The concept of "mean" roams far wider than the traditional arithmetic mean; does it stretch so far as to include the median? By analogy,
$$
\text{raw data} \overset{\text{id}}{\longrightarrow} \text{raw data} \overset{\text{mean}}{\longrightarrow} \text{raw mean} \overset{\text{id}^{-1}}{\longrightarrow} \text{arithmetic mean} \\
\text{raw data} \overset{\text{recip}}{\longrightarrow} \text{reciprocals} \overset{\text{mean}}{\longrightarrow} \text{mean reciprocal} \overset{\text{recip}^{-1}}{\longrightarrow} \text{harmonic mean} \\
\text{raw data} \overset{\text{log}}{\longrightarrow} \text{logs} \overset{\text{mean}}{\longrightarrow} \text{mean log} \overset{\text{log}^{-1}}{\longrightarrow} \text{geometric mean} \\
\text{raw data} \overset{\text{square}}{\longrightarrow} \text{squares} \overset{\text{mean}}{\longrightarrow} \text{mean square} \overset{\text{square}^{-1}}{\longrightarrow} \text{root mean square} \\
\text{raw data} \overset{\text{rank}}{\longrightarrow} \text{ranks} \overset{\text{mean}}{\longrightarrow} \text{mean rank} \overset{\text{rank}^{-1}}{\longrightarrow} \text{median}
$$
The analogy I am drawing is to the quasi-arithmetic mean, given by:
$$M_f(x_1,\dots,x_n)=f^{-1}\left(\frac{1}{n}\sum_{i=1}^{n}f(x_i) \right)$$
For comparison, when we say that the median of a five-item dataset is equal to the third item, we can see that as equivalent to ranking the data from one to five (which we might denote by a function $f$); taking the mean of the transformed data (which is three); and reading back off the value of the item of data that had rank three (a sort of $f^{-1}$).
In the examples of geometric mean, harmonic mean and RMS, $f$ was a fixed function that can be applied to any number in isolation. In contrast, either to assign a rank, or to work back from ranks to the original data (interpolating where necessary) requires knowledge of the entire data set. Moreover in definitions I have read of the quasi-arithmetic mean, $f$ is required to be continuous. Is the median ever considered as a special case of quasi-arithmetic mean, and if so how is the $f$ defined? Or is the median ever described as an instance of some other wider notion of "mean"? The quasi-arithmetic mean is certainly not the only generalization available.
Part of the issue is terminological (what does "mean" mean anyway, especially in contrast to "central tendency" or "average"?). For instance, in the literature for fuzzy control systems, an aggregation function $F:[a,b] \times [a,b] \to [a,b]$ is an increasing function with $F(a,a)=a$ and $F(b,b)=b$; an aggregation function for which $\min(x,y) \leq F(x,y) \leq \max(x,y)$ for all $x,y \in [a,b]$ is called a "mean" (in a general sense). Such a definition is, needless to say, incredibly broad! And in this context the median is indeed referred to as a type of mean.$^{[1]}$ But I am curious whether less broad characterisations of the mean can still extend far enough to encompass the median – the so-called generalized mean (which might better be described as the "power mean") and the Lehmer mean do not, but others may. For what it's worth, Wikipedia includes "median" in its list of "other means", but without further comment or citation.
$[1]$: Such a broad definition of mean, suitably extended for more than two inputs, seems standard in the field of fuzzy control and cropped up many times during internet searches for instances of the median being described as a median; I will cite e.g. Fodor, J. C., & Rudas, I. J. (2009), "On Some Classes of Aggregation Functions that are Migrative", IFSA/EUSFLAT Conf. (pp. 653-656). Incidentally, this paper notes that one of the earliest users of the term "mean" (moyenne) was Cauchy, in the Cours d'analyse de l'École royale polytechnique, 1ère partie; Analyse algébrique (1821). Later contributions of Aczél, Chisini, Kolmogorov and de Finetti in developing more general concepts of "mean" than Cauchy are acknowledged in Fodor, J., and Roubens, M. (1995), "On meaningfulness of means", Journal of Computational and Applied Mathematics, 64(1), 103-115.
Best Answer
Here's one way that you might regard a median as a "general sort of mean" -- first, carefully define your ordinary arithmetic mean in terms of order statistics:
$$\bar{x} = \sum_i w_i x_{(i)},\qquad w_i=\frac{_1}{^n}\,.$$
Then by replacing that ordinary average of order statistics with some other weight function, we get a notion of "generalized mean" that accounts for order.
In that case, a host of potential measures of center become "generalized sorts of means". In the case of the median, for odd $n$, $w_{(n+1)/2}=1$ and all others are 0, and for even $n$, $w_{\frac{n}{2}}=w_{\frac{n}{2}+1}=\frac{1}{2}$.
Similarly, if we look at M-estimation, location estimates might also be thought of as a generalization of the arithmetic mean (where for the mean, $\rho$ is quadratic, $\psi$ is linear, or the weight-function is flat), and the median falls also into this class of generalizations. This is a somewhat different generalization than the previous one.
There are a variety of other ways we might extend the notion of 'mean' that could include median.