In some collections of measurements, the first significant digit is much more likely to be one than nine. The imbalance can look suspicious if every digit is expected to appear equally often. Benford’s law describes a setting in which that expectation is wrong. [1]
Under the law, the probability of first significant digit d is:
Where D is the first significant digit, d is an integer from one to nine. For one, the formula gives approximately 30.1 per cent. For nine, it gives approximately 4.6 per cent.
Numbers beginning with one occupy intervals such as one to two, ten to twenty, and one hundred to two hundred. Each spans the same proportional change: doubling. On a base-ten logarithmic scale, that interval has length about 0.301. The corresponding interval from nine to ten has length about 0.046. These lengths give the different probabilities.
Choose a number U uniformly between zero and three, then set X equal to ten raised to U. This produces values spanning three orders of magnitude. In each unit interval of U, the portion producing a first digit of one has length approximately 0.301, so the overall proportion is also 0.301.
This construction is uniform in the logarithm. Sampling X uniformly between one and one thousand would produce a different first-digit distribution. A wide numerical range alone does not establish Benford’s law.
More generally, a positive random variable is Benford when the fractional part of its base-ten logarithm is uniform. Multiplying every value by a fixed positive constant adds a constant to that logarithm and preserves this uniformity modulo one. This explains the law’s scale invariance. [2]
A famous financial example comes from Enron, which collapsed in 2001. In his 2012 book, forensic accountant Mark Nigrini examined twelve headline figures Enron originally reported for 1997–2000. Seven had zero as their second significant digit. Reported revenues included $40.112 billion in 1999 and $100.789 billion in 2000, just above prominent round-number thresholds. Nigrini interpreted the pattern as a sign that management might have targeted figures that crossed these thresholds. This uses the law’s extension to second digits, where zero has an expected frequency of about twelve per cent, rather than the first-digit formula above. [3]
There are straightforward counterexamples to treating the law as universal. Measurements restricted between two hundred and three hundred will mostly begin with two. Assigned identifiers may reflect an administrative numbering scheme. Legitimate data can therefore depart from Benford’s predictions.
The Enron example was a retrospective analysis. Digit patterns can suggest where to investigate, but cannot establish fraud by themselves. An auditor still needs to examine the underlying transactions and decide whether the data should have followed Benford’s law in the first place. [3]
[1] Berger and Hill — The mathematics of Benford law
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