Mean and Median

Question

Why do we need to define two different measures of central tendency, the mean and the median? Feel free to give an example supporting your line of argument.

Answer 1

Just because a statistic or measure defines a measure of central tendency does not mean that all are the same, no pun intended.

Mean (or average) has one measure, a mid point where the value of the SUM of all the elements less than the mean are equal to the value of the SUM of all the elements above. This is also the expected value.

Median, is a measure where the NUMBER of elements (not the sum of the value of the elements) less than the median is the same as the NUMBER of elements above the median.

In some cases (binomial distribution, normal distribution, uniform distribution) the two are the same.

In other cases (beta distribution, gamma distribution) the distribution is no longer symmetric around a central measure and the mean and median will be different, and will tell different stories.

Example. Income in California, where you have a lot of agricultural workers and service workers who have low incomes and a few elite entertainers and sports personalities who who have huge incomes. Those few people with incomes in the millions will each have an income equal to hundreds of agricultural workers. The average income will reflect the income of the few rich more than it does the multitude of low income workers. Median however will give a value where fully half the population have incomes above , regardless of how much they make.

Thus mean and median have different meanings and different uses. Both are useful.