What can be used to evaluate "variance" of samples of categorical data?

Question

Suppose you are drawing samples from a population of categorical variables. Given samples s1 and s2, I want to calculate an index that would tell me how close I am to the ideal situation where all elements of the sample are equal to each other:

# Given the two samples 
pop <- c("A", "B", "C")
set.seed(10)

s1 <- sample(c("A", "B", "C"), 10, rep=T)
s2 <- sample(c("A", "B", "C"), 10, rep=T)

# Suppose also that s3 = c("A", "A", "A")
# and s4 = c("A", "B", "C")

Basically I'd like to find and index f that when applied to a vector with high "variance" such as s4 gives a maximum (or minimum) value and when applied to a vector of characters with low "variance" such as s3 gives a minimum (or maximum) value.

What can I use? I know that in R I can convert factors to numbers using ther number level encoding, then I could calculate the variance, but I'm not sure this is is the method I'd like to use.

Answer 1

With categorical data, the variance depends on the mean, which in this case is a proportion. So it is a bit misleading and imprecise to talk about variance when you can simply summarize the data with an intuitive measure: the proportion. If there are more than two categories, a multinomial probability model can be summarized for either sample separately, or the joint sampling distribution of s1 and s2. Here, the probability refers to the proportion of the sample which is A, B, or C in either s1 or s2 and it adds to 1.

You can test for differences in proportions which is also called association in a contingency table. This is done with the Pearson Chi-square test, the Cochrane Mantel Haenszel test, logistic regression, or other methods for independent data. You can also test for proportions of differences which is also called agreement. This is done with a percentage agreement, Cohen's Kappa, or a weighted Kappa if the values are ordinal.

Answer 2

This article proposes a "coefficient of unalikability"

For the case of a finite number of observations (n), a finite number of categories (m) and a finite number of objects, ki, within Category i, as previously illustrated the pattern of blocks will allow expression of the coefficient of unalikeablity as:

$$u_2=2\sum{p_ip_j}$$ or $$u_2=\sum{p_i(1-p_i)}$$ or $$1-\sum{p_i^2}$$ where $$p_i=\frac{k_i}{n}$$ The interpretation of u2 is that it represents the proportion of possible comparisons (pairings) which are unalike. Note that u2 includes comparisons of each response with itself.