Driving rain lashes the windows of a grey building in Washington. In the offices of the Internal Revenue Service⁵, agents hunched over piles of tax returns and computer printouts scrutinise every anomaly. On the wall, strange curves, statistics and percentages tell a story invisible to the layman. It is the late 1990s, and in this anonymous department of the US taxman, a quiet revolution is under way.

In 1996, as the IRS begins experimenting, mathematician and accounting professor Mark Nigrini publishes a landmark paper that puts Benford at the heart of fraud detection⁶⁷. Three years later, he sets out precise statistical tests, sampling methods and alert thresholds, turning intuition into scientific method⁸.

In his small university office, Nigrini has no idea he is changing the world of auditing for good. IRS agents devour his publications. Seminars are organised. Internal procedures are rewritten, helped along by the mass arrival of computers. Tax audits, once random, now target returns whose digit distribution betrays human tampering.

Method in the madness

One spring day, Nigrini gets a call from the US taxman. His methods have helped uncover a major fraud. Then another call. Then dozens. His theoretical work saves honest taxpayers millions of dollars. Academia meets the real world, and the synergy is perfect.

Because at the IRS they had understood it was magic, but had never found a methodology. Nigrini lays it out clearly:

• On the data
  Use financial statements covering several years (5 years or more), and look at trends
  Focus on key items: total assets, liabilities and equity, in particular
  Have at least 1,000 entries for the results to be significant
  Exclude artificial data or data determined by other factors (postcodes, phone numbers, etc.)
• On applying Benford’s law
  First Digit Test⁹: compare the frequency of the first digits of the accounting values
  Extract the first digit of each financial value.
  Calculate how often each digit (1 to 9) appears.
  Compare the actual frequencies with the frequencies expected under Benford.
  Identify abnormal gaps (for example, an over-representation of the digit 9 may suggest manipulation).
  First Two-Digit Test¹⁰: refine the analysis by checking the frequency of the first two digits of the accounting values.
  Extract the first two digits of each financial value.
  Calculate their frequency and compare it with the expected distribution.
  Identify anomalies (e.g. if “50” comes up too often, it may signal fraud).
  Chi-Square Test¹¹: determine whether the observed gaps are statistically significant.
  Calculate the value
  Compare it with a critical threshold value
  If the value is above the critical value, there is a statistically significant anomaly, potentially linked to fraud.

OK, it looks fiendishly complicated, but it really isn’t: the idea is simply to keep Benford’s distribution chart in mind and see whether your numbers produce the same thing or not. Above all, the method tells you whether, roughly speaking, there might be fraud or not, and, if the chi-square test flags it, to go and look at the transactions causing the gap and apply the usual procedures for suspected fraud.