In many systems, the mean really does summarize the whole. E.g. kinematics works just fine using the mean point mass for solids (i.e. the centre-of-gravity) in place of the whole.
That's quite true, and the same statement also points to where the arithmetic mean isn't a good choice: where the goal isn't so much to summarize a whole as to give an insight into populations. For instance, if you are trying to figure out the "average" income in a city, the median provides better intuition than the mean.
Furthermore, there are many situations where both the mean and median (and even the mode) fail to provide insight. For instance, the average person has one testicle, whether defined by mean or median (and even the mode is not much help here, with its equally useless value of 0). We should really think of summary stats as something to only be used when a distribution is well understood. In other cases, a simple frequency distribution diagram provides much better insight, and isn't very hard to do.
There are two options:
1) The ratio between women and men is not exactly 1. In this case, the median number of testicles is not 1 (it is either 0, if there are more women than men, or 2 if there are more men than women).
2) The ratio between women and men is exactly 1. This is not true in real life, but let's pretend it is. In this case, any number between 0 and 2 is a median (since any number between 0 and 2 minimizes the sum of absolute deviations), and not just 1.
TL;DR: The median number of testicles is almost surely not 1 (or, at least, not uniquely so).
If you look at ten people with 2, ten people with 0, and one person with 1, then the median will be 1.
If you make it twelve people with 2, still ten with 0 and one with 1, suddenly the median is 2.
Across an entire population, I would hazard a guess the median would be 0 or 2 unless the difference in numbers of each sex is small enough that people with 0<X<2 testicles might fall right in the middle of the distribution.
No. The median is not necessarily unique (just like the mode, and unlike the mean). For more information, you can check here: https://en.wikipedia.org/wiki/Central_tendency#Uniqueness
A median (and not necessarily the median) is any point that minimizes its distance from a set of points under the L1 norm and often you have several points that minimize such distance (such as in the example you provided, assuming perfect 50/50 split between men/women).
> If you look at ten people with 2, ten people with 0, and one person with 1, then the median will be 1.
Correct.
> If you make it twelve people with 2, still ten with 0 and one with 1, suddenly the median is 2.
Sure.
But, again... you are choosing cases where the median of a set of points is unique (odd number of points aligned along a single dimension)... this is the exception, rather than the rule (particularly in higher dimensions than one).
> Across an entire population, I would hazard a guess the median would be 0 or 2 unless the difference in numbers of each sex is small enough that people with 0<X<2 testicles might fall right in the middle of the distribution.
Yes... as I said. The actual median is not 1, but actually either 0 or 2 (because the ratio between men and women is not exactly 50/50). But, again, even in the (rather unlikely) case that the ratio between men and women would be exactly 50/50, the median would still not be uniquely defined as 1 (in such a pathological case, any number between 0 and 2 is a median).
EDIT: after re-reading your comment carefully... I have not considered the subpopulation of people with a single testicle (which is nonzero); so I stand corrected... the median could be 1 (but it is very unlikely that it is the case... it's probably either 0 or 2).
E.g. in this case you can calculate the motion of the center of mass from the forces applied to the object. The same can’t be said for other averages.