The study of statistics generally places considerable focus upon the distribution and measure of variability of quantitative variables. A discussion of the variability of qualitative--or categorical-- data can sometimes be absent. In such a discussion, we would consider the variability of qualitative data in terms of unlikeability. Unlikeability can be defined as the frequency with which observations differ from one another. Consider this in contrast to the variability of quantitative data, which ican be defined as the extent to which the values differ from the mean. In other words, the notion of "how far apart" does not make sense when evaluating qualitative data. Instead, we should focus on the unlikeability.

In qualitative research, two responses differ if they are in different categories and are the same if they are in the same category. Consider two polls with the simple parameters of "agree" or "disagree. " These polls question 100 respondents. The first poll results in 75 "agrees" while the second poll only results in 50 "agrees. " The first poll has less variability since more respondents answered similarly.

Index of Qualitative Variation

An index of qualitative variation (IQV) is a measure of statistical dispersion in nominal distributions--or those dealing with qualitative data. The following standardization properties are required 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.

Variation Ratio

The variation ratio is a simple measure of statistical dispersion in nominal distributions. It is the simplest measure of qualitative variation. It is defined as the proportion of cases which are not the mode:

$\displaystyle v=1-\frac{f_m}{N}$

Just as with the range or standard deviation, the larger the variation ratio, the more differentiated or dispersed the data are; and the smaller the variation ratio, the more concentrated and similar the data are.

For example, a group which is 55% female and 45% male has a proportion of 0.55 females and, therefore, a variation ratio of:

$1.0-0.55=0.45$

This group is more dispersed in terms of gender than a group which is 95% female and has a variation ratio of only 0.05. Similarly, a group which is 25% Catholic (where Catholic is the modal religious preference) has a variation ratio of 0.75. This group is much more dispersed, religiously, than a group which is 85% Catholic and has a variation ratio of only 0.15.