Math CentralQuandaries & Queries


Question from Shawna, a student:
If the median is greater than the mean on a set of test scores,
describe the situation.


The official answer is that the data are "skewed to the left", with a long tail of low scores pulling the mean down more than the median. There is one definition of skewness (Pearson's) by which this is the case by definition.

However, skewness is also, perhaps more often, defined in terms of the third moment (essentially the average of [datum - mean] cubed) and with this definition it need not be true. There is a good explanation of this in Wikipedia.

In practice, this is often because of a "natural boundary" at 100%, meaning that many scores are so high that they can't easily get much higher. In some cases (eg, first aid courses) mastery is expected and this is a standard situation; in other cases it may mean that the course is not challenging enough for most students.

Good Hunting!


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