# Index of Qualitative Variation

## IQV Measures the Distribution of Nominal Variables

The index of qualitative variation (IQV) is a measure of variability for nominal variables, such as race, ethnicity, or gender. These kinds of variables divide people by categories that cannot be ranked, unlike a variable measure of income or education, which can be measured from high to low. The IQV is based on the ratio of the total number of differences in the distribution to the maximum number of possible differences in the same distribution.

## Overview

Let’s say, for instance, that we are interested in looking at the racial diversity of a city over time in order to see if its population has gotten more or less racially diverse, or if it has stayed the same. The index of qualitative variation is a good tool for measuring this.

The index of qualitative variation can vary from 0.00 to 1.00. When all of the cases of the distribution are in one category, there is no diversity or variation, and the IQV is 0.00. For instance, if we have a distribution that consists entirely of Hispanic people, there is no diversity among the variable of race, and our IQV would be 0.00.

In contrast, when the cases in a distribution are distributed evenly across the categories, there is maximum variation or diversity, and the IQV is 1.00. For example, if we have a distribution of 100 people and 25 are Hispanic, 25 are White, 25 are Black, and 25 are Asian, our distribution is perfectly diverse and our IQV is 1.00.

So, if we are looking at the changing racial diversity of a city over time, we can examine the IQV year-over-year to see how diversity has evolved. Doing this will allow us to see when diversity was at its highest and at its lowest.

The IQV can also be expressed as a percentage rather than a proportion. To find the percentage, simply multiply the IQV by 100. If the IQV is expressed as a percentage, it would reflect the percentage of differences relative to the maximum possible differences in each distribution.

For example, if we were looking at the racial/ethnic distribution in Arizona and had an IQV of 0.85, we would multiply it by 100 to get 85 percent. This means that the number of racial/ethnic differences is 85 percent of the maximum possible differences.

## How To Calculate The IQV

The formula for the index of qualitative variation is:

IQV = K(1002 – ΣPct2) / 1002(K – 1)

Where K is the number of categories in the distribution and ΣPct2 is the sum of all squared percentages in the distribution. There are four steps, then, to calculating the IQV:

1. Construct a percentage distribution.
2. Square the percentages for each category.
3. Sum the squared percentages.
4. Calculate the IQV using the formula above.

Updated by Nicki Lisa Cole, Ph.D.

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