From a computer simulation I have built a histogram of the results and normalized it so that the probability of finding a point $X$ in bin $b_j$ is $\sum_j P(X \in b_j) =1$. From this I have calculated the histogram's Shannon entropy $H$ in order to have some way to quantify the "predictivity" of $P$.
Now, while I get a number easily enough, I'm having a hard time understanding what I should do with it. My first thought was to compare $H$ for $P$ versus $H$ for the uniform distribution over the same $X$-range, since this has the maximal entropy (we know $X$ must belong to a finite range). Or I could compare the $X$-range to some "effective volume" $\Delta X$, where $\Delta X$ is the range over which a uniform distribution with the same $H$ as my histogram has been defined. I freely admit these aren't wonderful comparisons, since my histograms don't look at all like uniform distributions.
I work in a field that does not regularly use $H$ as a statistic, so I can't just give my reader a number and be done with it. However, I know it's a valuable quantity for my histogram. My question is: How would you report, describe, and compare the Shannon entropy for experimental/simulated histograms?