Summary: An Invariant-Sum Characterization of Benford's Law
Pieter C. Allaart
Vrije Universiteit Amsterdam1
The accountant Nigrini remarked that in tables of data distributed according to
Benford's Law, the sum of all elements with rst digit d (d = 1 2 :: 9) is approxi-
mately constant. In this note, a mathematical formulation of Nigrini's observation
is given and it is shown that Benford's Law is the unique probability distribution
such that the expected sum of all elements with rst digits d1 :: dk is constant for
every xed k.
Keywords and phrases: First signi cant digit, Benford's law, mantissa function,
AMS 1990 subject classi cation: 60A10.
The main goal of this article is to give a mathematical proof of an empirical observation
of the accountant M. Nigrini. In his Ph.D. thesis (1992), Nigrini observed that tables of
unmanipulated accounting data closely follow Benford's Law (see x2 below), and that
in su ciently long lists of data for which Benford's Law holds,
the sum of all entries with leading digit d is constant for various d.
(cf. Nigrini, 1992, pp. 70/71).