The paper shows that in addition to the theoretical relevance of choosing between the Hirschman-Herfindahl index and the concentration ratio, both measures may provide empirically very different information to assess industry performance. It is shown that the correspondence between the two measures can be represented by a horn-shaped figure. The implications of this horn-shaped relationship are investigated in the context of specifying empirical profitability-concentration models. Among different specification effects tested, it is shown that the neglect of the horn-relationship may bias the results towards the identification of a critical concentration ratio.