Benford's Law

Benford's law gives first significant digit probabilities log₁₀(1 + 1/d). Some numerical processes and datasets approximate this pattern, while many legitimate datasets do not.

Benford's law assigns first significant digit d the probability log₁₀(1 + 1/d), making one more common than nine under the model. A connection to logarithms explains why digit patterns can differ from a uniform distribution, but spanning many orders of magnitude is not sufficient by itself to establish Benford behavior.

Applicability depends on how numbers are generated, selected, rounded, capped, and labeled. Assigned identifiers and narrow permitted ranges often have no reason to follow the law. A departure can motivate a data-quality question when a justified benchmark exists; it is not proof of fabrication, and agreement does not certify that records are genuine.

When to use it

When analyzing data; when building predictive models; when evaluating statistical claims; when quantifying uncertainty.

How it can help

Examine how numbers were generated and selected before choosing a digit benchmark. Investigate deviations with sampling uncertainty and legitimate alternatives in view.