Mean,mode,median

Question

In a mathematics class half of the students scored 42 on an achievement test. With the exception of a few students who scored 78, The remaining students scored 53. Which of the following statements is true about the distribution of scores?

A. The mean and the median are the same

B. The mean is less than the median

C. The mean is greater than the median

D. The mean is less than the mode

Answer 1

The median and mean are the same when the distribution is symmetric in frequency and values. It can be contrived to make the mean and the median the same in an asymmetric distribution. The mean is sensitive to any skewing of the data (i.e. the 78 scores which we don't know how many there are).

Normally we would say the mean is affected more than the median and the mean > median.

If equal numbers scored 42 and 53, the mean and median are the same: 47.5

Let's add a point: 78 (and take away a 53 point)

We go from 5 42s and 5 53s to 5 42s, 4 53s, and 1 78, The median does not move, but the mean moves upward. Therefore mean > median. So the answer is C.

However, note that the statement that half the points are 42 is critical because adding a 78 point to 5 42s, 5 53s results in the median jumping to 53! (greater than the mean) This is why thinking through an example can be clarifying. Another way to think of it is that the mean always moves up with higher points but if 50% are one lower score, the median cannot move unless the one data point above that score moves.

I hope that helps. (Oh, the mode, by default, is 42 because it is the most common. D is always wrong)

Answer 2

If half the scores are 42, then 42 is the median. Since the remaining scores are greater than the median, the mean will be greater than the median (C).