Benford's Law Calculator

Benford's Law, also known as the First-Digit Law, states that in many naturally occurring numerical datasets, the number 1 will be the leading significant digit about 30.1% of the time, while larger digits appear with progressively lower frequency. Mathematically, the expected probability P(d) of any leading digit d (from 1 to 9) is calculated using the formula: P(d) = log10(1 + 1/d). When analyzing a dataset, the calculator tallies the leading digits of your input numbers, computes their empirical percentages, and compares them against these theoretical logarithmic probabilities.

Imagine you are auditing a company expense report containing 1,000 transaction amounts. You extract the leading significant digit of every number. Suppose you find that the leading digit '1' appears 305 times, giving an empirical frequency of 30.5%. According to Benford's Law, the expected frequency for '1' is log10(1 + 1/1), which equals approximately 30.1%. By evaluating all digits from 1 to 9, the calculator aggregates these discrepancies to show whether your dataset aligns closely with the expected mathematical baseline or deviates significantly.

To ensure accurate results when using this calculator, make sure your dataset spans several orders of magnitude—ideally covering numbers from the single digits up to the thousands or millions. Avoid applying Benford's Law to datasets that have built-in minimum or maximum thresholds, such as assigned human-generated IDs, phone numbers, or pre-filtered price lists, as these artificial constraints naturally disrupt the logarithmic distribution.