First Things First Log Tables, Random Numbers, and Tax Fraud Ahbel, copyright 2001

The First Digit Phenomenon “Everyday numbers obey a law so unexpected it is hard to believe it's true.” New Scientist - 10 July 1999 Ahbel, copyright 2001

Simon Newcomb Astronomer and mathematician Log tables 1881 American Journal of Mathematics “The law of probability of the occurrence of numbers is such that all mantissae of their logarithms are equally likely.” Newcomb’s Conjecture P=log 10 (1+1/d)

Ahbel, copyright 2001 Graph of the Distribution

Ahbel, copyright 2001 Frank Benford 57 years later (1938) G.E. Physicist p=log 10 (1+1/d) Proceedings of the American Philosophical Society 20,229 observations Areas of 335 rivers, American League baseball statistics, specific heats of 1389 chemical compounds, numbers appearing in reader’s digest articles and newspapers.

Ahbel, copyright 2001 Newspapers – Benford in Census – Mark Nigrini Dow Jones ( ) – Eduardo Ley

Ahbel, copyright 2001 An Intuitive Explanation Plant 1 tree per year Every tree grows 1 cm per year What happens to the distribution of first digits of heights over time?

Ahbel, copyright 2001 Roger Pinkham Rutgers 1961 Base invariance Scale invariance Benford’s law is the only way!

Ahbel, copyright 2001 Scale Invariant Numbers seem to conform to a first digit phenomenon described by Benford’s Law Benford’s Law seems to be Scale Invariant If there is a first digit phenomenon that is scale invariant, then multiplying numbers by a constant will not change the distribution of the first digits.

Ahbel, copyright 2001 We can express any number in scientific notation: x*10 n, 1  x<10 If the distribution of x is scale-invariant, then the distribution of y = log 10 x must be “translation invariant”: log 10 ax = log 10 a + log 10 x = log 10 a + y If multiplying x by a constant leaves the distribution of first digits unchanged then adding a constant to y will leave the distribution of first digits unchanged constant

Ahbel, copyright 2001 x is on 1  x < 10 y is on log 10 1  y < log y is on 0  y < 1 Since the only distribution that is “translation invariant” is the uniform distribution, y must be distributed uniformly over its interval, 0  y < 1. This is the probability distribution function y=1.

Ahbel, copyright 2001 P(d 1 =1) = P(1  x < 2) = P(log 10 1  y < log 10 2) = P( 0  y < log 10 2) = log log 10 2  30.1% y=log 10 x P log 10 2 d 1 =1 log 10 2log 10 3 P d 1 =2 log 10 3-log 10 2  17.6%

Ahbel, copyright 2001 In general, P(d 1 =d) = P( d  x < d+1 ) = P(log 10 d  y < log 10 (d+1)) = log 10 (d+1) - log 10 d

Ahbel, copyright 2001 Theodore Hill Georgia Institute of Technology 1996 All is outcome of process normal, logarithmic, oscillating, etc. distributions “Now imagine grabbing random handfuls of data from a hotchpotch of such distributions. Hill proved that as you grab ever more of such numbers, the digits of these numbers will conform ever closer to a single, very specific law. This law is a kind of ultimate distribution, the “Distribution of Distributions.” And he showed that its mathematical form is…Benford’s Law.” New Scientist 10 July 1999

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Probability Distribution?

Ahbel, copyright 2001 First Digit Law Second Digit Law General Significant-Digit Law

Ahbel, copyright 2001 Tax Fraud Detection True Tax Data - Mark Nigrini’s 169,662 IRS model files Fraudulent Data King’s County, New York, Cash disbursement and payroll 743 freshmen’s responses to a request to write down a six-digit number at random

Ahbel, copyright 2001 Testing Mathematical Models Suppose past census data follows Benford’s Law closely A model that predicts the same census data should also follow Benford’s Law

Ahbel, copyright 2001 Computer Design Donald Knuth, Stanford University If the numbers computers process are not uniformly distributed then computers can be re-designed to: Minimize storage space Maximize calculation rate

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