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Scale-distortion inequalities for mantissas of finite data sets. (English) Zbl 1162.68306
Summary: 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.

MSC:
68M07 Mathematical problems of computer architecture
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