The income data is in interval format.
The data does not have individual income measurements. But is in {1,2...10}, that is in deciles, where 1 is the lowest income range and 10 is the highest.
What are the common ways I can compare the income distributions of two groups using this decile interval income data?
It depends on what you mean by "compare". Two possibilities come to mind.
Your data is fully described by two histograms, where the bins are defined by income deciles. There are ways to define distances between histograms. One I personally like is the earth mover's distance or Wasserstein metric. This would allow you to say, e.g., that distributions A and B were closer together than C and D. (Which would of course depend on the specific distance measure you choose.)
You could similarly compare both distributions to a reference distribution, like the uniform one. This naturally leads to the Gini coefficient.
These are purely descriptive measures.
If you want to do inferential statistics or similar on your data, then you can do different things. You first notice that your data is ordinal-data - we don't know the specific income of two individuals, but if they are in different deciles, then we know which individual has a higher income. So I'd recommend you read up on ways to analyze ordinal-data, e.g., by looking through our earlier questions carrying that tag. The Wikipedia page on ordinal regression may also be helpful and has a few references.
