Testing leading digits against the Benford shares
Benford law is the empirical observation that in naturally collected figures, smaller leading digits appear more often. The expected probability of leading digit d is log10(1 + 1/d): 30.10% for 1, 17.61% for 2, 12.49% for 3, and 4.58% for 9. Enter the observed counts for digits 1 through 9 and the tool builds the expected counts, then sums the squared difference between observed and expected divided by expected to get the chi-square statistic with 8 degrees of freedom.
The p-value is the upper tail of the chi-square distribution with 8 degrees of freedom computed numerically, so treat it as an approximation. The usual convention is to apply the test only when every expected count is at least 5; the expected share of leading digit 9 is the smallest at 4.58%, so that needs at least 110 observations, and below that the tool marks the reading as unreliable. A p-value is not the probability that the data was manipulated. It is the probability of a departure this large when the data really does follow Benford, and failing to reject is not proof that it does.
Benford fits data that spans several orders of magnitude and grows naturally. It does not apply to values with a fixed digit length, such as ZIP codes or heights. A poor fit is never on its own a finding of tampering. This tool does not test the fairness of prize draw winner counts. Written as of October 2026.
Frequently asked questions
The probability of leading digit d is log10(1 + 1/d), which gives 30.10% for 1 and 4.58% for 9.
No. A poor fit is only a signal worth a closer look, and ordinary data can depart from Benford on its own.
Values with a fixed number of digits, such as ZIP codes or heights, and values confined to a narrow range.