Statistical Inference: A Short Course

3.7 Relative Variation

How may we compare the dispersions of two (or more) distributions? Suppose we have distributions A and B and they have the same (or approximately the same) means and are expressed in the same units. Then the distribution having the larger standard deviation (a measure of absolute variation) has more variability or heterogeneity among its values. (The one with the smaller standard deviation has more homogeneity or uniformity among its values.)

It should be evident that if distributions A and B are expressed in different units (one might be expressed in cubic feet and the other in lumens), then there is no realistic basis for comparing their dispersions. Equally troublesome can be the instance in which their means display a marked difference, for example, we may ask: “is there more variability among the weights of elephants in a herd of elephants or among the weights of ants in an anthill?” While these two data sets can be converted to the same units, won't the standard deviation of the weights of the elephants “always” be a bigger number than the standard deviation of the weights of the ants?

This said, to induce comparability, let us convert the measures of absolute variation to “relative forms”—let us express each standard deviation as a percentage of the average about which the deviations are taken. This then leads us to define the coefficient of variation as:

(3.9a)

(3.9b)

Here v, say, is a pure number that is independent of units. Once v is determined, ...