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

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.

• When you say “all elements of the sample are equal to each other”, do you mean how far you are from a uniform distribution, or how far you are from the original/true/supposed distribution? Commented Jul 2, 2018 at 19:07
• I mean I would like to evaluate how far is the sample from the "ideal" sample. In my case the ideal sample would be a sample where all the categorical variable are equal to one another. Sample s3 in the example is an ideal sample as I have defined it. Commented Jul 2, 2018 at 19:13
• Could we edit your question into "What can be used to evaluate difference between an observed sample distribution and a theoretic distribution in the case of categorical data?" Commented Jul 2, 2018 at 20:03
• This issue is covered in a number of posts on site, including stats.stackexchange.com/questions/115453/… (and perhaps stats.stackexchange.com/questions/110600/…) Commented Jul 3, 2018 at 13:01

$$u_2=2\sum{p_ip_j}$$ 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.