The information content of an outcome of a variable is $h(x_i)=-\log_2 (p(x_i))$. I am interested in using this concept to provide more insight on the differences in the distributions of two independent variables.
I am aware of entropy and relative entropy for comparing the expected value of these variables, and I'm using these as needed. However, I am looking for a metric that captures the shape of the distributions that would be graphically captured by a histogram. Is there any reason that I can't take the sum of information content values for the various outcomes of a variable? Effectively, calculating: $$ h(X)=\sum(-log_2 (p(x_i))$$
These summed information content values give a measure of the shape of the distribution.
I've tried looking for examples of people using this metric, but have come up empty. I'm not sure what it would be called -- presumably just the information content of the variable. I can't see any mathematical reason why this can't be done, but I thought someone here might be able to fill me in if there's something I'm missing.