"The Benford's Law-based tests signal abnormal duplications. The mathematics of Ben-ford's Law gives us the expected or the normal duplications, and duplications above the norm are abnormal or excessive."1

Benford's Law forms part of many audit plans and is frequently used by auditors. However, it is not always well understood. The results of applying Benford's Law provide a starting point for the auditor.

Benford's Law analyzes the digits in numerical data, helps identifies anomalies, and detects systematic manipulation of data (that is, the making up of false numbers) based on the digital distribution in a natural population. Natural population could be almost anything, such as all the transactions in a general ledger, the transactions in accounts payable, or even the cubic meters of water in all the lakes in Africa.

Frank Benford published "The Law of Anomalous Numbers" in 1938; it postulated that the lower the leading digit of a number, the more often it would appear. Frank Benford used data from rivers, populations, newspapers, cost data, addresses, and many other categories to confirm his theory. Since then, many have furthered his research, particularly Mark J. Nigrini, PhD. He has written many articles and books on the subject, such as Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection.

In 1938, the research and calculations were performed manually, which was painstaking. Today, with computing power and the ease of accessing big data sets, one can see that Benford's Law of expected numbers is valid. One website (Testing-BenfordsLaw.com) applies a number of data sets against Benford's Law. It tests data such as Twitter users by followers' count, most common iPhone passcodes, population of Spanish cities, U.K. government spending, and even includes the first 652,066 Fibonacci numbers.

The expected values for any data set of the first leading digit and also for the first two leading digits are outlined in Table 5.1.

For the first digit test, the first leading digit output is depicted in the graph in Figure 5.1. For example, the leading digit 1 appears 30 percent of the time, whereas the leading digit 9 appears 4.6 percent of the time. The bars are the actual data counts and the lines are the lower and upper boundaries along with the expected count. This data set conforms to Benford's Law.

TABLE 5.1 Benford's Law First Digit Frequency and First Two Digits Frequency

First Digit Frequency

Second Digit Frequency

0

—

0.11968

1

0.30103

0.11389

2

0.17609

0.10882

3

0.12494

0.10433

4

0.09691

0.10031

5

0.07918

0.09668

6

0.06695

0.09337

7

0.05799

0.09035

8

0.05115

0.08757

9

0.04576

0.08500

FIGURE 5.1 Benford's Law First Digit Test

For Benford's Law to be applicable, certain conditions must be met.

- The numbers in the data set should describe the same object.

- There should be no built-in maximum or minimum to the numbers.

- The numbers should not be assigned, such as telephone numbers, bank account numbers, social insurance, or social security numbers.

- Does not apply to uniform distributions such as lottery balls where the uniform balls are selected and not the actual numbers.

Primary Benford's Law tests are the if rst digit, first two digits, first three digits, and second digit tests. Advanced Benford's Law tests are summation and second order. Associated tests are last two digits, number duplication, and distortion factor model. All but the last two tests can be automatically executed from within the IDEA software.

The number duplication test identifies specific numbers causing spikes or anomalies in primary and summation tests. Spikes in the primary tests are caused by some specific numbers occurring abnormally too often. Abnormally large numbers in value cause spikes in the summation test.

The distortion factor model shows whether the data has an excess of lower digits or higher digits. It assumes that the true number is changed to a false number in the same range or percentage as the true number.

Most presentations and articles discuss using Benford's Law to detect numbers near their authorization limits. For example, if someone's authorization limit is $10,000, then many first two digits in the 99, 98, and 97 area will be detected using Benford's Law if they are trying to maximize authorizing expenditures. Some other practical applications include:

- Accounts payable (expenses) data

- Estimations (accruals) in the general ledger - Sales

- Purchases

- Non-arm's-length transactions

- Customer refunds

- Bad debts

- Anti-money laundering

There is a potential to detect money laundering because money laundering flows money into the revenue stream that is not generated by the regular business. Since paying income tax on the false revenue is not desirable, corresponding expenses are made up to offset the false revenue. If there are enough of these made-up expenses or numbers, Benford's Law may detect the anomalies.

Not only is Benford's Law relevant to detecting anomalies in financial related data, it is applicable in other fields, too. A study was published in the New Zealand Journal of Marine and Freshwater Research entitled "Statistical Fraud Detection in a Commercial Lobster Fishery."2 The study tested the reliability of fisheries' data in Canada. The study was prompted by the fact that lobster sales formed a large part of the underground economy. The Royal Canadian Mounted Police proceeds-of-crime unit first thought that large money transfers into a bank branch was from drug money, but later found them to be from cash sales of lobster. From highly regulated lobster fishery areas, the data was found to conform with the distribution as expected by Benford's Law. Lobster and snow-crab data from different, less regulated areas did not conform.

Another research paper titled "Not the First Digit! Using Benford's Law to Detect Fraudulent Scientific Data,"3 found that there could be nonconformity with Benford's Law for the second or higher order tests for scientific data produced by researchers. Fabricated data may conform to the first digit test.

The paper "When Does the Second-Digit Benford's Law-Test Signal an Election Fraud? Facts or Misleading Test Results"4 focused on Benford's Law and the conformity or nonconformity of election results.

The first digit test is a high-level test and is suitable for use with less than 300 transactions. The first three digits test is too detailed and will result in the need to investigate too many anomalies. The first two digits test is the most practical to use. Examples of applying the first two digits test are shown in Figures 5.2, 5.3, 5.4, and 5.5.

FIGURE 5.2 Benford's Law First Two Digit Test Accounts Payable from a Large Corporation Example Showing Conformity to Benford's Law

There are significant spikes for the first two digits of 19, 20, 21, . . . 31 and 32 in Figure 5.3. As this is an auto manufacturer, they sell cars to dealerships where sales of $19,000 to $32,000 are normal. Knowledge of the business allows you to eliminate this area for additional review.

Figure 5.4 clearly shows that Benford's Law identified the contents of this file as fabricated to be used to demonstrate payment data with the IDEA software.

The author generated the data in Figure 5.5 using the Benford Wiz software download from members.tripod.com/benfordwiz . This is to demonstrate and make auditors aware that where there are tools to detect fraud, there are always tools developed to prevent or circumvent detection.

FIGURE 5.3 Benford's Law First Two Digits Test Accounts Receivable File of an Automotive Manufacturer Showing Nonconformity Spikes

FIGURE 5.4 Sample Payment File Included with the IDEA Software Showing Nonconformity to the First Two Digits of Benford's Law

FIGURE 5.5 Benford's Law First Two Digits Test on Data Generated with Benfordwiz Software Showing General Conformity

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