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BENFORD’S LAW

BENFORD’S LAW. BENFORD’S LAW. History What is Benford’s Law Types of Data That Conform Uses in Fraud Investigations Examples Other uses of Benford’s Law Cautions When Using Benford’s Law. HISTORY. Simon Newcomb – 1881 Frank Benford – 1938 Roger Pinkham – 1961 Theodore Hill – 1995.

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BENFORD’S LAW

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  1. BENFORD’S LAW

  2. BENFORD’S LAW • History • What is Benford’s Law • Types of Data That Conform • Uses in Fraud Investigations • Examples • Other uses of Benford’s Law • Cautions When Using Benford’s Law

  3. HISTORY • Simon Newcomb – 1881 • Frank Benford – 1938 • Roger Pinkham – 1961 • Theodore Hill – 1995

  4. History – Accounting Data • Benford’s Law was first used by accountants in late 1980 • Nigrini,

  5. WHAT IS BENFORD’S LAW • BENFORD’S LAW FORMULA The probability of any number “d” from 1 through 9 being the first digit is…. Log10 (1 + 1/d)

  6. WHAT IS BENFORD’S LAW? • Benford’s law gives the probability of obtaining digits 1 through 9 in each position of a number. • For example, 3879 • 3 - first digit • 8 - second digit • 7 - third digit • 9 – fourth digit

  7. WHAT IS BENFORD’S LAW • Most people assume the probability is 1/9 that the first digit will be 1 - 9 • This would mean digits are equally likely to occur, but this is not the case • According to Benford’s Law the probability of obtaining a 1 in the first digit position is 30.1%

  8. Expected Frequencies Based on Benford’s Law

  9. Logic Behind Benford’s Law • If a data entry begins with the digit 1 it has to double in size (100%)before it begins with the digit 2 • If a data entry begins with the digit 9 it only has to be increased by 11% in order for the first digit to be a 1

  10. Types of Data That Conform

  11. Types of Data That Do Not Conform

  12. Types of Data That Conform • Accounts payable data • Accounts receivable data • Estimations in the general ledger • Relative size of inventory unit prices among locations • New combinations of selling prices • Customer refunds • Duplicate payments

  13. Uses in Fraud Investigations • Invented or altered numbers are not likely to follow Benford’s Law • Human choices are not random • 1993, State of Arizona v.Wayne James Nelson

  14. Examples of Benford’s Law • Benford’s Law was used to analyze the first 2 digits of accounts payable data for a NASDAQ-listed software company

  15. Examples of Benford’s Law • Bank audit

  16. Small Business Owner Example • Small business owner expanded his one-store family-owned business into a four-store chain • Had to relinquish some hands-on control with the expansion • Concerned about bookkeeping errors or possibility of fraud • Owner used Excel program based on Benford’s Law to analyze the store’s disbursement data

  17. Small Business Owner Example(Source: Rose, 2003) • First Digits Test

  18. Small Business Owner Example • Second Digits Test

  19. Small Business Owner Example • First 2 Digits Test

  20. Small Business Owner Example • Benford’s Law Analysis • First Digits Test • Digits 5, 6, & 7 appear much more than expected, while the digit 1 appeared much less than expected • Second Digits Test • Again, the digits 6 & 7 appear much more often than expected, and 0 did not occur at all

  21. Small Business Owner Example • Benford’s Law Analysis cont. • First Two-Digits Test • 56 & 67 are the two digit combinations that appear more frequently than expected • Owner pulled sample of disbursements starting with the 56 & 67 sequences • Discovered pymts to unfamiliar vendor • Addtl invest revealed vendor did not exist – pymts going to personal acct

  22. Other Uses of Benford’s Law • Year 2000 problem • allocating computer disk space • detect irregularities in clinical trials

  23. Other Uses of Benford’s Law • Demographic models of population statistics & vital statistics • Custom’s officials • ballot fraud in Ukraine’s republic elections

  24. Cautions • Not necessarily fraud • False positives • Certain types of fraud will not be detected using Benford’s Law analysis

  25. Conclusion • Useful tool • Digital analysis

  26. Suggested Reading • Works Cited • “Amazing Applications of Probability and Statistics”, http://www.intuitor.com/statistics/Benford’s%20Law.html, accessed 9/21/2004. • Browne, Malcolm W. “Following Benford’s Law, or Looking Out for No. 1”, http://www.rexswain.com/benford.html, accessed 9/21/2004 (From The New York Times, Tuesday, August 4th, 1998). • Durtschi, Cindy and William Hillison and Carl Pachini. “The Effective Use of Benford’s Law to Assist in Detecting Fraud in Accounting Data”, Journal of Forensic Accounting 1524-5586/Vol.V(2004): 17-34. • “Findings and Observations”, http://www.nigrini.com/images/ForensicAccountantReport.html, accessed 9/22/2004. • Matthews, Robert. “Benford Bend”, http://www.euromoneyplc.com, accessed 12/1/2004 (From World Link; May/June 2000, 10-12). • Nigrini, Mark. “Taxpayer Compliance Application of Benford’s Law”, Journal of the American Taxation Association. 18(1), 1996: 72-92. • Nigrini, Mark. “I’ve Got Your Number”, http://www.aicpa.org/pubs/jofa/may1999/nigrini.html, accessed 9/22/04 (From Journal of Accountancy, May 1999). • Rose, Anna and Jacob Rose. “Turn Excel Into a Financial Sleuth”, http://web9.epnet.com.ezproxy.udayton.edu, accessed 12/1/2004 (From Journal of Accountancy, August 2003, Vol 196 Issue 2, 58-61). • Walthoe, Jon and Robert Hunt and Mike Pearson. “Looking Out For Number One”, http://pass.maths.org.uk/issue9/features/benford, accessed 11/24/2004 (From Millennium Mathematics Project, University of Cambridge, Sept. 1999).

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