How this calculator works
First-digit probabilities decline from digit 1 to digit 9. Data mode takes the first significant decimal digit of each absolute non-zero value, ignoring zeros and signs. The comparison statistic measures discrepancy but supplies no fraud verdict. Assigned identifiers, constrained ranges and minimum purchase prices often make Benford assumptions inappropriate.
Formula
P(D=d) = log10(1+1/d), d = 1,...,9; expected_d = N P(D=d); χ² = Σ(observed_d-expected_d)²/expected_d.
Worked example
Digit 1 has probability log10(2) = 0.301030. In 1,000 suitable observations its expected count is about 301.03, rather than 111.11.
Advertisement
Frequently asked questions
Does every data set follow this law?
No. Benford behaviour usually needs suitable scale variation and a generating process that does not assign the digits.
How are decimals and negatives handled?
Signs are ignored and the first non-zero significant digit is used. For example, -0.0042 contributes digit 4.
Does disagreement prove manipulation?
No. It may reflect an unsuitable model, rounding, sampling or ordinary variation.