Comparing Tutoring Gains With an Outlier
A math coach is comparing two fifth-grade classes after a four-week tutoring cycle. Both took the same 10-point quiz before and after. Class A scores before: 5, 6, 6, 7, 7, 8, 8, 9, 9, 10; after: 6, 7, 7, 8, 8, 9, 9, 10, 10, 10. Class B scores before: 1, 4, 5, 5, 6, 6, 7, 7, 8, 9; after: 4, 5, 6, 6, 7, 7, 8, 8, 9, 9. One Class B student was absent for most tutoring sessions and scored 1 before; the after score is 4. The coach wants to report which class made the stronger gain. The principal asks whether the conclusion changes if the absent student is excluded. The coach must also explain why median may tell a different story from mean.
1. Compute the mean and median gain for each class, using all 10 students.
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Class A before sum = 75, mean = 7.5. After sum = 84, mean = 8.4. Mean gain = 0.9. Median before = (7 + 8)/2 = 7.5; after = (8 + 9)/2 = 8.5; median gain = 1.0.
Class B before sum = 58, mean = 5.8. After sum = 69, mean = 6.9. Mean gain = 1.1. Median before = 6; after = 7; median gain = 1.0.
By mean, B improved more (1.1 vs 0.9); by median, the gains tie at 1.0.
2. Recompute Class B without the absent student and decide whether the stronger-gain conclusion changes.
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Removing the absent student leaves 9 Class B scores. Before sum = 58 − 1 = 57, mean = 57/9 ≈ 6.33. After sum = 69 − 4 = 65, mean = 65/9 ≈ 7.22. Mean gain ≈ 0.89.
Median before is still 6; after is still 7; median gain remains 1.0.
Without the outlier, Class B’s mean gain (0.89) is slightly below Class A’s 0.9, so the mean-based conclusion reverses. The median-based conclusion does not change: both classes gained about 1 point at the median.
3. Which summary should the coach report, and what caution should accompany it?
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Report both mean and median, and state whether the absent student is included. The mean is sensitive to the low score of 1; that one score pulls Class B’s before-mean down and makes the gain look larger. The median is resistant to that outlier and shows equal median gains.
If the absent student’s low score reflects missed tutoring rather than typical performance, excluding it gives a fairer comparison of the tutoring cycle, and then Class A’s mean gain is slightly larger. The coach should not claim B is clearly stronger without naming the outlier rule. The principal’s question changes the conclusion, so transparency matters.