Shauna, I absolutely love this question - it gets to the heart of where quantitative analysis can be useful and also really tricky when it comes to equity. Asking the right question, and disaggregating your groups according to the needs of each, is key to finding the answer to service distribution thresholds.
The mathematical threshold piece, is pretty simple, mathematically, distribution of services should be within some window (depending on n, I might say something inside 5% of the distribution of the population). But the mathematical threshold isn’t really the question here, the question is, according to these and additional demographic characteristics, which segments of the population should be receiving services.
This brings us to a second component, which I think is the root of your question. Generally, smaller segments of the population have less bargaining power, less sway over voting, and pay fewer taxes as a group, making it so that they receive (generally) fewer resources than the majority segment of the population. Taking this logic, there should be an inverse relationship between demographic group size and need for services.
Let’s break down the math problem with our set assumptions:

There is an inverse relationship between demographic group size and need for services

Services don’t target the full population of the county so let’s assume our goal is to get services to 40% of our full county population

Let’s assume, for math’s sake, that our full population is 10,000 ( with our targeted 40% at 4,000)
The table breakdown for your two scenarios looks like this:

Full population

scenario 1

scenario 2

Green

9161

3158

34%

3681.6

40%

Blue

605

524

87%

234

39%

Purple

234

318

136%

84.4

36%

To get to the answer, I would reverse engineer the problem to say, for example, 36% of the green population require services, 79% of the blue pop require services, and 96% of the purple pop require services (these pcts get to our goal of 40% of the full population) . We could also use percentages that represent a true inverse which I’ll include below too, though this method achieves well under our target of 40% of the full population, serving only 1566 individuals.

I love the additional dimension you’re bringing to the question. It is very helpful to me, and I am excited to share it with my team.

I don’t completely follow how you’ve calculated the math of inverting and reverse engineering based on the assumptions you defined. Perhaps some additional information would be helpful to build on the language we are using to communicate.

The percentage of the population receiving the mental health services we provide is orders of magnitude less than 40%. It ranges from 0.12% of 374,682 to 2.17% of 98,789. You’ve made me curious now to go find what percentage we should be serving.

That aside, I’m trying to understand the math you used to come up with the percentages of people within each color group who require services. Are there sources you would recommend about calculating needs for services?