Rough approximation quality revisited

In rough set theory, the approximation quality γ is the traditional measure to evaluate the classification success of attributes in terms of a numerical evaluation of the dependency properties generated by these attributes. In this paper we re-interpret the classical γ in terms of a classic measure based on sets, the Marczewski–Steinhaus metric, and also in terms of “proportional reduction of errors” (PRE) measures. We also exhibit infinitely many possibilities to define γ-like statistics which are meaningful in situations different from the classical one, and provide tools to ascertain the statistical significance of the proposed measures, which are valid for any kind of sample.