Benford’s law of digits and their frequency, with an application to invoice review

Benford’s law, named after statistician Frank Benford, holds that in most lists of numbers from real life, the digits in the first place should occur according to a known table of probabilities. For example, the digit 1 should occur 30.1 percent of the time as the right-most digit, the digit 2 should appear 17.7 percent of the time, 3 at 12.5 percent, and so on in declining frequency. This précis of the law comes from Len Fisher, The Perfect Swarm: The science of complexity in everyday life (Basic Books 2009) at 161.

Benford’s law underpins some forensic accounting, since fraudsters try to randomize numbers, but those faked patterns violate Benford’s law. What I would like to see is an analysis of time records from law firms. If the chargeable time was created long after the work was done and the person simply plugged in approximate numbers, the first digits will not conform to the normal and expected distribution (See my post of July 12, 2010: 60% of law firms reported reconstructive time tracking.).

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