There are many notions of average in mathematics and statistics. Well-known are the *mean*, *median* and *mode*.

Also well-known, amongst the means, are the *arithmetic mean* (A), *geometric mean* (G), *harmonic mean* (H) and *quadratic mean* or *root mean squared* (Q or RMS). Recently, I have become interested in the notion of a *generalised mean* (or *generalized mean*) over positive (non-negative and non-zero) real numbers .

### Common Means

Consider a finite collection of values. As a special case, also consider functions taking a pair of values .

i.e.

and

We might write some of these differently as:

These functions satisfy

;

strictly so if *X* contains at least two distinct values.

(It also so happens, in the special case of only a pair of values, that .)

### Power Means

There is a generalised mean called a *power mean*, defined thus:

Then, abusing this notation a little

Then: .

### Lehmer Means

However, I also noticed that if we write

then we also have:

— in the special case of a pair

which leads to an alternative generalised mean.

Unfortunately (for me), this idea is not new: it is called a *Lehmer mean*:

Then, again abusing notation a little

is the so-called *contraharmonic mean*

— in the special case of a pair

Similarly we have: .

(Many of these means also have weighted variants. It may also be possible to extend these definitions to cover infinite sets *X* [with suitable conditions], or even to continuous distributions.)

### Axiomatisations

(or axiomatizations)

There are many attempts to axiomatise (or axiomatize) the notion of a mean. Examples of axioms for a mean *M* of two values are:

identity/idempotent:

symmetrical:

linear:

left-monotonic:

right-monotonic:

bounded:

non-negative:

positive:

left-continuous; right-continuous; …

These are not independent, and different subsets of these might be considered. It is perfectly possible that some reasonable axioms might not even be consistent with all of the above.

(Other algebraic properties might not be appropriate for a mean function [such as transitivity].)

### Others?

If you know of other interesting generalisations of means or other notions of average, then please leave a comment.

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