Recall that a random variable X is not its image set A but a (measurable) function from a probability space (Ω,F,P) to a (measurable) set A. In your context, A={x1, x2, ... xn} and the distribution of the random variable X is characterized by a collection of nonnegative real numbers p(x1), p(x2), ..., and p(xn) summing to 1.
As @Chris said, your first task is to define a measure of uncertainty, suitable in your context. In other words, for every distribution p one must define a number U(p).
Two options are to use the variance
U(p)=M2(p)-M(p)2 with M2(p)=∑i xi2 p(xi) and M(p)=∑i xi p(xi),
or the entropy
U(p)=−∑i p(xi) log p(xi).
Both quantities are nonnegative, and zero only in the degenerate case when p(xi)=1 for a given i and p(xj)=0 for every other j.
Once the functional U is defined, if the distribution p changes with time t, one might consider the integral
∫0T U(p(t)) dt
of the functional U applied to p(t) on the time interval of interest.