Benford's Law

A Benford's Law is an empirical pattern that … Frequency Distribution that ...

• See: Binomial Distribution, Cauchy Distribution, Significant Digit, Power Law.

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References

2016

• (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/Benford's_law Retrieved:2016-1-28.
  • Benford's law, also called the first-digit law, is a phenomenological law about the frequency distribution of leading digits in many (but not all) real-life sets of numerical data. The law states that in many naturally occurring collections of numbers the small digits occur disproportionately often as leading significant digits. For example, in sets which obey the law the number would appear as the most significant digit about 30% of the time, while larger digits would occur in that position less frequently: would appear less than 5% of the time. If all digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also concerns the expected distribution for digits beyond the first, which approach a uniform distribution. It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude. The graph here shows Benford's law for base 10. There is a generalization of the law to numbers expressed in other bases (for example, base 16), and also a generalization from leading 1 digit to leading n digits. It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450
  • QUOTE: Benford's Law: The first digits of numbers in a collection of related multi-digit numbers (commonly cited examples are tables of logarithms, populations of towns, newspaper circulations) are not equally likely. Instead, the probability that the first digit is D is [math]\displaystyle{ \log_{10} \left( 1+ \frac{1}{D} \right) }[/math] Thus, in such collections, approximately 30% of numbers begin with the digit 1.

2004

• (Durtschi et al., 2004) ⇒ Cindy Durtschi, William Hillison, and Carl Pacini. (2004). “The Effective Use of Benford’s Law to Assist in Detecting Fraud in Accounting Data.” In: Journal of forensic accounting, 5(1).
  • ABSTRACT: Benford’s law has been promoted as providing the auditor with a tool that is simple and effective for the detection of fraud. The law is based on a peculiar observation that certain digits appear more frequently than others in data sets. For example, in certain data sets, it has been observed that more than 30% of numbers begin with the digit one. After discussing the background of the law and development of its use in auditing, we show where digital analysis based on Benford’s law can most effectively be used and where auditors should exercise caution.

    Specifically, we identify data sets which can be expected to follow Benford’s distribution, discuss the power of statistical tests, types of frauds that would be detected and not be detected by such analysis, the potential problems that arise when an account contains too few observations, as well as issues related to base rate of fraud. An actual example is provided demonstrating where Benford's law proved successful in identifying fraud in a population of accounting data.

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