Benford's Law Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
Benford's law predicts the expected frequency of a number's leading digit with P(d) = log₁₀(1 + 1/d). The digit 1 appears as the leading digit in about 30.1% of cases in many natural datasets, far more often than if all digits were equally likely (which would give 11.1% each).
Explanation
Intuitively, you might expect the leading digit of a number picked at random from a large dataset (city populations, invoice amounts, stock values) to be 1, 2, 3... 9 with the same frequency, about 11.1% each — the same intuition underlying our uniform distribution calculator, where every possible outcome is by definition equally likely. Yet that is not what happens in many natural numerical datasets: the digit 1 appears first in about 30% of cases, and this frequency decreases steadily to the digit 9, which appears first only in about 4.6% of cases. This phenomenon, discovered by the astronomer Simon Newcomb in 1881 then independently rediscovered by the physicist Frank Benford in 1938, is explained by the fact that many natural quantities grow multiplicatively rather than additively (exponential growth, distributions spanning several orders of magnitude): a number spends proportionally more time with a leading digit of 1 before reaching a leading digit of 2 than it spends with a leading digit of 9 before moving to the next order of magnitude. Benford's law has a notable concrete application in financial auditing and fraud detection: financial data fabricated or manipulated by a human (who tends, often unconsciously, to distribute digits more uniformly) frequently deviates from the Benford distribution expected for authentic data, which makes it a warning signal used by some auditors and investigators — without ever constituting, on its own, proof of fraud — a caveat valid for interpreting any isolated statistical result, including those of our simple probability calculator.
Example: expected frequency of the digit 1
Inputs
Leading significant digit: 1.
Calculation
P(1) = log₁₀(1 + 1/1) × 100 = log₁₀(2) × 100 ≈ 30.1%.
Result
According to Benford's law, about 30.1% of numbers in a natural dataset begin with the digit 1.
Frequently asked questions
Why is the digit 1 so much more frequent than the digit 9?
Because for a number to go from a leading digit of 1 to a leading digit of 2, it only needs to increase by 100% (double), whereas to go from a leading digit of 9 to the next leading digit of 1 (in the higher order of magnitude), it only needs to increase by about 11%. On a logarithmic scale, the interval covered by numbers starting with 1 is therefore proportionally much wider than that covered by numbers starting with 9.
Does Benford's law apply to any dataset?
No, it applies well to datasets spanning several orders of magnitude and resulting from multiplicative processes (population growth, stock prices, varied accounting data), but not to data bounded within a narrow range or assigned arbitrarily (phone numbers, postal codes, lottery numbers), where it simply does not apply by construction.
Does a deviation from Benford's law prove fraud?
No, a significant deviation from the Benford distribution is a statistical warning signal that justifies closer examination, not proof in itself. Many legitimate reasons can explain a deviation (the specific nature of the data, regulatory constraints on certain amounts, too small a sample): this law is one anomaly-detection tool among others, never a definitive conclusion on its own.