# Benford’s Law: An Important Tool in the Auditors Toolbag

When it comes to those who discover fraud the external auditor ranks as one of the lowest, at 3.8% according to the ACFE’s Report to The Nations 2016, and in the United States alone it is 4%. There are a number of factors for why the external auditor is typically ineffective in discovering fraud including breadth of the engagement, number of transactions, number of accounts, and, in cases such as Enron, the cunning of the perpetrators. Trying to overcome these factors has always been a hurdle for accounting firms, and despite the prescriptions of SAS 99, which requires auditors to employ analytical techniques during the planning phase, fraud is not typically the main focus of an audit. These failures by auditors exemplify the need for a quick and effective analysis that can shine light on potential wrongdoings or fraud. Benford’s analysis gives auditors the ability to digitally analyze transactions for the possible presence of fraud. It can shine a light on areas of potential wrongdoing in a relatively short period of time, making it attractive for auditors to employ as the number of transactions gets larger, which increases the probability that those transactions should conform to Benford’s law.

Benford’s law, or first digit law, was first discovered in 1881 by Simon Newcomb, an astronomer and mathematician, who observed that the pages of logarithm books were unevenly worn on pages that contained lower digits logarithms more so than pages with higher digits logarithms. Newcomb noted his discovery and created a mathematical formula to explain the formula that was published in the *American Journal of Mathematics.* Newcomb did not offer a theory to explain why the phenomenon existed, and his discovery went relatively unnoticed for 50 years. Ironically, in the 1930’s, Frank Benford discovered the same phenomenon while also searching through books of logarithms, exactly as Newcomb had. He spent years testing the hypothesis against more than 20,000 data samples, including populations of cities in the United States, surface areas of major rivers and streams, even the numbers appearing in Readers Digest. Benford’s findings were published in 1938, titled The Law of anomalous numbers in the publication *Proceedings of the American Philosophical Society.*

The application of Benford’s law is relatively simple, straightforward, and becomes easier with the use of a computer and software to aid in digital analysis. It is the analysis of first digit occurrences in natural data sets, for example sets of numbers that result from a mathematical combination of numbers such as accounts receivable (quantity sold multiplied by price), or transaction level data such as expenses. It is important to remember that Benford’s does not apply to small data sets and typically conforms better over larger data sets. This also means it is best to use all of the data in an account rather than sampling.

The expected distribution of any set of numbers is defined mathematically as:

P(d)=Log_{10}(1+1/d)

Where D is the digit (1,2…9);

And P is the Probability.

The expected probabilities for the first digit of any number based on Benford’s law are as follows.

1 – .30103

2 – .17609

3 – .12494

4 – .09691

5 – .07918

6 – .06695

7 – .05799

8 – .05115

9 – .04576

Practically speaking, this means that based on Benford’s law the distribution the number 1 as the first digit of all numbers in a tested set should be roughly 30.103%, the number 2 should be 17.609% and so forth. It is worth noting that Benford’s can be applied beyond the first digit to the second or third digits, as well it can apply to two digit combinations. Benford’s law is scale invariant, which means it is independent of the unites that the data is expressed in. An example is that the list of lengths should have the same distribution whether the unit of measurement is feet, yards, inches, and so on.

Although forensic data analysis is a relatively new concept, mostly with the advent of computers and software able to handle such processes, Benford’s analysis is an old an proven equation for successful audits. There are numerous reasons for an auditor to employ Benford’s analysis, and as suggested in this blog, to employ it early into the audit. Benford’s helps an auditor to maintain a vigilance towards discovering fraud, and puts into perspective whether or not fraud could exist outside of a Benford’s analysis as well. Benford’s is atypical because it does not conform to the human notion of randomness in transactions, and has the ability to catch fraudsters who have gone through the effort to try and hide their crimes with intentionally random transactions.

Here is a great article written by one of the foremost experts on utilizings Benford’s law to discover fraud from 1999 in the *Journal of Accountancy. *Please follow my blog and/or comment below, and feel free to email me any suggestions for topics you might like to see or other questions regarding Benford’s law.