## Mean or median?

16th September 2006

Mean is just a mathematical average of all the data point values; median is the value, which divides the continuum of data point values into two equal parts: values of the “left” part are all smaller than the median, values of the “right” part are all bigger than the median. Comparing mean and median, they usually say that

- mean is optimal for standard deviation calculation in the sense that it is the only measure of data points range, when standard deviation is the smallest possible;
- median is much less sensitive to the extreme values (outliers), which often are the result of error, and thus presents a more fair idea about the data set.

Here’s a popular comparison example for mean and median:

Suppose 19 paupers and one billionaire are in a room. Everyone removes all money from their pockets and puts it on a table. Each pauper puts $5 on the table; the billionaire puts $1 billion (that is, $10

^{9}) there. The total is then $1,000,000,095. If that money is divided equally among the 20 persons, each gets $50,000,004.75. That amount is the mean (or “average”) amount of money that the 20 persons brought into the room. But the median amount is $5, since one may divide the group into two groups of 10 persons each, and say that everyone in the first group brought in no more than $5, and each person in the second group brought in no less than $5. In a sense, the median is the amount that the typical person brought in. By contrast, the mean (or “average”) is not at all typical, since no one presentâ€”pauper or billionaireâ€”brought in an amount approximating $50,000,004.75.