History of the arithmetic mean

The arithmetic mean of a set of points $\{x_1, x_2, ..., x_n\}$ is defined by

$$\frac{1}{n}\sum_{i=1}^n x_i.$$

It is remarkable for its ubiquitous use and universal understanding. It represents a center of mass, a point of equilibrium, a fair ressource distribution, the center of a probability distribution, etc. It is also not only an elementary concept: its generalization by integrals is quite fundamental in higher mathematics.

What is the history of the arithmetic mean? When did it first arise and when did it become widely understood through its different interpretations?