nLab mean

Idea

The term “mean” or “average” usually refers to a value (for example a number) that lies between some given values. For example, the number $3$ is the average of $2$ and $4$.

The idea admits several generalizations in different fields such as geometry, analysis, and probability theory.

As algebraic operations, several notions of “mean” give rise to probability monads.

Definitions

Given two numbers or vectors $x,y$, their arithmetic mean or average is the number or vector

$$\frac{x+y}{2} .$$

More generally, given $x_1,\dots,x_n$, their arithmetic mean or average is given by

$$\frac{x_1+\dots+x_n}{n} .$$

Weighted averages

Given numbers or vectors $x_1,\dots,x_n$ and nonnegative real numbers $p_1,\dots,p_n$ such that $p_1+\dots+p_n=1$ (equivalently, a discrete probability distribution), the weighted average of the $x_i$ with weights $p_i$ is given by

$$\sum_i p_i\cdot x_i \;=\; p_1\cdot x_1 + \dots + p_n\cdot x_n .$$

In other words, a weighted average is the same as a convex combination, or as the expectation value of a discrete random variable?.

Continuous generalization

One can replace the sum with an integral and obtain a continuous analogue of weighted averages as follows. First of all, fix a Banach space $V$ (for example the real line).

• Instead of values $x_1,\dots,x_n\in V$ one can take a measurable space $X$ and a measurable function $f:X\to V$, which one can interpret as a selection of possibly infinitely many values $f(x)$ of $V$;
• Instead of weights $p_1,\dots,p_n$ one can take a probability measure on $X$;
• Instead of a sum one can now form the convex mixture
  $$\int f(x)\, p(d x)$$

  given by integration (for example, Bochner integration?).

In the language of probability theory, this is the expectation value of the random variable $f$ on the probability space $(X,p)$.

Nonlinear generalizations

Given numbers $x$ and $y$,

• Their geometric mean is given by

  $$\sqrt{xy} .$$

  In other words, one is taking the arithmetic mean of their logarithms.

• Their quadratic mean is given by

  $$\sqrt{\frac{x^2+y^2}{2}}$$

  In other words, one is taking the mean of the squares. This is used, for example, to compute the variance? in probability theory.

• Their harmonic mean is given by

  $$\frac{1}{\;\frac{1/x+1/y}{2}\;}$$

  In other words, one is taking the mean of the inverses.

One has the inequality

$$harmonic \le geometric \le arithmetic \le quadratic ,$$

and equality holds if and only if $x=y$.

The same can be extended to any tuple of numbers.

Similarly, in the continuous case, given a probability space $(X,\mu)$ and a random variable $f:X\to\mathbb{R}$, one can take the mean

$$\left( \int |f|^p d\mu \right)^{1/p}$$

for any $p$, which is equivalently the L^p norm of $f$.

Related concepts

• probability monad
• expectation value
• convex combination
• convex mixture
• L^p space
• mean value theorem

References

• Wikipedia, Mean