I want to describe a statistic on my data. For example, I have data on firms (companies), and I have a measure of $x$ for each of them. Now, firms differ in size, which could be understood in terms of number of employees, or value of sales, or other metric.
Say I want to compute $\bar x$. A simple arithmetic average will treat small and large firms identically. However, a weighted average (using a measure of relative size as weight) will give more weight to the large firm and less weight to the small firm. In a sense, a weighted average divides large firms into smaller units, making them all of equal size, and assigning an identical $x$ to them.
Now, because the unweighted mean is contained in the weighted one (in the sense that, if firms have identical size, the two are the same), it might seem logical to assume that the weighted average is unambiguously superior to the unweighted one. But is this the case?
I think a counterexample would suffice here. Is there any sense in which we are better off knowing the unweighted average of $x$ rather than a weighted one?
PS: surely, one might say "compute both". But say you are conducting a study were if you use both definitions, you are essentially duplicating the number of tables, graphs, and their analysis. In this case, it might be better just to focus on one definition of $\bar x$.