- Qualitative variation
An

**index of qualitative variation (IQV)**is a measure ofstatistical dispersion innominal distribution s. There are a variety of these, but they have been relatively little-studied in the statistics literature. The simplest is thevariation ratio , while the most sophisticated is theinformation entropy .**Properties**There are various indices of qualitative variation; a number are summarized and devised by Wilcox Harv|Wilcox|1967, Harv|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. [*This can only happen if the number of cases is a multiple of the number of 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.

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.

**Formulas**Wilcox gives a number of formulas for various indices of QV Harv|Wilcox|1973, the first, which he designates DM for "Deviation from the Mode", is a standardized form of the

variation ratio , and is analogous tovariance as deviation from the mean.One formula for IQV, [

*[*] given as M2 in Harv|Gibbs|1975|p=472 is::$ext\{IQV\}\; :=\; frac\{K\}\{K-1\}left(1-sum\_\{i=1\}^K\; p\_i^2\; ight)$where "K" is the number of categories, and $p\_i\; =\; f\_i/N$ is the proportion of observations that fall in a given category "i". The factor of $frac\{K\}\{K-1\}$ is for standardization.*http://www.xycoon.com/qualitative_variation.htm IQV at xycoon*]The unstandardized index, $left(1-sum\_\{i=1\}^K\; p\_i^2\; ight)$, denoted as M1 Harv|Gibbs|1975|p=471, can be interpreted as the likelihood that a random pair of samples will belong to the same category Harv|Lieberson|1969|p=851, so this formula for IQV is a standardized likelihood of a random pair falling in the same category. M1 and M2 can be interpreted in terms of variance of a

multinomial distribution Harv|Swanson|1976 (there called an "expanded binomial model").**Evaluation of indices**Different indices give different values of variation, and may be used for different purposes: several are used and critiqued in the sociology literature especially.

If one wishes to simply make ordinal comparisons between samples (is one sample more or less varied than another), the choice of IQV is relatively less important, as they will often give the same ordering.

In some cases it is useful to not standardize an index to run from 0 to 1, regardless of number of categories or samples Harv|Wilcox|1973|pp=338, but one generally so standardizes it.

**Notes****References*** Citation

last1=Gibbs

first1=Jack P.

last2=Poston, Jr.

first2=Dudley L.

title=The Division of Labor: Conceptualization and Related Measures

journal=Social Forces

volume=53

issue=3

year=1975

month=March

pages=468–476

doi=10.2307/2576589

id=JSTOR stable URL|0037-7732(197503)53%3A3%3C468%3ATDOLCA%3E2.0.CO%3B2-T* Citation

last=Lieberson

first=Stanley

title=Measuring Population Diversity

year=1969

month=December

volume=34

issue=6

journal=American Sociological Review

pages=850–862

doi=10.2307/2095977

id=JSTOR stable URL|0003-1224(196912)34%3A6%3C850%3AMPD%3E2.0.CO%3B2-O* Citation

last=Swanson

first=David A.

title=A Sampling Distribution and Significance Test for Differences in Qualitative Variation

journal=Social Forces

volume=55

issue=1

year=1976

month=September

pages=182–184

doi=10.2307/2577102

id=JSTOR stable URL|0037-7732%28197609%2955%3A1%3C182%3AASDAST%3E2.0.CO%3B2-U* Citation

last=Wilcox

first=Allen R.

title=Indices of qualitative variation

year=1967

url=http://www.ornl.gov/info/reports/1967/3445605133753.pdf* Citation

last=Wilcox

first=Allen R.

title=Indices of Qualitative Variation and Political Measurement

year=1973

month=June

volume=26

issue=2

journal=The Western Political Quarterly

pages=325–343

doi=10.2307/446831

id=JSTOR stable URL|0043-4078(197306)26%3A2%3C325%3AIOQVAP%3E2.0.CO%3B2-Z**See also***

statistical dispersion **Other measures of dispersion for nominal distributions***

Information entropy

*Variation ratio

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