  The general relationship between an arbitrary frequency distribution and the
expectation value of the frequency distributions of its samples is discussed. A
wide set of measurable quantities ("invariant moments") whose expectation value
does not in general depend on the size of the sample is constructed and
illustrated by applying the results to Ewens sampling formula. Invariant
moments are especially useful in the sampling of systems characterized by the
absence of an intrinsic scale. Distribution functions that may parametrize the
samples of scale-free distributions are considered and their invariant
expectation values are computed. The conditions under which the scaling limit
of such distributions may exist are described.
