Benford's Law

Also called the First Digit Law, First Digit Phenomenon, or Leading Digit Phenomenon. In listings, tables of statistics, etc., the Digit 1 tends to occur with Probability ~ 30%, much greater than the expected 10%. This can be observed, for instance, by examining tables of Logarithms and noting that the first pages are much more worn and smudged than later pages. The table below, taken from Benford (1938), shows the distribution of first digits taken from several disparate sources. Of the 54 million real constants in Plouffe's ``Inverse Symbolic Calculator'' database, 30% begin with the Digit 1.

		First Digit
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	$n^1$, $n^2\cdots n!$	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	$\pm 0.8$	$\pm 0.4$	$\pm 0.4$	$\pm 0.3$	$\pm 0.2$	$\pm 0.2$	$\pm 0.2$	$\pm 0.3$

In fact, the first Significant Digit seems to follow a Logarithmic Distribution, with

$$P(n)\approx \log(n+1)-\log n$$

for $n=1$, ..., 9. One explanation uses Central Limit-like theorems for the Mantissas of random variables under Multiplication. As the number of variables increases, the density function approaches that of a Logarithmic Distribution.

References

Benford, F. ``The Law of Anomalous Numbers.'' Proc. Amer. Phil. Soc. 78, 551-572, 1938.

Boyle, J. ``An Application of Fourier Series to the Most Significant Digit Problem.'' Amer. Math. Monthly 101, 879-886, 1994.

Hill, T. P. ``Base-Invariance Implies Benford's Law.'' Proc. Amer. Math. Soc. 12, 887-895, 1995.

Hill, T. P. ``The Significant-Digit Phenomenon.'' Amer. Math. Monthly 102, 322-327, 1995.

Hill, T. P. ``A Statistical Derivation of the Significant-Digit Law.'' Stat. Sci. 10, 354-363, 1996.

Hill, T. P. ``The First Digit Phenomenon.'' Amer. Sci. 86, 358-363, 1998.

Ley, E. ``On the Peculiar Distribution of the U.S. Stock Indices Digits.'' Amer. Stat. 50, 311-313, 1996.

Newcomb, S. ``Note on the Frequency of the Use of Digits in Natural Numbers.'' Amer. J. Math. 4, 39-40, 1881.

Nigrini, M. ``A Taxpayer Compliance Application of Benford's Law.'' J. Amer. Tax. Assoc. 18, 72-91, 1996.

Plouffe, S. ``Inverse Symbolic Calculator.'' http://www.cecm.sfu.ca/projects/ISC/.

Raimi, R. A. ``The Peculiar Distribution of First Digits.'' Sci. Amer. 221, 109-119, Dec. 1969.

Raimi, R. A. ``The First Digit Phenomenon.'' Amer. Math. Monthly 83, 521-538, 1976.