Post
by **kenm** » Thu Nov 12, 2009 11:19 am

In mathematics, there are several sorts of mean. I can think of three.

Arithmetic: the sum of the values divided by the number of them;

Geometric: the product of the values divided by the number of them;

Harmonic: the reciprocal of the arithmetic mean of the reciprocals.

I discover that there is a fourth, because I remembered it (incorrectly) as the name of the last: Logarithmic, which I won't give here, because it is rather complicated (masochists see Wikipedia).

In statistics, I have never seen any but the first used, so that the adjective is usually dropped. "Average" is often used as a synonym for it, but the Wikipedia author disapproves, because of the risk of confusion with the mode and the median.

In statistics, the median is the value that splits the data set into two sets with equal numbers of data points, and with values greater on one side and smaller on the other. The mode is the value that occurs the most frequently in a data set or a probability distribution. However, it is also used loosely as the name for a local maximum, and a curve that has two of these is called "bi-modal". Such a shape is a warning that two different populations have been mixed.

The normal distribution, often called the "bell" curve (no capital letter) has only one peak and has identical values for mean, mode and median. Widely different values for any two of these* are a warning that some of the convenient numerical manipulations used to summarise the characteristics of a population are invalid, since the distribution of the measured value does not approximate closely to the normal.

* This will always occur with a bi-modal distribution, but also with skewed distributions; e.g. reading age, that can't have values less than zero but can have a "long tail" of rare high values.

"... the innovator has as enemies all those who have done well under the old regime, and only lukewarm allies among those who may do well under the new." Niccolo Macchiavelli, "The Prince", Chapter 6