Benford's law

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Given a finite set of numerical data, it would be expected that the digits 1 through 9 would occur as first digits with roughly equal probability. However, in many cases, the digit 1 occurs as the first digit most often, with roughly 30% probability, and the other digits occur with decreasing frequency.

This is called Benford's law, after Frank Benford, who noticed a curious pattern of usage in logarithm tables. Benford went on to also notice this unexpected disparity in many different sets of data, including atomic weights, baseball statistics and street addresses.

A selection of house numerals at Brooks Lumber in Detroit. Notice the newer packaging for the 1s, 2s and 7s.

The effects of Benford's law can be observed even in those cases that might not necessarily reflect a difference in usage between first digits and other digits of a number. Presumably, building contractors know about Benford's law and know whether it applies to the addresses of the properties they're building. Hardware stores, on the other hand, might not know for what addresses replacement house numerals are being bought, and a given numeral can just as easily be used for the last digit of a house address as it can be for the first. It is still nevertheless a good idea to stock more 1s and 2s than other numerals.

Prime numbers and Benford's law

Note: The prime numbers do not satisfy Benford's law (Daniel I. A. Cohen and Talbot M. Katz, "Prime numbers and the first digit phenomenon," J. Number Theory 18 (1984), 261-268; A. Berger and T. P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.)

Street addresses are different from prime numbers in that, even if we include all the world's addresses, we are still dealing with a finite set, while there are infinitely many primes. So, to simplify things for ourselves, in this article, we will only consider primes up to some threshold. And, so as to give the digits 2 to 9 a fair shake, we will set that threshold at a power of 10.

We see that going only up to 10, 1 as a first digit gets off to a lousy start, since we don't consider 1 prime anymore (for reasons that are beyond the scope of this article). One might be tempted to temporarily regard 1 as prime for the sake of a handicap, but just going up to 100 will show that to be unnecessary.

	1	2	3	4	5	6	7	8	9
Up to 10	0	1	1	0	1	0	1	0	0
Up to 100	4	3	3	3	3	2	4	2	1
Up to 1000	25	19	19	20	17	18	18	17	15
Up to 10000	160	146	139	139	131	135	125	127	127
Up to 10 5	1193	1129	1097	1069	1055	1013	1027	1003	1006
Up to 10 6	9585	9142	8960	8747	8615	8458	8435	8326	8230
Up to 10 7	80020	77025	75290	74114	72951	72257	71564	71038	70320
Up to 10 8	686048	664277	651085	641594	633932	628206	622882	618610	614821
Up to 10 9	6003530	5837665	5735086	5661135	5602768	5556434	5516130	5481646	5453140
Cf.:	A073517	A073516	A073515	A073514	A073513	A073512	A073511	A073510	A073509

However, as we go further up, the advantage of 1 as a first digit gradually erodes.

Another way to look at this is to see what is the 50th prime number to start with a given digit $d$. Now, 50 is a bit arbitrary: the choice was arrived at by a rough estimate of term visibility for the relevant OEIS sequence entries.

$d$	50th $p$
1	1171	A045707
2	2243	A045708
3	3259	A045709
4	4231	A045710
5	5297	A045711
6	6269	A045712
7	7309	A045713
8	8291	A045714
9	9311	A045715

If for these 50th primes we chop off the leading digit, we see that the resulting number for 9 is significantly higher than the one for 1.

But some reflection upon the prime number theorem suggests that as we look at higher and higher powers of 10, the distribution of first digits will become more or less uniform, leading to the conclusion that Benford's law does not actually apply to prime numbers.

References

• Daniel I. A. Cohen and Talbot M. Katz, "Prime numbers and the first digit phenomenon," J. Number Theory 18 (1984), 261-268.