Choosing a Typical Value with a Verified Outlier
Mr. Okafor’s class records the number of books read last month by nine students: 2, 3, 3, 4, 4, 5, 5, 6, and 32. The 32 is a verified value, not a typo, from a student who read a long series. The principal asks for a one-sentence summary of the typical number of books read. One group reports the mean, about 7.1 books, and says it is best because it uses every data point. Another reports the median, 4 books, and says it is more typical. A third student says the mean should be rounded to 7 and that this is close enough to the median. Mr. Okafor wants students to justify a choice of summary measure using the shape of the distribution and the audience’s purpose, not just compute both.
1. Compute the mean and median, and explain how the outlier affects each. Show the arithmetic.
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The sum is 2 + 3 + 3 + 4 + 4 + 5 + 5 + 6 + 32 = 64. The mean is 64 ÷ 9 ≈ 7.11 books. With nine ordered values, the median is the fifth value: 2, 3, 3, 4, 4, 5, 5, 6, 32, so the median is 4 books. The 32 is much larger than the other eight values, which range from 2 to 6. Because the mean uses the numerical size of every observation, that one value pulls it upward by about 3.1 books relative to the median. The median depends on position, not on how far the largest value is from the center, so it stays at 4. The outlier changes the mean substantially but does not change the median.
2. Evaluate the claim that the mean is best because it uses every data point. Is that sufficient justification? What additional information matters?
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Using every data point is a real property of the mean, but it is not sufficient justification for calling the mean typical. Sensitivity to extreme values can be a disadvantage when the goal is a summary of a typical student. The distribution is strongly right-skewed because of the verified 32, so the mean sits above almost all the data: only one student read more than 7 books. The median and mean answer different questions. The mean estimates a total-per-student share if the total number of books were redistributed equally; the median describes the middle student. The audience and purpose matter. For a one-sentence description of a typical student, the median is more defensible; for a total-resource calculation, the mean may be appropriate. Rounding 7.1 to 7 does not make it close to the median in a meaningful sense.
3. Write the one-sentence summary Mr. Okafor should send and justify the choice. Include a caveat about the outlier and what the class should avoid saying.
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A defensible summary is: ‘The median number of books read was 4; the mean was about 7.1 because one student read 32 books, far more than the others.’ This sentence reports a typical value and immediately explains why the mean is higher, so the principal is not misled. The median is chosen because the distribution is skewed by a verified extreme value and the purpose is to describe a typical student. The class should avoid saying ‘the typical student read about 7 books,’ because that describes a redistributed average rather than a middle observation. It should also avoid deleting the 32 as an outlier without justification; the value is verified and belongs in the data. If the principal needs a total, the class can report that the nine students read 64 books in all.