“Benford's Law”的概念、定义、习题解题方法及翻译-数学辞典

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

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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.

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.