  Inference on overall ranking of a set of entities, such as athletes or
players, schools and universities, hospitals, cities, restaurants, movies or
books, companies, states, countries or subpopulations, based on appropriate
characteristics or performances, is an important problem. Estimation of ranks
based on point estimates of means does not account for the uncertainty in those
estimates. Treating estimated ranks without any regard for uncertainty is
problematic. We propose a novel solution using the Bayesian approach. It is
easily implementable, competitive with a popular frequentist method, more
effective and informative. Using suitable joint credible sets for entity means,
we appropriately create {\it credible distributions} (CDs, a phrase we coin),
which are probability distributions, for the rank vector of entities. As a
byproduct, the supports of the CDs are credible sets for overall ranking. We
evaluate our proposed procedure in terms of accuracy and stability using a
number of applications and a simulation study. While the frequentist approach
cannot utilize covariates, the proposed method handles them routinely to its
benefit.
