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I want to evaluate sets of numbers by scoring how "spread out" they are compared to other sets in a collection. So let's say we have $\mathcal{X} = \{X_i\}$ a collection of sets of numbers. I want a scoring rule that measures how spread out the numbers are. These sets might be uncountably infinite. I talk about maxima and minima, but what I meant was suprema and infima. I can't be bothered to change all the mentions of max and min...

I'd like the rule to have the following properties:

  • If the spread is 0, the score is 0. That is, if $\max X_i = \min X_i$, the score is zero
  • If all the sets in the collection have the same spread, the score of each set is zero. This rules out just taking $\max X_i -\min X_i$ as the rule. So if $\max X_i - \min X_i = \max X_j -\min X_j$ for all $i,j$ then the score is zero.
  • The rule should "scale". If I multiply all the numbers in all the sets in the collection by $a$ then the score of each should also be multiplied. So $f_{\mathcal{X}}(aX)=af_{\mathcal{X}}(X)$ (The subscript shows that the function is relativised to a particular collection.)

I don't want to make any assumption about the sets being measurable, so I want to rely only on their maxima and minima to do this. I've been playing around with various possibilities, but I can't seem to find anything that works. Am I being dumb? What do functions that satisfy these constraints look like?

To clarify, these sets are "unstructured". They could contain uncountably many numbers, but I don't want to assume I have any kind of distribution function or measure over them. This means (I think) that most of orthodox statistics is inapplicable: I cannot determine a variance for the sets since I don't have a probability distribution over them. Hence no expectation. Hence no variance. [Hence rolling back the edit that added the tag.]

I am interested in finding a function (a function of the extrema, probably) that has the properties I listed above. I guess the extrema are the only useful characteristics of the set that I can safely make use of.

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What sort of numbers are these? Are the $X_i$ finite? If not minimal and maximal elements might not exist. Either way it seems as though you want to have a measure on some collection of sets (that is have a measure on the space of $\mathcal X$'s) and just have the "size" of $X_i$ as $\max X_i-\min X_i$. –  Asaf Karagila Jul 26 '11 at 12:07
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What's wrong with $S_j=A_j-\min_i A_i$ where $A_i=\max X_i-\min X_i$? –  fedja Jul 26 '11 at 12:14
    
expectation of $X^2$? –  Alice Jul 26 '11 at 12:31
    
What do you mean by a score? In your first bullet you give a score to a single set, or is this supposed to be qualified 'if for all'? In your second bullet you give a score to the collection of sets, and I'm not sure how this is the same concept. For example the second bullet applied to a collection containing a single set would give a zero score (because all the spreads would be equal). I think you have not quite pinned down the concept you are looking for yet. –  Mark Bennet Jul 26 '11 at 14:40
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@fedja what's wrong with writing that as an answer? –  Seamus Jul 26 '11 at 17:06

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