Benford’s Law

Definition

Benford’s law (also called the first digit law) states that the leading digits in a collection of data sets 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. [1]

Benford’s Law states that the first digits found in a data set are expected to be arranged in a way that the lowest digit, one, appears the most frequently, followed by two, three, etc. This law can be utilized to detect patterns, or lack thereof, in naturally occurring data sets, which can be used to help catch anomalies or fraud in data. [2]

Benford’s Law, named for physicist Frank Benford, who worked on the theory in 1938, is the mathematical theory of leading digits. Specifically, in data sets, the leading digit(s) is (are) distributed in a specific, nonuniform way. While one might think that the number 1 would appear as the first digit 11 percent of the time (i.e., one of nine possible numbers), it actually appears about 30 percent of the time (see Figure 1). Nine, on the other hand, is the first digit less than 5 percent of the time. The theory covers the first digit, second digit, first two digits, last digit and other combinations of digits because the theory is based on a logarithm of probability of occurrence of digits. [3]

*Figure I—Benford’s Law Distribution Leading Digit*

Who

One of the most common uses of Benford’s Law is in accounting. To ensure that the financial records are accurate, auditors will often use Benford’s Law to check for errors. This is because the distribution of digits in financial records should follow the pattern predicted by Benford’s Law. [4]

Accountants often compare the leading digits of financial transaction data, such as ledger entries, to a Benford curve to spot anomalies that may indicate fraud. The same technique can be used to detect irregular network activity and other data that may indicate malicious insider activity. [6]

When it comes to fighting fraud, there’s a tried-and-true statistical precept that remains as relevant and widely accepted as ever. “Benford’s Law” is often used by forensic accountants to spot dubious digits — and catch even the most sophisticated thieves. [7]

What

Put simply, Benford’s law says that the leading digit in a number is more likely to be a small number like 1, 2, or 3 than a large number like 7, 8, or 9. It also states that one is the most likely leading digit to occur by far.

Benford’s Law can be applied in a number of ways. Its main use is in detecting fraud or human tampering with data sets such as tax returns or financial records. It can also be used to predict the distribution of digits in data sets that have not yet been collected. [4]

Why

See Theoretical Knowledge Vs Practical Application.

How

Here are the basic things to remember: [5]

It relies on the idea that the distribution of digits in multi-digit natural numbers is not random; instead, it follows a predictable pattern. More on that later.
It applies only to “natural numbers.” The definition for natural numbers and non-natural numbers in a fraud examination are different than they are in math, so don’t let that throw you off. Here’s how we define them for Benford’s Law.
- Natural numbers are those numbers that are not ordered in a particular numbering scheme and are not human-generated or generated from a random number system.
- Non-natural numbers are designed systematically to convey information that restricts the natural nature of the number (e.g., postal codes and telephone numbers).

The assumptions regarding the data to be examined by Benford’s Law are:

Numeric data
Randomly generated numbers:
– Not restricted by maximums or minimums
– Not assigned numbers
Large sets of data
Magnitude of orders (e.g., numbers migrate up through 10, 100, 1,000, 10,000, etc.) (Other assumptions exist that are unimportant in applying Benford’s Law in IT audits.)

The mathematical theory has always been applied to digital analysis, i.e., a logarithmic study of the occurrence of digits by position in a number. [10]

Please use the following web pages for examples on using Benford’s Law.

Benford’s Law Explained with Examples
Benford’s law
TESTING BENFORD’S LAW
Benford’s Law in Excel (1 Solved Example)

Benford’s Law, however, isn’t infallible. The law may not work in cases that involve smaller sets of numbers that don’t follow the rules of randomness or numbers that have been rounded (resulting in different first digits). Also, smaller numbers (1, 2, etc.) are more likely to occur simply because they’re smaller and the logical place to begin a count.

Assigned numbers, such as those on invoices, are also iffy. On a similar note, uniform distributions — such as lotteries where every number painted on a ball has an equal likelihood of selection — may not suit a Benford’s Law analysis. And prices involving the numbers 95 and 99 (often used because of marketing strategies) may call for a different approach.

In addition, the principle may be ineffective for sets of numbers with built-in ceilings and floors. For example, expense reports where receipts are required for meals costing $25 or more will reveal many claims just under the limit, in amounts such as $24.90. [7]

Then there is the fact that Benford Law seems to apply only to certain types of data. Physicists have found that it crops up in an amazing variety of data sets. Here are just a few: the areas of lakes, the lengths of rivers, the physical constants, stock market indices, file sizes in a personal computer and so on. However, there are many data sets that do not follow Benford’s law, such as lottery and telephone numbers. [8]

One application of the Law is in fraud detection. For example, people are more likely to claim amounts of money starting with a 9, because it’s the maximum amount they can claim for a certain number of digits — $992 looks much less than $1002 because it’s got fewer digits, even though it’s only about 1% less. But Benford’s Law says that on true expenses claims, this pattern should not happen. This provides a way of checking for suspected fraudulent claims — although, by itself, it’s not proof of anything, of course. [9]

Many of the References and Additional Reading websites and Videos will assist you with understating and applying,

As some professors say: “It is intuitively obvious to even the most casual observer.”

References

[1] “Benford’s Law (The First Digit Law): Simple Definition, Examples”. 2023. Statistics How To. https://www.statisticshowto.com/benfords-law/.

Science, technology and business publications are not above suspicion and second screening today. It’s an unfortunate but true development. In this era of Big Data, we are inundated by statistical models and conclusions. These studies, models, and their conclusions affect our society deeply, in an all-encompassing way from healthcare to economics to social interaction and technological research. It is even more disheartening to see that basic science research publications, which are supposed to report the pure objective truth, are not above such dishonesty and fraud. We need effective screens and sound analytical techniques to judge the veracity of these models and research publications. In this article, we discussed one such statistical law and its utility for detecting anomalies in number patterns.

[2] “What Is Benford’s Law and Why Is It Important?”. 2023. Built In. https://builtin.com/data-science/benfords-law.

[3] “Understanding and Applying Benford’s Law”. 2023. ISACA. https://www.isaca.org/resources/isaca-journal/past-issues/2011/understanding-and-applying-benfords-law.

[4] Mahr, Nathan. “Benford’s Law Process, Uses and Examples”. 2023. Study.com. https://study.com/learn/lesson/benfords-law-process-uses-examples.html.

[5] Gill, John. 2019. “What Is Benford’s Law and Why Do Fraud Examiners Use It? — ACFE Insights”. ACFE Insights. https://www.acfeinsights.com/acfe-insights/what-is-benfords-law.

[6] Kessel, Emily. “Benford’s Law: Potential Applications for Insider Threat Detection”. 2020. SEI Blog. https://insights.sei.cmu.edu/blog/benfords-law-potential-applications-insider-threat-detection/.

[7] “HOW FORENSIC INVESTIGATORS USE DIGITS TO FIND FRAUD”. 2023. SSACPA. https://www.ssacpa.com/how-forensic-investigators-use-digits-to-find-fraud/.

[8] “Benford’s Law and a Theory of Everything”. 2023. MIT Technology Review. https://www.technologyreview.com/2010/05/07/203468/benfords-law-and-a-theory-of-everything/.

[9] Willetts, Ben. “Benford’s law (with Vi Hart, 1 of 2)”. 2013. https://www.khanacademy.org/math/algebra-home/alg-exp-and-log/alg-logarithmic-scale/v/vi-and-sal-talk-about-the-mysteries-of-benford-s-law.

[10] ⭐ “Understanding and Applying Benford’s Law”. 2023. ISACA. https://www.isaca.org/resources/isaca-journal/past-issues/2011/understanding-and-applying-benfords-law.

Additional Reading

“Benford’s Law — From Wolfram MathWorld”. 2023. mathworld.wolfram.com. https://mathworld.wolfram.com/BenfordsLaw.html.

“Benford’s Law – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Benford%27s_law.

⭐ “Benford’s Law | Brilliant Math & Science Wiki”. 2023. brilliant.org. https://brilliant.org/wiki/benfords-law/.

“First Digits Rule! (Benford’s Law)”. 2023. mathsisfun.com. https://www.mathsisfun.com/numbers/benfords-law.html.

“Using Excel and Benford’s Law to detect fraud”. 2017. Journal Of Accountancy. https://www.journalofaccountancy.com/issues/2017/apr/excel-and-benfords-law-to-detect-fraud.html.

Frost, Jim. “Benford’s Law Explained with Examples”. 2023. Statistics By Jim. https://statisticsbyjim.com/probability/benfords-law/.

Gonsalves, Robert. “Benford’s Law — A Simple Explanation”. 2022. Medium. https://towardsdatascience.com/benfords-law-a-simple-explanation-341e17abbe75.

“How Reliable Is Benford’s Law In Forecasting Crises?”. 2011. Quantitative Finance Stack Exchange. https://quant.stackexchange.com/questions/3412/how-reliable-is-benfords-law-in-forecasting-crises.

Keep in mind that Benford’s law is not a universal or natural law. A violation of Benford’s law is neither a necessary nor a sufficient condition to prove a flaw or a quality issue in the data. At the best, it can give you a hint, but it should not be trusted blindly. Moreover, note that for some types of data the law will not work at all, such as e.g., Likert scale variables or binary variables. It is also problematic to apply Benford’s law to macroeconomic data. Structural breaks which can typically be found in economic data series can result in a rejection of Benford’s law.

Marchand, Chase and Dalton Maahs, “Benford’s Law and COVID-19 Data | CHANCE”. 2023. chance.amstat.org. https://chance.amstat.org/2021/04/benfords-law/.

Murtagh, Jack. 2023. “What Is Benford’s Law? Why This Unexpected Pattern of Numbers Is Everywhere”. Scientific American. https://www.scientificamerican.com/article/what-is-benfords-law-why-this-unexpected-pattern-of-numbers-is-everywhere/.

Spivey, Mike. “Real Life Usage Of Benford’s Law”. 2010. Mathematics Stack Exchange. https://math.stackexchange.com/questions/58/real-life-usage-of-benfords-law/8100#8100.

Videos

Benford’s law (with Vi Hart, 1 of 2)

Benford’s law (with Vi Hart, 2 of 2)

Benford’s law explanation (sequel to mysteries of Benford’s law)

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

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