# What can I use to compute a similarity (or diversity) index for a sample with “multidimensional” attributes?

Current problem: We have a batch of $$n$$ items for which we capture their details with $$m$$ attributes. It could look something like this: The goal is to compute an "index" that says how "similar" this batch is (or how "diverse" it is ($$1-similarity$$?))

I've looked at things like the Simpson's index but that requires things to be grouped as "species" and I don't have such a thing in my domain. Even the Jaccard index doesn't seem to cater to my need since it compares 2 populations and I don't have such discrete sets. I'm not sure if entropy indexes would help but am open to suggestions and would like to see some examples of problem/computation.

What's are some "indexes" that I could explore prior to rolling my own (enjoyably pursuing actively but nothing concrete/satisfying yet)? Any references/articles/pointers to examples that are close to what I'm trying to achieve and could "map" my problem to that?

Requirement: If every item is the "same" across all $$m$$ attributes, then $$diversity = 0$$ (similarity is $$1$$) else if all are different then $$diversity = 1$$ (or similarity is $$0$$) and other combinations somewhere in between.

The problem could be equivalently reformulated as "Each entity has a particular DNA string representation - how similar is the group?", if that helps.

UPDATE: After mulling further I was wondering if I could use the Simpson's index by auto-collating "species" in my data set - rows with the "same values" across dimensions are grouped together and maybe assumed as a species. The count of the number of "organisms" in a group $$n$$ and total number of such groups $$N$$ could feed into the formula below:

$$D = 1-\frac{\sum n(n-1)}{N(N-1)}$$

Would this work?