Benford's Law Calculator

Benford's Law describes a surprising pattern in many naturally occurring collections of numbers: the first significant digit is not uniformly distributed. The digit 1 appears first about 30.1% of the time, while 9 appears first only about 4.6% of the time. The expected probability for a digit d is log10(1 + 1/d). This logarithmic pattern often emerges when values span several orders of magnitude and arise from multiplicative processes rather than from a deliberately assigned range. This calculator extracts the first nonzero digit from every valid number and compares the observed proportions for digits 1 through 9 with Benford's theoretical proportions. It reports the sample size, a mean absolute deviation, and a chi-square statistic. Mean absolute deviation summarizes the average gap between observed and expected proportions. The chi-square statistic weights squared count differences by the expected count, so larger departures contribute more strongly. Both measurements are descriptive here; interpreting formal significance also requires attention to sample size, dependence, and how the data was selected. Benford analysis is commonly used as a screening technique in forensic accounting, auditing, election-data exploration, scientific quality control, and anomaly detection. An unusual distribution can help investigators identify records that deserve closer review. It does not prove fraud or fabrication. Legitimate datasets may fail to follow the law because values are capped, rounded, assigned, clustered in a narrow interval, or governed by a minimum or maximum. Invoice numbers, postal codes, telephone numbers, and other identifiers are especially unsuitable because their digits are assigned rather than naturally generated. For a useful comparison, provide a reasonably large dataset whose positive magnitudes cover multiple powers of ten. Negative values can be included because their absolute magnitudes have the same leading digits, while zeros have no first significant digit and are omitted. Avoid mixing fundamentally unrelated populations merely to increase the sample size. If the result shows a large discrepancy, inspect the collection method, range restrictions, rounding conventions, and subgroup behavior before drawing conclusions. Benford's Law works best as one diagnostic signal combined with domain knowledge, data validation, and other statistical evidence.