Mean, Median, and a Misleading Average
A tutoring program reports that the average score increase for nine students was 12 points. The individual increases were 2, 4, 5, 6, 8, 10, 12, 15, and 46. A brochure says, 'Students improve by an average of 12 points.' A parent asks whether 12 represents a typical student. The director must decide whether to use the mean, the median, or both in the brochure. The program serves small groups and wants to be accurate without misleading families. No other students are in the data set.
1. Compute the mean and median. Explain why they differ so much.
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Mean = (2+4+5+6+8+10+12+15+46)/9 = 108/9 = 12. Ordered data: 2, 4, 5, 6, 8, 10, 12, 15, 46. Median = 8 (the fifth value). They differ because the 46-point gain is an outlier: it pulls the mean upward. The mean is mathematically correct, but the median better represents a typical student in this small group.
2. The brochure says 'average of 12 points.' Is that misleading? Advise the director.
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It is technically true if 'average' means mean, but it is potentially misleading because no typical student is near 12: the median is 8, and eight of nine students gained 15 or fewer points. A family could reasonably expect 12 as typical. The director should either report both measures ('mean gain 12, median gain 8') or use the median with a note about the outlier. Accuracy and context matter more than the larger number.
3. Write a one-sentence brochure statement that is accurate and fair.
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Example: 'In a small group of nine students, the median score increase was 8 points and the mean was 12 points; one unusually large gain raised the average.' This gives both measures, identifies the small sample, and explains the outlier without hiding the higher mean.