What statistical measures are recommended for mixing of population samples? I want to know what statistics should be used to measure the mixing of languages in cities in a country/state. I have samples from individuals and know the language they speak and the city they live in. The cities are not all the same size and there are 3 different language groups.
The samples are taken uniformly from the total population so that larger cities have a proportionally larger number of samples in the data set.
I want to avoid naive approaches such as; taking the value from A/B since 100/50 is the same as 2/1 and have equal contributions.  
 A: A humble Chi sqare test is probably all you need to test the null hypothesis that there is no relationship between mother tongue and city ie that the proportion of speakers is the same in each city (note that this is not the same as all proportions are a third).
As per my comments, I'm not sure this is that useful a question, depending on the context.  After all, you would expect different cities to have different proportions of languages wouldn't you, on historical, geographical and cultural grounds?  So you will almost certainly reject a null hypothesis of equal proportions.  
But the test would be something like the below.  The numbers in the table represent the number from a sample reporting that language as their mother tongue (made-up data).
> x <- data.frame(
+ row.names=c("London", "New York", "Hanover"),
+ english=c(100,100,10),
+ german=c(5,8,60),
+ french=c(7,4,12))
> x
         english german french
London       100      5      7
New York     100      8      4
Hanover       10     60     12
> 
> # inbuilt chi square test:
> chisq.test(x)

        Pearson's Chi-squared test

data:  x 
X-squared = 174.4, df = 4, p-value < 2.2e-16

> 
> # or, by hand:
> # First, what are the "expected" values if there 
> # is no relationship between city and language
> e <- apply(x,1,sum) %o% apply(x,2,sum)/sum(x)
> e
         english german french
London     76.86  26.72  8.418
New York   76.86  26.72  8.418
Hanover    56.27  19.56  6.163
> sum((x-e)^2/e)
[1] 174.4
> 

A: If you are only looking for a score you can take inspiration from the question
How does one measure the non-uniformity of a distribution?.
If you have a perfect mixing, the distribution of your languages should be uniform in every city (i.e. 1/3 each language group). If mixing is imperfect, it will not be uniform.
The answer to this post suggests using the $\chi^2$ metric, the entropy or the Kullback-Leibler divergence. I would actually use the last one, which easily allows you to normalize for unequal language distribution in the whole population.
