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If I have a function that calculates the mean value of a set of elements that is an arbitrary subset of some set $X$, does the mean, by definition, have to also be in $X$? (In other words, if the mean can't be in $X$, does that imply that "mean" cannot be defined for $X$?) If not, is there a counter example?

Example: If I can calculate the mean of a set of floobles, does the mean value also need to be a flooble for me to be able to claim that I can actually calculate the mean?

Edit - Possibly More Concrete Example: If I claimed I could calculate the average value of a set of words, but I defined that to mean "the average length of a word in the set", could you then argue that since the average value was not a word, then it is not actually an "average" value of those words?

Sorry if that was confusing, I am having trouble expressing this in words.

Jason C
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    Yes, if you want an average flooble. No, if you want just an average (in whatever way you defined it), not necessarily a flooble. – dtldarek Mar 19 '14 at 01:16

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I think what you are asking is the following, does it make sense to talk about the mean of $\{1,2,3,5\}$ as a subset of $\mathbb{N}$. The answer is no. There is no number in $\mathbb{N}$ that does what you want the mean to do.

Stella Biderman
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There is no universal definition of average. Where averages are usually mentioned you're working with at least the real numbers. There's even a sort-of average taken in group theory, forgot what it was.... ah wait, here we go:

Burnside's lemma

So when something is labeled with the term average there is usually a "summation" operator, although this maybe some non-everyday arithmetic operator like set union for instance, and a final normalizing factor that you scale the sum by.

Daniel Donnelly
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  • I see; I didn't realize "mean" was such a fuzzy concept. "Maximum" and "minimum" have concrete definitions based on $\geq$ and $\leq$, for example, but I guess the question I was really asking was if "mean" had a concrete definition and properties as well. Just to confirm, the answer to that is "no", correct? "Mean" is a fuzzy term that differs depending on context, as long as it somehow represents some concept of a center-y (not sure what word to use) value that is meaningful to whatever problem is being solved? – Jason C Mar 19 '14 at 01:33
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    Yes, correct. That's the way I see it but maybe an expert Category theorist could find an all-connecting generalization of "mean". However I think that's akin to saying gravitational orbits are similar to electron orbits, so the relatedness of the two phenomena is not really interesting yet, unless you're from Sedona, AZ, where I live :D and everything is connected. – Daniel Donnelly Mar 19 '14 at 03:34