How is a mean defined?

Question

How exactly is a mean defined? There are obviously some similarities I can see between different types of means like the arithmetic and geometric means, but I would like to formalize the idea of a mean. So far, here is my progress:

Let $S=\{(x,n):n\in\mathbf N\}$ be nonempty. Essentially, $x$ represents a data point and $n$ represents the number of repetitions of that data point. Then we can represent the arithmetic mean as $$A(S)=\frac{1}{|S|}\sum_{(x,n)\in S}nx$$ and the geometric mean (assuming $x\geq0\,\forall (x,n)\in S$) as $$G(S)=\bigg(\prod_{(x,n)\in S}x^n\bigg)^{1/|S|}.$$ It is easy to see that if $|S|=1$, i.e. $S$ contains only one data point regardless of repetition, that both means should equal the value of that data point. What other properties are needed to define a mean?

Answer 1

As you can observe, the geometric and harmonic and RMS means are equivalent to the arithmetic mean modulo a nonlinear, invertible transformation:

$$\log G(S)=A(\log S),$$

$$G(S)=e^{A(\log S)},$$ and $$\frac1{H(S)}=A\left(\frac1S\right),$$ $$H(S)=\frac1{A\left(\dfrac1S\right)},$$

and

$$RMS^2(S)=A\left(S^2\right),$$ $$RMS(S)=\sqrt{A\left(S^2\right)}.$$

with a somewhat condensed notation.

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An empirical mean is homogeneous (same dimension as the data), insensitive to a permutation, representative of a central trend and comprised between the extreme values.

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We can invent the "exponential mean", such that

$$e^{E(S)}=A(e^S),$$

$$E(S)=\log A(e^S).$$

Answer 2

Let us assume the $x$ are samples from a probability distribution $p$. (Let $X\sim p$ be such a random variable.) For some function $m$ to be a mean we could define it using following properties:

In general you probably want $m$ to be a consistent and unbiased estimator of the expected value $E[X]$.