A novel, non-trivial, probabilistic upper bound on the entropy of an unknown
one-dimensional distribution, given the support of the distribution and a
sample from that distribution, is presented. No knowledge beyond the support of
the unknown distribution is required, nor is the distribution required to have
a density. Previous distribution-free bounds on the cumulative distribution
function of a random variable given a sample of that variable are used to
construct the bound. A simple, fast, and intuitive algorithm for computing the
entropy bound from a sample is provided
