Coefficient of variation: the second-order alternative

The coefficient of variation V introduced by Karl Pearson over 100 years ago is one of the most important and widely used moment-based summary statistics. The V does, however, have some significant limitations: (1) it becomes problematic when the data are both positive and negative, (2) it generally lacks an upper bound so that interpretations of V-values become difficult, (3) it lacks an intuitive and meaningful interpretation, (4) it is highly sensitive to outliers, and (5) it is very much affected by the mean and errors or changes in the mean. In order to overcome such limitations, this paper introduces the second-order coefficient of variation as a simple alternative to Pearson's V. While V is based on a ratio of second-order to first-order moments, is a ratio of second-order moments. The is well defined for all real-valued variables and data. The sample form of the new coefficient is also a ratio of (Euclidean) distances and takes on values between 0 and 1, making interpretations intuitively simple and meaningful. The various properties of are discussed and compared with those of V. Statistical inference procedures are derived both for the case when the underlying random variable can reasonably be assumed to have a normal distribution and for the case when no such distribution assumption is required.