Detecting the dubious digits: Benford's law in forensic accounting

摘要

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, 14289, 19621. The last pages held the logarithms of numbers like 95712 and 98 124. The first half of the book—corresponding to numbers beginning 2, 3 and 4—was also dirtier than the second half, where the logarithms of numbers beginning with 6, 7 and 8 were found. Obviously people were using the first pages more than they were the last ones. This flew in the face of common sense. Surely numbers 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 common? Newcomb noted his bizarre observation, but got no further.