I'm trying to evaluate a clustering method by looking at the ratio of the mean intra-clustering distance (the average distance between points in the same cluster) to the mean inter-cluster distance (in my case, defined as the average between all pairs of points).
This metric would belong to the class of "internal" clustering methods that try to evaluate clustering without knowledge of ground-truth labels. My basic question is: is there a name for this ratio?
When I look at lists of internal clustering methods (e.g. https://en.wikipedia.org/wiki/Cluster_analysis#Internal_evaluation), I can't seem to find this specific quantity. There are related, but more complicated, quantities, like Dunn's index or the Silhouette coefficient, but what about just this simple ratio?
I'm wondering -- perhaps this simple quantity is not used? If so, why not? If it is used, does it have a name?