping on all ten scores; the median is describing the typical student.
Judgment: the chair's "modest 3.8-point gap" is a statement about means. On the median, the gap is 11.5 points. His word "modest" is an artifact of the mean, not a property of the two classes.
2. The dean proposes deleting the 10 before comparing sections. Give the strongest argument for and against that move, and compute Section B's mean without the score.
Read the worked answer
For removing it: the score apparently reflects a one-time medical event, not ten weeks of instruction. A test score is evidence about learning only if the testing conditions are comparable; here they plainly were not.
Against removing it: deleting a score because it is inconvenient is how comparisons get rigged. The student was enrolled, the score is real, and if the question is "how did each section perform as a group," the group included that student. A defensible removal needs the criterion set in advance (e.g., documented medical accommodation), not discovered after seeing the number.
Recomputed mean for B: (828 − 10) ÷ 9 = 818 ÷ 9 = 90.9 (up from 82.8).
Key observation: removing the score changes the size of the gap, not its direction. Section B leads under every measure computed here - so the dean's factual point does not rescue Section A. It only makes the chair's "modest" label look worse.
3. The chair wants to claim that Section B was taught more effectively. What is the single most important piece of missing information, and why is it more important than the outlier debate?
Read the worked answer
Most important missing information: how the two sections compared before the course - prior math background, placement scores, or the same group's scores on a common earlier assessment.
Why it outweighs the outlier debate:
- A final-exam mean gap of 3.8 points (or 11.5 median points) is a description of outcomes, not of teaching. If Section B entered with stronger preparation, the gap may be entirely a selection effect. If Section A entered stronger and finished near B, A could be the better-taught section.
- The outlier question shifts the gap by about 8 points, but a prior-achievement difference could erase or reverse its meaning entirely. Magnitude arguments cannot settle a causal question.
Secondary needs: same instructor or different instructors, attendance rates, whether the exam was curved, and whether the hospital student had an accommodation. Each bears on whether the comparison is fair, but only prior achievement bears on what the comparison means.