We show that size-rank distributions with power-law decay (often only over a limited extent) observed in a vast number of instances in a widespread family of systems obey Tsallis statistics. The theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the Lyapunov exponent is negligible or vanishes. The relevant statistical-mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints, and the resulting duality of entropy indexes is seen to portray physically relevant information. Whereas the value of the index α fixes the distribution's power-law exponent, that for the dual index 2 − α ensures the extensivity of the deformed entropy.rank-ordered data | generalized entropies Z ipf's law refers to the (approximate) power law obeyed by sets of data when these are sorted out and displayed by rank in relation to magnitude or rate of recurrence (1). The sets of data originate from many different fields: astrophysical, geophysical, ecological, biological, technological, financial, urban, social, etc., suggesting some kind of universality. Over the years this circumstance has attracted much attention and the rationalization of this empirical law has become a common endeavor in the study of complex systems (2, 3). Here we pursue further the view (4, 5) that an understanding of the omnipresence of this type of rank distribution hints at an underlying structure similar to that which confers systems with many degrees of freedom the familiar macroscopic properties described by thermodynamics. That is, the quantities used in describing this empirical law obey expressions derived from principles akin to a statistical-mechanical formalism (4, 5). The most salient result presented here is that the reproduction of the data via a maximum entropy principle indicates that access to its configurational space is severely hindered to a point that the allowed configurational space has a vanishing measure. This feature appears to be responsible for the entropy expression not to be of the Boltzmann-Gibbs or Shannon type but instead to take that of the Tsallis form (6), while the extensivity of entropy is preserved. It is perhaps worth clarifying that our study is set in discrete space and it does not consider any formal Hamiltonian system.In Fig. 1 we show three examples of ranked data that appear to display power-law behavior along a considerably large interval of rank values. Fig. 1 (Top) shows data for the wealth of billionaires in the United States (7), Fig. 1 (Middle) shows data for the energy released by earthquakes in California (8), and Fig. 1 (Bottom) shows data for the intensity of solar flares (9). In Fig. 1 (Left), logarithmic scales are used for both size and rank, whereas Fig. 1 (Right) shows the same data in log-linear scales. Fig. 1 (Left) indicates approximate power-law decay for large rank and a clear deviation from this for small to moderate rank. As we shall sho...