As far as we know, there is no good privacy metric for quantify- ing how privacy-efficient an anonymity system is. This paper discusses first the features needed for defining such a metric and proposes a new metric based on information theory and named DR for Discrimination Rate. The DR is the first metric enabling some fine grained measurements down to the attribute level to quantify the attributes identification capacity with a score scaling from 0 to 1 for any given anonymity system. The DR can be easily applied in practice thanks to the algorithms provided in the paper. The DR measurement onto attributes enables to reflect the attacker's capacity, and to evaluate how much the attribute is able to refine the anonymity set. The formalization brought by the DR permits to give more accurate definitions of identifiers and to introduce new notions like sketchy-identifiers, zero-identifiers and partial-identifiers. Finally, the usefulness and practical dimensions of the DR are illustrated through evaluation and comparison of the k-anonymity and l-diversity mechanisms over a dataset