The Misleading Average
A tutoring center advertises: "Our students improve by an average of 40 points!" A parent asks to see the data. The center serves 11 students this term. Ten students improved by 10 points each. One student improved by 340 points after a retake in which the original score had been lost and the student was scored on a different test form. The center computed the average by adding all gains and dividing by 11. The parent's child is considering enrolling and wants a realistic expectation.
1. Compute the advertised average and explain why it misrepresents a typical student's experience.
Read the worked answer
Total gain = (10 × 10) + 340 = 100 + 340 = 440. Divide by 11 students: 440 ÷ 11 = 40 points, matching the ad.
Why it misleads: A single extreme value (the 340-point outlier) pulls the mean far above what almost everyone experienced. Ten of eleven students gained 10 points; the median gain is 10. The mean of 40 is technically correct arithmetic but describes almost no actual student. The outlier is also questionable on its own terms (lost score, different test form), so it may not even be a valid data point.
2. What statistic should the parent ask for, and what realistic expectation should she set?
Read the worked answer
She should ask for the median gain and the distribution (how many students gained how much), plus how many students were excluded and why.
With these data, the median is 10 points, and 10 of 11 students gained exactly 10. A realistic expectation is roughly 10 points, not 40. She should also ask whether the 340-point case is a valid comparable measurement at all - if the test form changed and the original score was lost, that gain may not be comparable to the others and should probably be excluded or reported separately.
Judgment tested: recognizing when an outlier distorts a mean and choosing a measure that reflects typical performance.