Why have we defined "average" as $a_1 + a_2 + \cdots + a_N\over N$ only? I am graduate student in Physics and I have taken a graduate level course called Probability and Statistics as an elective. The professor told me that in this course will take a Measure theoretic perspective to the theory of Probability and Statistics. In the first lecture, he posed a problem (among others) that why do we define "averages" as $a_1 + a_2 + \cdots + a_N\over N$ ? One may say that is our intuition, but is it really the average? 
By an Average, we tend to find a number that roughly tells us the "general" picture about the data we have at hand. With averages, of course, we need to have another information about the variance, that is how spread our data is? But does our formal definition of "sum divided by the number of data points" really does what it is intended to do?
Why don't we have any of the following as the definition of averages?


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*$\sqrt{a_1^2 + a_2^2 + \cdots + a_N^2\over N}$ (we use this in physics a lot)

*$\left(\frac{a_1^p + a_2^p + \dots + a_N^p}{N}\right)^{1/p}$

*$(a_1a_2\cdots a_N)^{1/N}$

*$\exp{(\ln{a_1}\ln{a_2}\cdots\ln{a_N})^{1/N}}$
I assume that there must be a way to quantify the effectiveness of these averages and there must be a more effective measure of our intutive average than just 
$a_1 + a_2 + \dots + a_N\over N$ . 
Can someone help me out with this? Thank you in advance!
 A: There are many "averages" of interest in specific contexts, but authors ought to make clear which one they mean. We choose an average for which we can prove a useful result. (In fact, it tends to be a case of choosing a nice average/dispersion pair.) Let's look at a few examples:


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*The classical central limit theorem considers the arithmetic mean of independent samples from a common finite-variance univariate distribution. We call this the sample mean. Its arithmetic mean and variance are easily obtained, and the theorem states that for large samples the sample mean's distribution is approximately Normal. This is very useful because univariate Normal distributions are completely characterized by their arithmetic mean and variance, and all univariate Normal distributions are equivalent up to linear transformations. So if you know how to compute the CDF and inverse CDF of an $N(0,\,1)$ distribution to an adequate level of precision, you can approximate the sample mean's probability distribution. And there are other CLTs that do something similar for, say, a sum of lots of variables satisfying certain niceness conditions. This motivates the prevalence (though by no means universality) of approximately Normal distributions in many natural phenomena. (For the Cauchy distribution, whose variance is undefined, you'd prefer the sample median to the sample mean, because it "settles down" for large samples in a similar way to the sample mean for a finite-variance distribution.)

*The utility of the above exercise hinges on knowing, roughly, what the mean and variance of the underlying distribution are. Let's say you think you know the underlying distribution up to parameters. How do you estimate these parameters? That's a big question, because whatever function of a dataset you propose as a parameter estimate, there are a number of desirable features for such so-called estimators, and sometimes they're at odds with one another. Again, when Normal approximations work, they make our priorities a bit clearer. For example, you can easily determine the parameter values in a Normal distribution that maximize the dataset's likelihood, and you can thereby obtain "unbiased" estimators of the parameters. This means the estimator's arithmetic mean is the true parameter value. One can even find the unbiased estimator with least variance among them all, a nice example of a soluble constrained optimization problem. And given the especial interest in means and variances that comes from Normal distributions, it's a welcome result too.

*On a related note, what if you want to split your money between multiple investments whose historical log-returns have known arithmetic means and covariances (the former concept being necessary to define the latter)? Then you can maximize mean performance subject to a tolerable risk (defined in terms of variance), or minimize risk for a desired mean performance; see here for details. Again, we like these constrained optimization problems because we know how to solve them. You're welcome to instead ponder how to maximize median performance for a preferred entropy, but good luck working out how.

*On the other hand, you wouldn't be as interested in the arithmetic mean if something more useful turned up. As you referenced, if you consider the isotropic kinetics of non-relativistic particles, the Pythagorean theorem gives a particle speed in terms of the velocity's Cartesian components that motivates the root-mean square. And sometimes you'll find the geometric or harmonic mean more useful when you want to learn something. Again, it comes down to what you can solve.

*Sometimes we choose a particular average because it's less misleading than another one. For example, sociologists prefer to consider median income rather than the arithmetic mean income. Why? Because the very non-normal distribution of income is so heavily right-skewed the arithmetic mean is misleadingly high. In fact, data is usually a reasonable fit for a Pareto distribution, which might not even have a finite variance judging by the parameter estimates we get. The most direct inequality measure is discernible from the probability of being below a certain fraction of the median. This is a popular way of defining poverty lines.

*In a first-past-the-post election, the seat winner is the "average" vote, by which I mean the modal one. If each voter chooses one party in a proportional representation system, the seats accorded to a party nationally approximate the arithmetic mean of its $1$-if-voted-for-$0$-if-not performance per voter. In practice, PR has a more complex way of polling voter preferences, which may make yet another average interesting. Meanwhile, the national consequences of FPTP can be really hard to map onto statistical averages we've all heard of, thanks to large numbers of parties.

A: You can try (and will succeed) in pinning down predicates that characterize this particular function. Of course, that is just like the act of choosing axioms.
Here's a book I recently dived into, on the many alternatives, discussion of their properties and recommendations for when they are applicable. Take a look at 350 pages strong Table Of Content to get a feeling for all the properties you can abstract away from the classical mean:

