Properties
There are various indices of qualitative variation; a number are summarized and devised by Wilcox (Wilcox 1967), (Wilcox 1973), 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 category.
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 in this sense complementary to information entropy, which is maximized when all cases belong to a single category and minimized in a uniform distribution, but they are not complementary in the sense of a particular IQV equaling 1 minus entropy. 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.
Read more about this topic: Qualitative Variation
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