I need a metric that not only gives me the statistical distance between two distributions, but that also is comparable to another distance between two completely different distributions, calculated with the same metric. I have looked at the 1-Wasserstein distance and done some small-scale experiments, that seem to show it scales with the sqaure root of the variance of both distributions combined.
So, the metric I came up with is simply
$$ W(U,V)/\sigma([U,V]) $$
where $W$ is the 1-Wasserstein distance and $\sigma$ is the square root of the variance, and $U$ and $V$ are the two distributions in question.
Simply looking at the distributions visually, and the values this metric produces, it seems like a good metric. However, I would like some confirmation on this, as I am not a mathematician. I also couldn't find this metric online. Does it exist / does it have a name?