Question

Benford's Law is often used in fraud investigation to determine whether additional analysis and investigation is warranted. When (rather than if) you are assigned to perform a routine data analysis as part of an anti-fraud program, why would you use Benford's law? Your manager never heard of this law and is very excited by its potential after you describe its uses to her. It looks like you're on the hook to find fraud using Benford's Law. What further discussion should you have with the manager ?

Answer 1

Benford’s law (also called the first digit law) states that the leading digits in a collection of data set are probably going to be small. For example, most numbers in a set (about 30%) will have a leading digit of 1, when the expected probability is 11.1% (i.e. one out of nine digits). This is followed by about 17.5% starting with a number 2. This is an unexpected phenomenon; If all leading numbers (0 through 9) had equal probability, each would occur 11.1% of the time. To put it simply, Benford’s law is a probability distribution for the likelihood of the first digit in a set of numbers.

The law only works for the significand^* S(x), which is essentially any number placed into a standard format. To find the significand:

• Find the first non-zero digit,
• Move the decimal point to immediate right of that digit,
• Ignore the sign.

Benford’s law can also be used to make predictions about what the second, third and subsequent leading digits will be as well as the probabilities for certain number combinations.

Benford’s law doesn’t apply to every set of numbers, but it usually applies to large sets of naturally occurring numbers with some connection like:

• Companies’ stock market values,
• Data found in texts — like the Reader’s Digest, or a copy of Newsweek.
• Demographic data, including state and city populations,
• Income tax data,
• Mathematical tables, like logarithms,
• River drainage rates,
• Scientific data.

The law usually doesn’t apply to data sets that have a stated minimum and maximum, like interest rates or hourly wages. If numbers are assigned, rather than naturally occurring, they will also not follow the law. Examples of assigned numbers include: zip codes, telephone numbers and Social Security numbers.

Uses

One practical use for Benford’s law is fraud and error detection. It’s expected that a large set of numbers will follow the law, so accountants, auditors, economists and tax professionals have a benchmark what the normal levels of any particular number in a set are.

• In the latter half of the 1990s, accountant Mark Nigrini found that Benford’s law can be an effective red-flag test for fabricated tax returns; True tax data usually follows Benford’s law, whereas made-up returns do not.
• The law was used in 2001 to study economic data from Greece, with the implication that the country may have manipulated numbers to join the European Union.
• Ponzi schemes can be detected using the law. Unrealistic returns, such as those purported by the Maddoff scam, fall far from the expected Benford probability distribution (Frunza, 2015).