june2007 81 Detecting the dubious digits: Detecting the dubious digits: Benford’s law in forensic Benford’s law in forensic accounting accounting Kuldeep Kumar and Sukanto Bhattacharya introduce an unexpected quality in natural numbers that can help to detect faked data In 1881, an American mathematician by the name of Simon Newcomb noticed, rather to his surprise, that the first few pages of a book of log-tables were much dirtier and more dog-eared than the later pages. The first pages contained the logarithms of numbers beginning with 1, such as 15 792, 14 289, 19 621. The last pages held the logarithms of numbers like 95 712 and 98 124. The first half of the book—correspond- ing to numbers beginning 2, 3 and 4—was also dirtier than the second half, where the loga- rithms of numbers beginning with 6, 7 and 8 were found. Obviously people were using the first pages more than they were the last ones 1 . This flew in the face of common sense. Surely num- bers were equally and evenly distributed across the spectrum, so people should be looking them up equally evenly. The chance that a number picked at random starting with 1 and the chance of it starts with 9, or with any digit in between should be the same: one-ninth, or approximately 11%; but Newcomb found that this did not seem to be true. People were looking up the 1s far more frequently. Is it in the nature of things that numbers beginning with 1 are much more com- mon? Newcomb noted his bizarre observation, but got no further. Almost half a century after Newcomb’s dis- covery, Frank Benford, a mathematical physicist, happened to stumble upon the same phenom- enon while going through a large assemblage of numerical data from disparate sources. Un- like his predecessor, Benford vigorously pursued this phenomenon and published his findings in a number of scholastic papers. Thus the phenom- enon came to be known as “Benford’s law” and Newcomb’s name sank into obscurity. Benford collected numerical data on a wide variety of subjects including the area of rivers, population concentration, physical and math- ematical constants, birth and death rates, and even numbers which appeared in articles in the Readers’ Digest—all of which appeared to cor- roborate the phenomenon. More numbers began with 1 than with anything else. The probability the first digit was less than 5 was significantly greater than that it was greater than 5; and the probability of occurrence diminished logarithmi- cally from the digits 1 to 9. Having gone through a vast volume of data, Benford derived a loga- rithmic expression for the probability distribu- tion function and estimated the individual prob- abilities that each of the nine digits was the first significant digit in a randomly observed naturally occurring number as in Table 1 and Figure 1 2 . The defining characteristic of Benford’s law is that it is observable only in naturally occurring numbers—not in numbers that have been artifi- cially concocted. Thus, Benford’s law may be re- garded as a veritable signature of Nature—some- thing that cannot be replicated manually. The human brain assumes that natural numbers come distributed linearly. The human brain is wrong. Apart from mathematical nicety, this actually makes Benford’s law extremely useful in detect- ing fraudulent data, in finance or in science. Mark Nigrini has pioneered application of Benford’s law to cases of tax evasion and other types of financial fraud. One of his examples from the stock market is illustrated in Figure 2 3 . As- sume that, in the course of a bull run, a market index starts with 1000 and grows at 20% a year. For the next 15 years the market index values will be as shown in Table 2. As may be seen, 40% of the data starts with the digit 1. Although the observed relative fre- quencies do not correspond exactly to those predicted by Benford’s law, it is quite obvious that they are strongly biased towards the lower digits. In the context of financial fraud detection, the more an observed set of accounting data de- The St Lawrence river: 1900 miles and 25th longest river in the world. River lengths obey Benford’s law Downloaded from https://academic.oup.com/jrssig/article/4/2/81/7029600 by guest on 08 February 2023