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 theexpected 10%. This can be observed, for instance, by examining tables of Logarithms and noting that thefirst pages are much more worn and smudged than later pages. The table below, taken from Benford (1938), shows thedistribution of first digits taken from several disparate sources. Of the 54 million real constants in Plouffe's ``InverseSymbolic 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 |  ,  | 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 |  |  | |  |  | | | | |||
In fact, the first Significant Digit seems to follow a Logarithmic Distribution, with
for
, ..., 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.
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