Can we represent a population by its mean? Can we represent a population by its mean?
Is the answer simply "yes, unless there are outliers or the data is skewed"?


Can we represent a population by its mean?

Theoretically
Yes, unless the variance is non-zero.
Populations can be expressed by moments, cumulants, and other type of statistics. In the special case that the variance is zero, then all the moments/cumulants are zero, except for the mean, and the population is entirely determined by the mean.
This special case, when the variance is zero, is a degenerate distribution. In most cases the variance is not non-zero and this theoretic case is not applicable.
Practically
Yes, unless other properties are important.
For many practical cases it is only the mean that is important, or mostly important.
However, this is a bit of a pitfall. Example: Say, you design seats for an airplane, then you should not use the average size of the population, but instead some quantile, like the x% largest people. You want to design the seats such that only x% may have problems with it. You don't care for the mean in such a case, but the mean is often used as a default statistic to represent a population.
