# Index of Qualitative Variation Calculator (IQV)

URL: https://speedycalculator.com/calculators/statistics/index-of-qualitative-variation/
Last updated: 2026-10-30

> The index of qualitative variation calculator measures how evenly cases are spread over the categories of a nominal variable, from category counts or percentages or from the sum of squared percentages, on a scale from 0 (all in one category) to 1 (perfectly even). It also draws a pie chart.

## Inputs

- Enter
- Number of categories (K)
- Sum of squared percentages Σp² (%²)
- Categories – one per line: name and count (or %)

## Results

- Index of qualitative variation (IQV)
- Interpretation
- Pie chart of the categories
- Gibbs–Martin / Blau index 1 − Σπ²
- Largest possible 1 − Σπ² for K categories
- Sum of squared percentages Σp²
- Number of categories K
- Category table

## Formula

```
IQV = K × (100² − Σp²) ÷ (100² × (K − 1)), p in percent · equivalently IQV = (1 − Σπ²) ÷ (1 − 1/K) with proportions π
```

## Explanation

Variance and standard deviation need numbers, but many variables are categories: favourite flavour, blood type, religion, party voted for. The index of qualitative variation (IQV) gives such nominal variables a measure of spread. It asks how close the distribution is to the most even one possible with that many categories.

IQV is 0 when every case falls into the same category – there is no variation at all – and 1 when all K categories are equally common. In between, a value of 0.9 means the distribution is 90% as diverse as it could be. Because it is divided by its own maximum, IQV can compare variables with different numbers of categories, such as the ethnic mix of two cities or the spread of answers to two survey questions.

The unscaled part, 1 − Σπ², is known as the Gibbs–Martin index of diversity or Blau's index (and as the Gini–Simpson index in ecology): the probability that two randomly chosen cases fall in different categories. The calculator reports it alongside IQV, together with a table of percentages and a pie chart of your categories.

### How to use the Index of Qualitative Variation Calculator

- **Enter**
- **Number of categories (K)**
- **Sum of squared percentages Σp² (%²)**
- **Categories – one per line: name and count (or %)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Type one category per line with its count or percentage (counts are converted to percentages), or choose the summary mode and enter the number of categories K and the sum of squared percentages Σp². The calculator then applies IQV = K(10,000 − Σp²) ÷ (10,000(K − 1)).

### Worked example

An ice-cream shop sells Vanilla 40%, Chocolate 30%, Strawberry 20% and Pistachio 10%. The squared percentages are 1,600, 900, 400 and 100, so Σp² = 3,000. With K = 4, IQV = 4 × (10,000 − 3,000) ÷ (10,000 × 3) = 28,000 ÷ 30,000 = 0.933. Sales are 93% as evenly spread as they could be. If all four flavours sold 25%, Σp² would be 2,500 and IQV = 1; if only vanilla sold, Σp² = 10,000 and IQV = 0.

### How to read IQV

IQVMeaning0Every case in one category0.2 – 0.5One category clearly dominates0.5 – 0.8Moderate variety0.8 – 0.95High variety1All categories equally common

## Worked example

Inputs — Enter: Each category with its count or %; Categories – one per line: name and count (or %): Vanilla 40
Chocolate 30
Strawberry 20
Pistachio 10.

Results — **Index of qualitative variation (IQV) = 0.9333**; Interpretation = High variation; Gibbs–Martin / Blau index 1 − Σπ² = 0.7; Largest possible 1 − Σπ² for K categories = 0.75; Sum of squared percentages Σp² = 3,000; Number of categories K = 4.

## Key terms

- **Nominal variable**: A variable whose values are names or labels with no natural order.
- **IQV**: The diversity of a nominal distribution divided by the maximum diversity possible with K categories.
- **Gibbs–Martin (Blau) index**: 1 − Σπ², the probability that two random cases fall in different categories.

## FAQ

**What is a good IQV value?**
There is no "good" or "bad" – IQV describes diversity. Values above about 0.9 mean the categories are nearly equally common; values below 0.5 mean one or two categories dominate.

**How do I calculate IQV by hand?**
Convert each category to a percentage, square the percentages and add them (Σp²), then compute K(10,000 − Σp²) ÷ (10,000(K − 1)). For 40/30/20/10%: Σp² = 3,000 and IQV = 4 × 7,000 ÷ 30,000 = 0.933.

**What does IQV = 1 mean?**
Every category has the same share – for four flavours, 25% each. Then Σp² = 4 × 625 = 2,500 = 10,000 ÷ K.

**What does IQV = 0 mean?**
All cases are in a single category, so Σp² = 100² = 10,000 and there is no variation.

**Can I use proportions instead of percentages?**
Yes: IQV = (1 − Σπ²) ÷ (1 − 1/K) with proportions π that add to 1. It gives exactly the same number.

**Why does the number of categories matter?**
With more categories more variety is possible. Dividing by the maximum, 1 − 1/K, puts every variable on the same 0–1 scale.

**Should empty categories be counted in K?**
Count every category that could occur. Leaving out an empty but possible category raises IQV, so state your choice when you report it.


## Sources

- [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e)
- [Wikipedia – Diversity index (Simpson index)](https://en.wikipedia.org/wiki/Diversity_index)
- [Penn State STAT 500 – Applied Statistics](https://online.stat.psu.edu/stat500/)