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Every day cognitive and experimental researchers attempt to find evidence in support of their hypotheses in terms of statistical differences or similarities among groups. The most typical cases involve quantifying the difference of two samples in terms of their mean values using the t statistic or other measures, such as Cohen's d or U metrics. In both cases the aim is to quantify how large such differences have to be in order to be classified as notable effects. These issues are particularly relevant when dealing with experimental and applied psychological research. However, most of these standard measures require some distributional assumptions to be correctly used, such as symmetry, unimodality, and well-established parametric forms. Although these assumptions guarantee that asymptotic properties for inference are satisfied, they can often limit the validity and interpretability of results. In this article we illustrate the use of a distribution-free overlapping measure as an alternative way to quantify sample differences and assess research hypotheses expressed in terms of Bayesian evidence. The main features and potentials of the overlapping index are illustrated by means of three empirical applications. Results suggest that using this index can considerably improve the interpretability of data analysis results in psychological research, as well as the reliability of conclusions that researchers can draw from their studies.