It is a well established fact that in some circumstances, lists of what appear to be random numbers, show a striking non-uniform distribution of digits. In many instances, these numbers arise relative to a system of units. In such cases there is an underlying assumption of scale invariance, by which is meant that the choice of units may well be arbitrary. In this paper we consider the general problem of scale invariance from the point of view of certain means on suitable function spaces. This is then applied to give a simple explanation for the distribution of first significant digits.