It was a satire article that sparked this question, in which the median income was stated to be \$$0$ and the unemployment rate to be $100\%$ in a particular area (the Mariana Trench in the western Pacific Ocean) where the population is clearly zero humans. I began to wonder if that was correct and if the calculations/statistics can even be performed and if they make sense.
I know that any sum over the empty set is $0$, i.e. $\displaystyle\sum_{x\in\emptyset} f(x) =0$. One can understand this because $0$ is the identity of the addition operation. Similarly, the product over the empty set is 1, i.e. $\displaystyle\prod_{x\in\emptyset} f(x) = 1$ due to the identity of multiplication being $1$.
To my knowledge, the median of the empty set is simply undefined.
If instead, we are interested in the average salary and define it as $\frac{1}{|A|}\sum_{x\in A}S(x)$ where $A$ is the empty set and $S(x)$ is the salary of individual $x$ then we end up with $\frac{0}{0}$ which is undefined.
Interestingly (but vacuously), it is true that there is no resident of the Mariana Trench with a salary below \$$1,000,000,000$. It is vacuously true that they are the richest people in the world!
If we define the unemployment rate as the number of unemployed residents divided by the total number of residents, then we again get $\frac{0}{0}$. Again, it is vacuously true that all residents have great jobs and that all residents are unemployed.
So my conclusion is that the satire article makes math errors.
Now maybe the questions are: (1) Am I making any mistakes? (2) Is there a way to redefine average or median or to calculate the unemployment rate so as to get a different result? Maybe by using only empty products we can get a numerical answer for the average, but we would want this slick definition to be consistent with the standard calculations, i.e. that it also works as expected with a non-empty set.