The Mean Versus the Median

One difficulty many Americans have is the inability to master some of the most basic quantitative concepts. If you mention to someone that you teach mathematics or statistics, their usual reaction is to roll their eyes, or their body seizes up, or they may even go into full panic or anxiety attack mode. My favorite example of this difficulty is many people's failure to distinguish between the mean and the median of a collection of numbers.

The mean, or average, of a bunch of numbers is obtained by adding them all up and then dividing what we get by the number of numbers we just added up. Thus the average of 0, 2, 4, 6, 10 is 22 divided by 5, which is 4.4. The average of 0, 1.5, 2, 2, 4, 6, 10, 11 is 36.5 divided by 8, which is 4.5625. We are all familiar with concepts such as Batting Average, Average Salary and Life Expectancy, which is an average.

The median of a bunch of numbers, on the other hand, is the middle number, after we have written the numbers in increasing order. For the first bunch above, the median is 4. The "position" of the median is 3rd, since there are five numbers. For the second bunch, there is an even number of numbers, so it is customary to take the average of the middle two numbers. Thus the median is 4 + 6, divided by 2, which is 5, and the its position is 4.5th. (For large bunches of numbers, this concern over what to do when the size of the bunch is an even number is of no practical significance.) One often hears the term "median income". For scores on an SAT or ACT exam, the 50th percentile is the median score: 50% of the scores are lower and 50% are higher.

George W. Bush was accused of misleading people during the 2000 Presidential campaign, when he stated that the vast majority of the help from his proposed tax cut would go to the people at the bottom end of the economic ladder.

So was his statement misleading? . It depends on what Bush meant by "majority of the help" If by "help" he meant the size of your tax cut, and if we line up all the tax cuts in increasing order, then the median tax cut belongs to someone at the lower end of the economic ladder, so the "majority" -the position of the median plus one- of the "help"- tax cuts- goes to people at the lower end of the economic ladder. Now most people hearing Bush's statement probably thought "average tax cut" rather than "median tax cut".. Because of the skewed nature of the distribution of the tax cuts in favor of the rich, the average tax cut would undoubtedly go to someone whose taxable income exceeded $100,000. Here the meaning of the word "distribution" is as in "distributing food", except that in this case it is tax cuts that are being distributed. If Bush thought that people would think "average" rather than "median", then he was being misleading. Otherwise, he was not.