What this calculator does
The index of qualitative variation measures how evenly observations are spread across categories. It runs from 0, when everything falls in one category, to 1, when every category holds exactly the same count.
It exists because the usual measures of spread need numbers to subtract, and categories have none. There is no mean religion, no standard deviation of eye colour. What can still be asked is whether the cases are concentrated or spread out, and this answers precisely that.
The formula
Each category share is expressed as a percentage and squared, and those squares are summed. The index is K times 10,000 minus that sum, divided by 10,000 times (K−1), where K is the number of categories. The sum of squared shares is largest when one category dominates and smallest when all are equal, which is what drives the scale.
| Term | Meaning |
|---|---|
| IQV | Index of qualitative variation, from 0 to 1. |
| Categorical data | Data in unordered groups, where arithmetic is meaningless. |
| Maximum variation | Every category holding an equal share, giving exactly 1. |
| K | The number of categories, which the formula normalises by. |
The inputs explained
| Field | What to enter |
|---|---|
| Category frequencies (comma separated) | Category frequencies, comma separated. One number per category. The categories themselves need no names, since only the counts matter. |
When to use it
Measuring demographic diversity
Comparing how evenly a population is spread across groups, between regions or over time.
Assessing market concentration
Whether share is spread across many competitors or concentrated in one.
Summarising survey responses
Whether answers to a multiple-choice question cluster on one option or spread across all of them.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the index respond to concentration?
The same total spread across categories in different ways.
| Category frequencies | Index of qualitative variation (IQV) | Sum of squared percentages (Σp²) | Number of categories (K) |
|---|---|---|---|
| 10, 10, 10, 10 | 1.000 | 2,500.00 | 4 |
| 12, 8, 15, 5 | 0.9517 | 2,862.50 | 4 |
| 20, 12, 5, 3 | 0.8517 | 3,612.50 | 4 |
| 37, 1, 1, 1 | 0.1900 | 8,575.00 | 4 |
Questions
What does an IQV of 1 mean?
That the cases are spread perfectly evenly across the categories, which is the maximum possible variation. Ten cases in each of four categories gives exactly 1.000. Any departure from equal counts reduces it.
Why not just use the standard deviation?
Because categorical data has no numerical values to take a mean of or deviations from. Eye colour cannot be averaged. The IQV works entirely from the counts and how evenly they are distributed, which is the only meaningful sense of spread for unordered categories.
Does the number of categories matter?
The formula normalises by K so the scale stays 0 to 1 regardless, but comparisons across different numbers of categories should still be made carefully. A perfectly even split across three categories and across ten both give 1.000, yet the ten-category case is the more diverse situation in any ordinary sense.
How does this relate to other diversity indices?
It is closely related to Simpson diversity index, which is built on the same sum of squared shares. The IQV rescales it so the maximum is exactly 1 for any number of categories. Shannon entropy is a different approach to the same question, weighting rare categories more heavily.
For a related ecological measure, see the Simpson’s diversity index calculator. For category shares as percentages, see the relative frequency calculator.