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Organizations are accumulating large amounts of data and internal auditors are rapidly increasing their mining for, and use of, these sizable data sets. This proliferation of data raises the question of how to extract meaning from it all. Benford’s Law relates to the frequency distribution of leading digits in large sets of numerical data and it states that in many naturally occurring collections of numbers, the number one appears as the leading digit about 30 percent of the time, while nine appears as the most significant digit less than 5 percent of the time.

On the other hand, if the digits were distributed uniformly, they would each occur about 11.1 percent of the time (100 percent divided by nine possible digits). These nine digits are one through nine and exclude zero, because zero as a leading digit is negligible. 

Simon Newcomb, an astronomer, shared his findings in the late 1880s after noticing that in logarithm tables, the earlier pages, which started with the digit one, were much more worn than subsequent pages. The phenomenon was again noted in the 1930s by the physicist Frank Benford, who tested it, provided thousands of observations to prove his findings, and was credited for it.

Uses in Internal Auditing

Benford’s Law can be used to detect possible fraud and data manipulation because people who make up figures tend to follow patterns and generally distribute their numbers uniformly. Also, people usually attempt to circumvent established authorization protocols and the data will show this manipulation. For example: An employee submitting false payment vouchers may follow a pattern to keep consistent or organized records in an attempt to conceal the abuse. Similarly, an employee subject to approval thresholds may split transactions just below such threshold to circumvent the approval limits. When this happens, there is a higher than expected number of transactions with the leading digits, which would be noticeable during a comparison of the first-digit frequency distribution from the data with the expected Benford’s Law distribution. The result would indicate anomalous results and point internal auditors toward further investigation. 

Benford’s Law can be applied in forensic analytics to detect procurement and accounts payable fraud, errors, waste and abuse. Other areas within organizations where Benford’s Law is particularly suitable include:

  • Credit card transactions
  • Purchase orders
  • Loan data
  • Customer balances
  • Journal entries
  • Stock prices
  • Accounts payable transactions
  • Inventory prices
  • Customer refunds

Benford's Law can be calculated in Excel, IDEA and ACL, among other data analytics tools. It also makes predictions about the distribution of second, third and subsequent digits. IDEA, a software tool widely used by internal auditors, can perform tests up to the fourth digit. ACL can do the same up to six digits.

The expected occurrence of the first leading digits can be illustrated as shown in the chart below: 

First Leading Digit Distribution

Screen Shot 2018-07-03 at 11.35.09 AM

If the data analyzed produces a chart where the bars for the leading digits are approximately the same height, or fail to show a pattern like the one shown in the graph, it suggests the underlying data may be fabricated.

Benford’s Law is expected to apply to certain data sets and not others.

Expected to Apply

  • Numbers that result from mathematical combination of numbers (e.g. quantity × price)
  • Transaction data (e.g. disbursements, sales)

Not Expected to Apply

  • Where numbers are assigned sequentially (e.g. check numbers, purchase orders, invoice numbers)
  • Where numbers are the result of human assignment (e.g. prices set by marketing or psychological thresholds like $4.99)
  • Accounts containing specific figures (e.g. accounts set up to record $25 refunds)
  • Accounts with built-in minimum or maximum amounts
  • Airline passenger counts per plane
  • Telephone numbers
  • Data sets with 1,000 or fewer transactions
  • Data generated by formulas (e.g., YYMM#### as in an insurance policy or loan number)
  • Data restricted by a maximum or minimum number (e.g., hourly wage rates)

Benford’s Law is a versatile tool applicable to a variety of transactions and it provides an analytical tool to detect anomalous transactions. Internal auditors should consider its applicability during assurance and consulting engagements to identify transactions for further review.

Hernan Murdock
Vice President, Audit Division
Dr. Hernan Murdock is Vice President, Audit Division for MIS Training Institute. Before joining MIS Training Institute he was the Director of Training at Control Solutions International, where he oversaw the company's training and employee development program. Previously he was a Senior Project Manager leading audit and consulting projects for clients in the manufacturing, transportation, high tech, education, insurance and power generation industries. Dr. Murdock also worked at Arthur Andersen, Liberty Mutual and KeyCorp. Dr. Murdock is a senior lecturer at Northeastern University where he teaches management, leadership and ethics. He is the author of Operational Auditing: Principles and Techniques for a Changing World, 10 Key Techniques to Improve Team Productivity, and Using Surveys in Internal Audits.

On the other hand, if the digits were distributed uniformly, they would each occur about 11.1 percent of the time (100 percent divided by nine possible digits). These nine digits are one through nine and exclude zero, because zero as a leading digit is negligible. 

Simon Newcomb, an astronomer, shared his findings in the late 1880s after noticing that in logarithm tables, the earlier pages, which started with the digit one, were much more worn than subsequent pages. The phenomenon was again noted in the 1930s by the physicist Frank Benford, who tested it, provided thousands of observations to prove his findings, and was credited for it.

Uses in Internal Auditing

Benford’s Law can be used to detect possible fraud and data manipulation because people who make up figures tend to follow patterns and generally distribute their numbers uniformly. Also, people usually attempt to circumvent established authorization protocols and the data will show this manipulation. For example: An employee submitting false payment vouchers may follow a pattern to keep consistent or organized records in an attempt to conceal the abuse. Similarly, an employee subject to approval thresholds may split transactions just below such threshold to circumvent the approval limits. When this happens, there is a higher than expected number of transactions with the leading digits, which would be noticeable during a comparison of the first-digit frequency distribution from the data with the expected Benford’s Law distribution. The result would indicate anomalous results and point internal auditors toward further investigation. 

Benford’s Law can be applied in forensic analytics to detect procurement and accounts payable fraud, errors, waste and abuse. Other areas within organizations where Benford’s Law is particularly suitable include:

  • Credit card transactions
  • Purchase orders
  • Loan data
  • Customer balances
  • Journal entries
  • Stock prices
  • Accounts payable transactions
  • Inventory prices
  • Customer refunds

Benford's Law can be calculated in Excel, IDEA and ACL, among other data analytics tools. It also makes predictions about the distribution of second, third and subsequent digits. IDEA, a software tool widely used by internal auditors, can perform tests up to the fourth digit. ACL can do the same up to six digits.

The expected occurrence of the first leading digits can be illustrated as shown in the chart below: 

First Leading Digit Distribution

Screen Shot 2018-07-03 at 11.35.09 AM

If the data analyzed produces a chart where the bars for the leading digits are approximately the same height, or fail to show a pattern like the one shown in the graph, it suggests the underlying data may be fabricated.

Benford’s Law is expected to apply to certain data sets and not others.

Expected to Apply

  • Numbers that result from mathematical combination of numbers (e.g. quantity × price)
  • Transaction data (e.g. disbursements, sales)

Not Expected to Apply

  • Where numbers are assigned sequentially (e.g. check numbers, purchase orders, invoice numbers)
  • Where numbers are the result of human assignment (e.g. prices set by marketing or psychological thresholds like $4.99)
  • Accounts containing specific figures (e.g. accounts set up to record $25 refunds)
  • Accounts with built-in minimum or maximum amounts
  • Airline passenger counts per plane
  • Telephone numbers
  • Data sets with 1,000 or fewer transactions
  • Data generated by formulas (e.g., YYMM#### as in an insurance policy or loan number)
  • Data restricted by a maximum or minimum number (e.g., hourly wage rates)

Benford’s Law is a versatile tool applicable to a variety of transactions and it provides an analytical tool to detect anomalous transactions. Internal auditors should consider its applicability during assurance and consulting engagements to identify transactions for further review.

Interested in learning more about this and other tools and techniques? Join Dr. Murdock when he teaches Lean Six Sigma Skills for AuditorsInternal Audit School, and High-Impact Skills for Developing and Leading Your Audit Team 

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Hernan Murdock
Vice President, Audit Division
Dr. Hernan Murdock is Vice President, Audit Division for MIS Training Institute. Before joining MIS Training Institute he was the Director of Training at Control Solutions International, where he oversaw the company's training and employee development program. Previously he was a Senior Project Manager leading audit and consulting projects for clients in the manufacturing, transportation, high tech, education, insurance and power generation industries. Dr. Murdock also worked at Arthur Andersen, Liberty Mutual and KeyCorp. Dr. Murdock is a senior lecturer at Northeastern University where he teaches management, leadership and ethics. He is the author of Operational Auditing: Principles and Techniques for a Changing World, 10 Key Techniques to Improve Team Productivity, and Using Surveys in Internal Audits.