Qualitative variation

An index of qualitative variation (IQV) is a measure of statistical dispersion in nominal distributions. There are a variety of these, but they have been relatively little-studied in the statistics literature. The simplest is the variation ratio, while more complex indices include the information entropy. There are several types of indices used for the analysis of nominal data. Several are standard statistics that are used elsewhere - range, standard deviation, variance, mean deviation, coefficient of variation, median absolute deviation, interquartile range and quartile deviation. In addition to these several statistics have been developed with nominal data in mind. A number have been summarized and devised by Wilcox , , who requires the following standardization properties to be satisfied: Variation varies between 0 and 1. Variation is 0 if and only if all cases belong to a single category. Variation is 1 if and only if cases are evenly divided across all categories. In particular, the value of these standardized indices does not depend on the number of categories or number of samples. For any index, the closer to uniform the distribution, the larger the variance, and the larger the differences in frequencies across categories, the smaller the variance. Indices of qualitative variation are then analogous to information entropy, which is minimized when all cases belong to a single category and maximized in a uniform distribution. Indeed, information entropy can be used as an index of qualitative variation. One characterization of a particular index of qualitative variation (IQV) is as a ratio of observed differences to maximum differences. Wilcox gives a number of formulae for various indices of QV , the first, which he designates DM for "Deviation from the Mode", is a standardized form of the variation ratio, and is analogous to variance as deviation from the mean. The formula for the variation around the mode (ModVR) is derived as follows: where fm is the modal frequency, K is the number of categories and fi is the frequency of the ith group.