1. Correlation ellipses are very helpful: you can clearly see the "weight" of the dependency between the categories. Here is example of my R script:

Code R script to model correlation ellipses showing correlation b...

2. Correlogramms are also useful:

Code R script to plot statistical correlogram (Pearson method), M...

You could use a Kruskal Wallis for nonnormal data.

Hope this helps,

Matt

In my opinion the best way to account for differences in population of each racial group would be pre-sampling, by using a stratified sampling method instead a simple random sampling. This way, smaller populations would contribute less for your sample than larger population, i.e. the sampled races size would be proportionally similar to the population races size. If this is not possible to accomplish, then I think that using a linear regression model with weighted observations based on stratum size (population size) would be a good way to go.

For an example of using weights with lm() and glm() funtions in R you can refer, for instance, to:

https://www.r-bloggers.com/linear-models-with-weighted-observations/

Why are you interested in weights? Generally, weighting involves estimating the group total or equivalently the group mean within a fixed finite population context. Also note that income is highly skewed with heavy tails & the mean will not be representative of the group. (For this reason the median is often used in income differential analysis.) Additionally, I am not sure the group totals have a meaningful real world interpretation. You seem to be mixing up sampling & modelling concepts. You might be thinking of a weighted model approach but you really need to think about what you are modelling.

I think If you want to account for the size of the population in your regression, you can add logs of population as regressors.