Benford's Law

Benford's law describes a distribution of significant digits in which the leading decimal digit has probability for , ..., 9. It is also called the first digit law, first digit phenomenon, or leading digit phenomenon. In particular, the leading digit 1 has probability about 30.1%, rather than the 11.1% that would result if all nine possible leading digits were equally likely. Many empirical data sets approximately follow this law, but it is not a universal law for numerical data. Newcomb (1881) described the greater wear on the first pages of tables of logarithms as evidence of unequal leading-digit frequencies. Benford (1938) studied a wide variety of data sets, and Hill (1995c, 1998) gave a statistical explanation under specified sampling hypotheses.

Benford's law was used by the character Charlie Eppes as an analogy to help solve a series of high burglaries in the Season 2 "The Running Man" episode (2006) of the television crime drama NUMB3RS.

A positive random variable has a Benford distribution of significant digits when the fractional part of has a uniform distribution on . This condition does not require that the data have physical dimensions. For example, a dimensionless random variable , where has a uniform distribution on , follows Benford's law exactly.

Scale invariance of significant digits must be distinguished from invariance of an entire probability density function on the positive real line. A scale-invariant probability density function would have the form , which cannot be normalized over (Hill 1995c). A proper log uniform distribution on a finite interval spanning whole decimal decades instead has

for , where and are integers with , and has density zero outside this interval. Each decade contributes the same leading-digit probabilities, so this distribution follows Benford's law exactly.

The probability that the leading decimal digit is is

for , ..., 9, illustrated above and tabulated below.

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

Products and mixtures of random variables do not automatically follow Benford's law. For example, a product of variables all equal to 10 always has leading digit 1, as does any mixture of distributions supported on . Hill's statistical derivation (1995c) uses random sampling from randomly selected distributions, with a sampling mechanism whose expected distribution of significant digits is invariant under changes of scale. The individual sampled distributions need not have this invariance. Under these hypotheses, the empirical significant-digit frequencies converge to Benford's law.

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. A later tabulation of 200125088 entries in Plouffe's Inverter likewise showed a first-four-significant-digit distribution closely following Benford's law (Groce 2005). Taking data from several disparate sources, the table below shows the distribution of first digits as compiled by Benford (1938) in his original paper.

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	,	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	,	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

The following table lists four sequences of leading digits intended to approximate Benford's law. The method labels follow the corresponding OEIS entries. For the Hare-quota sequence, the allocation is iterative. The th term is the digit maximizing , where counts occurrences of in the preceding terms. This rule differs from recomputing an ordinary largest-remainder allocation for each . For example, at the iterative counts are , whereas ordinary largest-remainder allocation gives (OEIS A055441).