Benford’s Law
In many real-world datasets, the leading digit is not uniformly distributed. The digit 1 appears as the first digit about 30% of the time, while 9 appears less than 5%. This pattern — Benford’s Law — emerges in data spanning multiple orders of magnitude: populations, stock prices, physical constants, and mathematical sequences. Uniform random numbers, by contrast, show no such bias.
P(d) = log₁₀(1 + 1/d) for d = 1, 2, …, 9
The law
Benford’s Law (also called the first-digit law) states that in many naturally occurring collections of numbers, the leading digit d appears with probability P(d) = log₁₀(1 + 1/d). This means 1 appears as the leading digit about 30.1% of the time, while 9 appears only 4.6% of the time.
Why it works
If a dataset spans several orders of magnitude and is scale-invariant (its distribution does not change when multiplied by a constant), then the leading digits must follow Benford’s distribution. This is because logarithmic intervals between successive powers of 10 are not equally spaced in linear terms — numbers beginning with 1 occupy a wider portion of each order of magnitude than numbers beginning with 9.
Datasets that follow it
Powers of 2 follow Benford’s Law perfectly because successive powers sample the logarithmic scale uniformly. Fibonacci numbers converge to it as the sequence grows. Factorials and population data follow it because they span many orders of magnitude. Uniform random numbers between 1 and N do not follow it, because they lack the multi-order-of-magnitude spread.
Forensic accounting
Benford’s Law is used in fraud detection. Fabricated financial data often has an approximately uniform first-digit distribution, because humans intuitively expect digits to be equally likely. Deviations from Benford’s prediction can flag suspicious datasets.