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Characterizing hydrodynamic turbulence in terms of the so-called 2-point velocity difference statistics is a basic aspect of fluid dynamics, both from the theoretical and experimental viewpoints. The issue concerns in particular the first-principle construction of the corresponding 2-point velocity difference probability density function (PDF) as well as the transport equation which determines its time evolution. The goal of this paper is to point out that the problem can be solved by suitably prescribing the ensemble-averaging operator to be employed in the statistical description of fluids. As a result, qualitative properties of the 2-point PDF which are usually ascribed to the occurrence of turbulence are shown to follow from the ab initio approachpresented here and to characterize both deterministic and stochastic fluid descriptions.