Postprint version. Published in Journal of Theoretical Probability, Volume 21, January 1, 2008, pages 97-117. Copyright © 2008 Springer. The original publication is available at http://dx.doi.org/10.1007/s10959-007-0112-z.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
In scientific computations using floating point arithmetic, rescaling a data set multiplicatively (e.g., corresponding to a conversion from dollars to euros) changes the distribution of the mantissas, or fraction parts, of the data. A scale-distortion factor for probability distributions is defined, based on the Kantorovich distance between distributions. Sharp lower bounds are found for the scale-distortion of n-point data sets, and the unique data set of size n with the least scale-distortion is identified for each positive integer n. A sequence of real numbers is shown to follow Benford’s Law (base b) if and only if the scale-distortion (base b) of the first n data points tends zero as n goes to infinity. These results complement the known fact that Benford’s Law is the unique scale-invariant probability distribution on mantissas.