Journal of Theoretical Probability

, Volume 21, Issue 1, pp 97–117 | Cite as

Scale-Distortion Inequalities for Mantissas of Finite Data Sets

  • Arno Berger
  • Theodore P. Hill
  • Kent E. Morrison


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.


Benford’s Law Scale-invariance Scale-distortion Mantissa distribution Kantorovich metric 


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Arno Berger
    • 1
  • Theodore P. Hill
    • 2
  • Kent E. Morrison
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of MathematicsCalifornia Polytechnic State UniversitySan Luis ObispoUSA

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