A phenomenological law also called the primary digit law, first digit phenomenon, or leading digit phenomenon. Benford’s law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability ∼30%, much greater than the expected 11.1% (i.e., one digit out of 9). Benford’s law are often observed, as an example , by examining tables of logarithms and noting that the primary pages are far more worn and smudged than later pages (Newcomb 1881). While Benford’s law unquestionably applies to several situations within the world , a satisfactory explanation has been given only recently through the work of Hill (1998).

Benford’s law was employed by the character Charlie Eppes as an analogy to assist solve a series of high burglaries within the Season 2 “The Running Man” episode (2006) of the tv crime drama NUMB3RS.

Benford’s law applies to data that aren’t dimensionless, therefore the numerical values of the info depend upon the units. If there exists a universal probability distribution P(x) over such numbers, then it must be invariant under a change of scale, so

P(kx)=f(k)P(x).

(1)

If intP(x)dx=1, then intP(kx)dx=1/k, and normalization implies f(k)=1/k. Differentiating with reference to k and setting k=1 gives

xP^'(x)=-P(x),

(2)

having solution P(x)=1/x. Although this is often not a correct probability distribution (since it diverges), both the laws of physics and human convention impose cutoffs. for instance , randomly selected street addresses obey something on the brink of Benford’s law .

BenfordsLaw

If many powers of 10 lie between the cutoffs, then the probability that the primary (decimal) digit is D is given by a logarithmic distribution

P_D=(int_D^(D+1)P(x)dx)/(int_1^(10)P(x)dx)=log_(10)(1+1/D)

(3)

for D=1, …, 9, illustrated above and tabulated below.

D P_D D P_D

1 0.30103 6 0.0669468

2 0.176091 7 0.0579919

3 0.124939 8 0.0511525

4 0.09691 9 0.0457575

5 0.0791812

However, Benford’s law applies not only to scale-invariant data, but also to numbers chosen from a spread of various sources. Explaining this fact requires a more rigorous investigation of central limit-like theorems for the mantissas of random variables under multiplication. because the number of variables increases, the density function approaches that of the above logarithmic distribution. Hill (1998) rigorously demonstrated that the “distribution of distributions” given by random samples taken from a spread of various distributions is, in fact, Benford’s law (Matthews).

One striking example of Benford’s law is given by the 54 million real constants in Plouffe’s “Inverse Symbolic Calculator” database, 30% of which begin with the digit 1. Taking data from several disparate sources, the table below shows the distribution of first digits as compiled by Benford

col. title 1 2 3 4 5 6 7 8 9 samples

A Rivers, Area 31.0 16.4 10.7 11.3 7.2 8.6 5.5 4.2 5.1 335

B Population 33.9 20.4 14.2 8.1 7.2 6.2 4.1 3.7 2.2 3259

C Constants 41.3 14.4 4.8 8.6 10.6 5.8 1.0 2.9 10.6 104

D Newspapers 30.0 18.0 12.0 10.0 8.0 6.0 6.0 5.0 5.0 100

E Specific Heat 24.0 18.4 16.2 14.6 10.6 4.1 3.2 4.8 4.1 1389

F Pressure 29.6 18.3 12.8 9.8 8.3 6.4 5.7 4.4 4.7 703

G H.P. Lost 30.0 18.4 11.9 10.8 8.1 7.0 5.1 5.1 3.6 690

H Mol. Wgt. 26.7 25.2 15.4 10.8 6.7 5.1 4.1 2.8 3.2 1800

I Drainage 27.1 23.9 13.8 12.6 8.2 5.0 5.0 2.5 1.9 159

J Atomic Wgt. 47.2 18.7 5.5 4.4 6.6 4.4 3.3 4.4 5.5 91

K n^(-1), sqrt(n) 25.7 20.3 9.7 6.8 6.6 6.8 7.2 8.0 8.9 5000

L Design 26.8 14.8 14.3 7.5 8.3 8.4 7.0 7.3 5.6 560

M Reader’s Digest 33.4 18.5 12.4 7.5 7.1 6.5 5.5 4.9 4.2 308

N Cost Data 32.4 18.8 10.1 10.1 9.8 5.5 4.7 5.5 3.1 741

O X-Ray Volts 27.9 17.5 14.4 9.0 8.1 7.4 5.1 5.8 4.8 707

P Am. League 32.7 17.6 12.6 9.8 7.4 6.4 4.9 5.6 3.0 1458

Q Blackbody 31.0 17.3 14.1 8.7 6.6 7.0 5.2 4.7 5.4 1165

R Addresses 28.9 19.2 12.6 8.8 8.5 6.4 5.6 5.0 5.0 342

S n^1, n^2…n! 25.3 16.0 12.0 10.0 8.5 8.8 6.8 7.1 5.5 900

T Death Rate 27.0 18.6 15.7 9.4 6.7 6.5 7.2 4.8 4.1 418

Average 30.6 18.5 12.4 9.4 8.0 6.4 5.1 4.9 4.7 1011

Probable Error +/-0.8 +/-0.4 +/-0.4 +/-0.3 +/-0.2 +/-0.2 +/-0.2 +/-0.3

The following table gives the distribution of the first digit of the mantissa following Benford’s Law using a number of different methods.

method OEIS sequence

Sainte-Lague A055439 1, 2, 3, 1, 4, 5, 6, 1, 2, 7, 8, 9, …

d’Hondt A055440 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 3, 1, …

largest remainder, Hare quotas A055441 1, 2, 3, 4, 1, 5, 6, 7, 1, 2, 8, 1, …

largest remainder, Droop quotas A055442 1, 2, 3, 1, 4, 5, 6, 1, 2, 7, 8, 1, …