Can we represent a population by its mean?

Question

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

Answer 1

If I tell you that the average height of my friends is 173 centimeters, does it tell you anything about my friends? It tells you what the average height is, but doesn't tell you almost anything about the distribution of their heights, etc. You cannot make any reasonable guesses about the heights of my friends. The number "represents" the population only if you want to limit the representation to a single number with all the limitations of such representation.

As for outliers, it's not that simple. In fact, the sensitivity of the mean is a feature, not a bug, as you can learn from the If mean is so sensitive, why use it in the first place? thread. So the choice of the point estimate for the central tendency would be more complicated than deciding if there are outliers.

Answer 2

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.

Answer 3

Easiest Example

It may help to look at a visual example, which is always good practice anyway when trying to understand where your data is centrally distributed. The easiest example is simulating a normal distribution. Here I have done so, including a mean:

x <- rnorm(n=1000,
           mean=20,
           sd=10)

The mean should be obvious and plotting it makes it easy to see why.

plot(density(x), main = "Density of X Variable")

From here I will plot the mean in red and the median in blue. Here you can see they are exactly the same, which make sense because the data is pretty easily seen to be mostly distributed in the center of the graph.

When Means Become Less Helpful

Now the most obvious example of when means become less helpful for understanding your data is when the data is heavily skewed. Here I have plotted a chi square distribution with mean and medians.

z <- rnorm(n=1000, mean=20, sd=10)^4
plot(density(z),main="Density of Z Variable")
mean(z)
median(z)
abline(v = mean(z),                    
       col = "red",
       lwd = 3)
abline(v = median(z),                    
       col = "blue",
       lwd = 3)

You can see that the median in blue is a much better approximation of where the center of the distribution is, as the mean is being dragged away by the data points on the right.

So to answer your question, the mean in useful when it's practical purposes is useful. If the population does not fit the mean well, then forgive the pun, the mean may not mean so much.