Why is the average so nice?

Question

The arithmetic mean is the most common kind of mean in statistics, and is the only thing people think of when they hear "the average". What are the reasons for this?

What nice properties does the arithmetic mean have?

Among them are:

• The arithmetic mean of a data set $S$ is the unique real number $\mu$ such that the deviations $\mu - x_i$ (where $x_i$ is a data value in $S$) all cancel each other out, or, in other words, add up to $0$
• $\mu$ is always in $[\min S, \max S]$, which would make little sense had it been otherwise

...

Edit: with 9 properties listed in just 3 answers, I feel like the big-list tag is fitting.

Answer 1

I can think of three properties (although there would be many others) that make arithmetic mean special

1) Lets say you perform an experiment to measure some quantity and take N measurements. Each measurement contains the quantity plus some noise or uncertainty. If you take the mean of all these measurements then the result would be the "best estimate" of that quantity under the assumption that the noise is Gaussian distributed. "Best" in the sense that it would minimize mean square error.

2) Markov's Inequality (for lists)

If the mean of a list of numbers is M, and the list contains no negative number then [fraction of numbers in the list that are greater than or equal to x] ≤ M/x. (ref: http://www.stat.berkeley.edu/~stark/SticiGui/Text/location.htm).

3) Chebychev's inequality (for lists)

If the mean of a list of numbers is M and the standard deviation of the list is SD, then for every positive number k,

[the fraction of numbers in the list that are k×SD or further from M] ≤ $1/k^2$. (ref: http://www.stat.berkeley.edu/~stark/SticiGui/Text/location.htm).

Answer 2

Some obvious ones:

The mean is the number for which if a set of the same size had all that number, it'd total the same.

The mean is directly proportional to the total.

It can be calculated quickly.

It is easy to understand.

Answer 3

• The sum of the squared deviations of a set values taken about their mean is minimum.
• If $\bar{x_{1}},\bar{x_{2}},\cdots,\bar{x_{k}}$ are the averages of
  $k$ samples with respective sizes $n_{1},n_{2},\cdots,n_{k}$, then
  $\bar{x}$ of all the samples with size $N=\sum_{i=1}^{k}n_{i}$ is
  given by $$\bar{x}=\dfrac{1}{N}\sum_{i=1}^{k}n_{i}\bar{x_{i}}.$$