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Edit: Ignore this Most of this doesn't make sense and is beyond edits but its too late to delete. If I make too many edits I could be kicked out of this sub like I was in math stack exchange.

I don't have a statistical background but I believe my ideas can be explained using statistics. For now, I will explain using pure mathematics.

Motivation

I want a way to determine how evenly distributed a set of values are.

For example, I believe $\left\{0/6,1/6,2/6,3/6,4/6,5/6,1\right\}$, should be one of many sets with the most even distribution since the differences between consecutive elements are the same throughout.


Method for determining Even Distribution

Suppose we have a set of values

$\left\{a_1,a_2,...,a_n\right\}$

Take the difference between consecutive values:

$\left\{a_2-a_1,a_3-a_2,...,a_{n}-a_{n-1}\right\}$

Out of the set of differences, we take of the largest difference, the largest plus second-largest difference, the largest plus the second-largest plus third largest difference, and continue until we get the largest difference plus all the way to the n-th largest difference (aka the smallest difference).

Finally, we take the mean of these values.

The closer the result is to $1/2$ the more evenly distributed the original set of values are.

Even if $n\to\infty$, there should be instances, for certain sets of values, where my results can be determined.

Question

Is there a statistical definition for this? If or if not, how do express this in terms of mathematics/statisitcs?

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    $\begingroup$ Have you tried applying your "Method for determining Even Distribution" to your example "{0/6,1/6,2/6,3/6,4/6,5/6,1}"? $\endgroup$
    – user20637
    Commented Aug 15, 2021 at 19:25
  • $\begingroup$ @user20637 I realized my definition makes no sense but it's too late to delete it. $\endgroup$
    – Arbuja
    Commented Aug 15, 2021 at 19:35
  • $\begingroup$ Is this the question you want to ask? How does one measure the non-uniformity of a distribution? $\endgroup$ Commented Aug 25, 2021 at 0:45

1 Answer 1

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Maybe not exactly what have been asked for, but the first things come to my mind are

Gini coefficient

and Shannon index

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