Nonparametric (or distribution-free) statistics have been widely used in psychological research because behavioral data can be messy and inconsistent with the Gaussian model for measurement error. Distribution-free procedures only use categorical or rank information, so they avoid the problems of outliers and violations of distributional assumptions. Yet frequentist nonparametric procedures are still subject to the limitation of relative frequency theory, which stems from the founding assumption that population parameters cannot be represented by probability distributions. Bayesian statistical methods, by contrast, allow for prior and posterior probability distributions for population parameters, so they rigorously provide experimental scientists with a probability representation of the population parameters of interest. The Bayesian counterpart for a set of distribution-free statistical methods is a relatively recent development. This paper is a detailed discussion of the DFBA package of R functions, which is designed for doing distribution-free Bayesian analyses for the common nonparametric procedures. Included in the package are functions that enable the user to explore the relative power for computer-based data that can be sampled from nine different probability models. The distribution-free procedures have almost the same power as the t test when the data are normally distributed, but for eight other alternative probability models, the distribution-free Bayesian procedures have greater power than the frequentist t.