Measures of randomness

Does anyone have any ideas on how to quantify randomness? More, specifically, I need a measure of clustering.

Let's say I have 2 groups within a dataset. I plot their values based on two measures and color-code by group. When I do the same using other measures, I get a different scatter of red and black observations.

Is there a way to quantify whether the colors are more clustered/scattered (i.e., less able to distinguish between groups) from one plot to another?

In reality, I have multiple groups. And I'm looking to find the measures that lead to the most "even mixing" of the groups.

If you can implement something in R, I'd be even more appreciative!
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Well, I would use an F-test (especially because your have several groups). The F-test measures true difference between means, and their variance. My thinking is that the more variance, the more randomness. So in essence, in your situation here, you want F as low as possible. Your situation is very different from any other (normal one), since we usually use the F-test for the exact opposite of what you want here.

Truth is I'm not sure about this method, but it's my shot at it.

The F-test in R is var.test(group1, group2, group3).


Less is more. Stay pure. Stay poor.
In addition, perhaps something along the lines of an r-square and partial r-square value for the groups?
maybe also looking at data with histograms?