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Chapter 34: Explaining Benford's Law

The Power of Signal Processing

Benford's law has never been viewed as a major mathematical problem, only a minor mystery. Nevertheless, many bright and creative people have spent time trying to understand it. The primary goal of this chapter has been to demonstrate the power of DSP in nontraditional applications. In the case of Benford's law this power is clear; signal processing has succeeded where other mathematical techniques have failed.

Nowhere is this more apparent than a review article published in 1976 by mathematician Ralph Raimi. He examined the many approaches in explicit mathematical detail, and his paper has become a landmark in the history of this problem. Buried in the detailed math, Raimi makes the brief comment: "...many writers ... have said vaguely that Benford's law holds better when the distribution ... covers several orders of magnitude." As we now know, this is the root of the phenomenon. In one of the most colorful events of this history, a small error in logic prompted Raimi to argue that this could not be correct. [Specifically, scaling a distribution does not change how many orders of magnitude it covers.] While this slight misdirection probably made no difference, it shows just how little success had been achieved by traditional mathematics. An understanding of the basic operation of Benfordís law was nowhere on the horizon.

Lastly, this discussion would be incomplete without mentioning the practical applications of Benford's law Next time you file your income tax return or other financial report, consider what happens to the distribution of leading digits if you fabricate some of the numbers. I'm not going to help you cheat, so I won't give the details away. Simply put, the numbers you make up will probably not follow Benford's law, making your fraudulent report distinguishable from an accurate one. I'll let you imagine who might be interested in this.