Benford's Law - Generalization To Digits Beyond The First

Generalization To Digits Beyond The First

It is possible to extend the law to digits beyond the first. In particular, the probability of encountering a number starting with the string of digits n is given by:

(For example, the probability that a number starts with the digits 3, 1, 4 is log₁₀(1 + 1/314) ≈ 0.0014.) This result can be used to find the probability that a particular digit occurs at a given position within a number. For instance, the probability that a "2" is encountered as the second digit is

And the probability that d (d = 0, 1, ..., 9) is encountered as the n-th (n > 1) digit is

The distribution of the n-th digit, as n increases, rapidly approaches a uniform distribution with 10% for each of the ten digits.