This is a very good question.

@Behnam Farid's suggested use of entropy to measure the variability of a set of sets is excellent!   

Here is an example.   Instead of writing set of sets, I write powerset P(A) for the set of sets of a set A.

Let P(A) be the powerset of a nonempty set A of points on the surface of an n-sphere.   For each subset B of P(A), find the convex hull convB.   Let B1,...,Bn be subsets with the same cardinality and let X = {|convB1|,...,|convBn|} be the set of cardinalities of the convex hulls of B1,...,Bn.   Then |minX - maxX| is a measure of the variation of the convex hulls of the subsets. 

Another possible measure of variability would be the distribution of the sizes of the boundaries of the subsets of a set of open sets.   

For other sources of measurement of the variability of a powerset, see

https://arxiv.org/pdf/1405.2238.pdf

http://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0419723-X/S0002-9939-1976-0419723-X.pdf

http://dx.doi.org/10.1016/j.ins.2013.05.013

https://www.researchgate.net/publication/306082616_Equivalence_and_correct_operations_for_soft_sets

Hope these two references will help you.

Working Paper Equivalence and correct operations for soft sets