 |
|
A phenomenological law also called the first 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  , much greater than the expected 11.1% (i.e., one
digit out of 9). Benford's law 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 (Newcomb 1881). While
Benford's law unquestionably applies to many situations in the real world, a
satisfactory explanation has been given only recently through the work of Hill
(1996).
Benford's law applies to data that are not dimensionless, so the
numerical values of the data depend on 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
 |
(1) |
If  , then
 , and
normalization implies  . Differentiating with respect to k and setting k = 1
gives
 |
(2) |
having solution  . Although this is not a proper probability distribution (since it
diverges), both the laws of physics and human convention impose cutoffs. For
example, if street addresses are distributed uniformly over the range of 1 to
some maximum cutoff value, then they'll obey something close to Benford's law.
If many powers of 10 lie between the cutoffs, then the probability that the
first (decimal) digit is D is given by the logarithmic distribution
 |
(3) |
for D = 1, ..., 9, illustrated above and tabulated below.
| D |
 |
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 variety of different sources. Explaining this fact
requires a more rigorous investigation of 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. Hill (1996)
rigorously demonstrated that the "distribution of distributions" given by random
samples taken from a variety of different distributions is, in fact, Benford's
law (Matthews 1999).
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
compiles by Benford (1938) in his original paper.
| |
|
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 |
 ,
 |
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 |
? 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 |
Sloane |
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,
... |
References
Barlow, J. L. and Bareiss, E. H. "On Roundoff Error Distributions
in Floating Point and Logarithmic Arithmetic." Computing 34,
325-347, 1985.
Benford, F. "The Law of Anomalous Numbers." Proc. Amer. Phil. Soc.
78, 551-572, 1938.
Bogomolny, A. "Benford's Law and Zipf's Law." http://www.cut-the-knot.org/do_you_know/zipfLaw.shtml.
Boyle, J. "An Application of Fourier Series to the Most Significant Digit
Problem." Amer. Math. Monthly 101, 879-886, 1994.
Flehinger, B. J. "On the Probability that a Random Integer Has Initial
Digit A." Amer. Math. Monthly 73, 1056-1061, 1966.
Franel, J. Naturforschende Gesellschaft, Vierteljahrsschrift (Z?rich)
62, 286-295, 1917.
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.
Knuth, D. E. "The Fraction Parts." ?4.2.4B in The Art
of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed.
Reading, MA: Addison-Wesley, pp. 254-262, 1998.
Ley, E. "On the Peculiar Distribution of the U.S. Stock Indices Digits."
Amer. Stat. 50, 311-313, 1996.
 Matthews,
R. "The Power of One." http://www.newscientist.com/ns/19990710/thepowerof.html.
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.
Nigrini, M. "I've Got Your Number." J. Accountancy 187,
pp. 79-83, May 1999. http://www.aicpa.org/pubs/jofa/may1999/nigrini.htm.
Plouffe, S. "Graph of the Number of Entries in Plouffe's Inverter." http://www.lacim.uqam.ca/~plouffe/statistics.html.
Raimi, R. A. "The Peculiar Distribution of First Digits." Sci.
Amer. 221, 109-119, Dec. 1969.
Raimi, R. A. "On the Distribution of First Significant Digits." Amer.
Math. Monthly 76, 342-348, 1969.
Raimi, R. A. "The First Digit Phenomenon." Amer. Math. Monthly
83, 521-538, 1976.
Schatte, P. "Zur Verteilung der Mantisse in der Gleitkommadarstellung einer
Zufallsgr??e." Z. Angew. Math. Mech. 53, 553-565, 1973.
Schatte, P. "On Mantissa Distributions in Computing and Benford's Law." J.
Inform. Process. Cybernet. 24, 443-455, 1988.
Sloane, N. J. A. Sequences A055439, A055440, A055441, and
A055442 in
"The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.
Eric W. Weisstein
? 1999 CRC Press LLC, ? 1999-2003 Wolfram Research, Inc.
|
