Benford's law, also called the first-digit law, states that in lists of numbers from many real-life sources of data, the leading digit 1 occurs much more often than the others (namely about 30% of the time). Furthermore, the higher the digit, the less likely it is to occur as the leading digit of a number. This applies to figures related to the natural world or of social significance; be it numbers taken from electricity bills, newspaper articles, street addresses, stock prices, population numbers, death rates, areas or lengths of rivers or color=#0000ffphysical and color=#0000ffmathematical constants.
Mathematical statement
More precisely, Benford's Law states that the leading digit n (n = 1, ..., 9) occurs with color=#0000ffprobability color=#0000fflog10(n + 1) - log10(n), or
Leading digit |
Probability |
1 |
30.1 % |
2 |
17.6 % |
3 |
12.5 % |
4 |
9.7 % |
5 |
7.9 % |
6 |
6.7 % |
7 |
5.8 % |
8 |
5.1 % |
9 |
4.6 % |
One can also formulate a law for the first two digits: the probability that the first two-digit block is equal to n (n = 10, ..., 99) is log10(n+1) - log10(n), and similarly for three-blocks without leading zeros and longer blocks.
Explanation
That in general the leading digit 1 should be more common than the other digits can be understood as follows: start counting from 1: 1, 2, 3, ... As you reach 9, every digit will have been equally likely. But then, from 10 to 19, you only have the leading digit 1, so 1 gets a huge head start. Only when you reach 99 will all digits be equally likely again. But then 1 gets another huge head start from 100 to 199. And so it continues: 1 has always a lead, except for very rare exceptions (9, 99, 999, 9999, ...). This is not particularly satisfactory as an explanation, unless some probability of stopping counting at some point is also included