I was wondering if there is some general notion of central tendency. I had in mind the following axiomatization: A measure of central tendency on a totally ordered set $X$ is a family of functions $\mu_n : X^n \to X$ such that
- If $a_i \geq b_i$ for all $i \in \{1, \ldots, n\}$, then $\mu_n(a_1, \ldots, a_n) \geq \mu_n(b_1, \ldots, b_n)$.
- If $a_{n+1} \geq \mu_n(a_1, \ldots, a_n)$, then $\mu_{n + 1} (a_1, \ldots, a_{n + 1}) \geq \mu_n (a_1, \ldots, a_n)$. The same statement holds with $\geq$ replaced by $\leq$ or $=$.
- $\min(a_1, \ldots, a_n) \leq \mu_n(a_1, \ldots, a_n) \leq \max(a_1, \ldots, a_n)$
- If $\sigma \in S_n$, then $\mu_n(a_1, \ldots, a_n) = \mu_n(a_{\sigma(1)}, \ldots, a_{\sigma(n)})$ (the $\mu_n$ are symmetric).
It seems like everything we call a measure of central tendency satisfies these axioms. For example, the arithmetic, geometric, and harmonic means; median; and mode (when it is unique) all satisfy these axioms. Based on this, these axioms seem general enough, but are they too general? Do they allow functions which we wouldn't call measures of central tendency?
I'm not that invested in the axiomatization I've layed out above. I'm mostly interested in whether or not this sort of thing has been written about. How have others attempted to generalize the notion of "central tendency?"