Statistics Question

Question

You have recently took a statistics exam and your instructor just returned your graded exam. The instructor announced that 70% of the students scored below the mean. How do you reconcile this with the fact that, in a normal distribution, half of the scores should be below the mean and the other half should fall above the mean?

Answer 1

The grades are likely not normally distributed. When the mean is greater than the median, as it is here (since 70% are less than the mean, 50% are less than a smaller value), the distribution is skewed to the right. This could be due to the general shape being skewed to the right, or there could be one or a few positive outliers. This latter case occurs, for example, if a couple of people have a much better background for a course (or perhaps there is an item on the test that calls for a greater knowledge than can be reasonably expected.)

A trivial example that meets this mean - median rule given here: 70% of test-takers make a 90 and 30% make a 100. Then mean is 93. 70% are below the mean, and 30% are above. Such a distribution (imagine a 70%-30% mixture of 2 normal distributions about 90 and 100 with a very small variance in each case) is referred to as multimodal.