  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. While the value of the index $\alpha $ fixes
the distribution's power-law exponent, that for the dual index $2-\alpha $
ensures the extensivity of the deformed entropy.
