Arithmetic Mean - Problems

Problems

The arithmetic mean may be misinterpreted as the median to imply that most values are higher or lower than is actually the case. If elements in the sample space increase arithmetically, when placed in some order, then the median and arithmetic average are equal. For example, consider the sample space {1,2,3,4}. The average is 2.5, as is the median. However, when we consider a sample space that cannot be arranged into an arithmetic progression, such as {1,2,4,8,16}, the median and arithmetic average can differ significantly. In this case the arithmetic average is 6.2 and the median is 4. When one looks at the arithmetic average of a sample space, one must note that the average value can vary significantly from most values in the sample space.

There are applications of this phenomenon in many fields. For example, since the 1980s in the United States median income has increased more slowly than the arithmetic average of income. Researchers dealing with frequency data must also be careful when reporting summary statistics such as means or median. Where a phenomenon is rare in general (for example, emergency room visits among the general population), but occurs frequently in some people (for example, daredevils), then the mean value may be much lower than the median.