BENFORDS LAW

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BENFORDS LAW
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BENFORDS LAW

• History
• What is Benfords Law
• Types of Data That Conform
• Uses in Fraud Investigations
• Examples
• Other uses of Benfords Law
• Cautions When Using Benfords Law

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HISTORY

• Simon Newcomb 1881
• Frank Benford 1938
• Roger Pinkham 1961
• Theodore Hill 1995

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History Accounting Data

• Benfords Law was first used by accountants in
  late 1980
• Nigrini,

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WHAT IS BENFORDS LAW

• BENFORDS LAW FORMULA
• The probability of any number d from
• 1 through 9 being the first digit is.
• Log10 (1 1/d)

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WHAT IS BENFORDS LAW?

• Benfords 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

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WHAT IS BENFORDS 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 Benfords Law the probability of
  obtaining a 1 in the first digit position is
  30.1

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Expected Frequencies Based on Benfords Law
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Logic Behind Benfords 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

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Types of Data That Conform
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Types of Data That Do Not Conform
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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

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Uses in Fraud Investigations

• Invented or altered numbers are not likely to
  follow Benfords Law
• Human choices are not random
• 1993, State of Arizona v.Wayne James Nelson

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Examples of Benfords Law

• Benfords Law was used to analyze the first 2
  digits of accounts payable data for a
  NASDAQ-listed software company

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Examples of Benfords Law

• Bank audit

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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 Benfords Law
  to analyze the stores disbursement data

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Small Business Owner Example(Source Rose, 2003)

• First Digits Test

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Small Business Owner Example

• Second Digits Test

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Small Business Owner Example

• First 2 Digits Test

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Small Business Owner Example

• Benfords 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

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Small Business Owner Example

• Benfords 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

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Other Uses of Benfords Law

• Year 2000 problem
• allocating computer disk space
• detect irregularities in clinical trials

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Other Uses of Benfords Law

• Demographic models of population statistics
  vital statistics
• Customs officials
• ballot fraud in Ukraines republic elections

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Cautions

• Not necessarily fraud
• False positives
• Certain types of fraud will not be detected using
  Benfords Law analysis

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Conclusion

• Useful tool
• Digital analysis

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Suggested Reading

• Works Cited
• Amazing Applications of Probability and
  Statistics, http//www.intuitor.com/
  statistics/Benfords20Law.html, accessed
  9/21/2004.
• Browne, Malcolm W. Following Benfords Law, or
  Looking Out for No. 1, http//www.rexswain.com/be
  nford.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 Benfords
  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.c
  om/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 Benfords Law, Journal of the American
  Taxation Association. 18(1), 1996 72-92.
• Nigrini, Mark. Ive Got Your Number,
  http//www.aicpa.org/pubs/jofa/may1999/nigrini.htm
  l, 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.u
  dayton.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.or
  g.uk/issue9/features/benford, accessed 11/24/2004
  (From Millennium Mathematics Project, University
  of Cambridge, Sept. 1999).