The paradigm that our societies should provide equal access to opportunities and a sufficient level of well-being to citizens regardless of their personal characteristics (gender, race, religion...) is one of the pillars, perhaps an axiom, of our political systems. And the link detected between the possession of income and wealth and access to opportunities means that the measurement of economic inequality (or gap) between different social groups is an indirect indicator to evaluate the degree of compliance with the previous axiom; that is, the degree to which said groups have equal access to opportunities and well-being.The mathematical characterization of the gap between groups is therefore key in this assessment. However, most proposals to assess economic gaps between groups use mathematical characterizations of the income or wealth of each group to be compared, whose inadequacy has already been demonstrated. Thus, to make progress on this issue, this text presents an alternative characterization of the income or wealth of the groups, based on the economic concept of equivalent average income, which states a relationship between average income, economic inequality, and generated welfare. This allows us to contrast the optimality of the different possible characterizations against the current axiomatic for inequality indexes. Likewise, the conceptualization itself implies the formulation of an inequality index which, compared to the most commonly used indexes (Gini, Pietra, Atkinson, Theil), provides the advantage of being able to compute zero or negative income or wealth values.