BENFORD’S LAWBENFORD’S LAW

BENFORD’S LAWBENFORD’S LAW

HistoryHistory What is Benford’s LawWhat is Benford’s Law Types of Data That ConformTypes of Data That Conform Uses in Fraud InvestigationsUses in Fraud Investigations ExamplesExamples Other uses of Benford’s LawOther uses of Benford’s Law Cautions When Using Benford’s Law Cautions When Using Benford’s Law

HISTORYHISTORY

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

History – Accounting DataHistory – Accounting Data

Benford’s Law was first used by Benford’s Law was first used by accountants in late 1980accountants in late 1980

Nigrini, Nigrini,

WHAT IS BENFORD’S LAWWHAT IS BENFORD’S LAW

BENFORD’S LAW FORMULABENFORD’S LAW FORMULA

The probability of any number “d” fromThe probability of any number “d” from

1 through 9 being the first digit is….1 through 9 being the first digit is….

LogLog10 10 (1 + 1/d)(1 + 1/d)

WHAT IS BENFORD’S LAW?WHAT IS BENFORD’S LAW?

Benford’s law gives the probability of Benford’s law gives the probability of obtaining digits 1 through 9 in each obtaining digits 1 through 9 in each position of a number. position of a number.

For example, 3879For example, 3879 3 - first digit3 - first digit 8 - second digit8 - second digit 7 - third digit7 - third digit 9 – fourth digit9 – fourth digit

WHAT IS BENFORD’S LAWWHAT IS BENFORD’S LAW

Most people assume the probability is 1/9 Most people assume the probability is 1/9 that the first digit will be 1 - 9that the first digit will be 1 - 9

This would mean digits are equally likely to This would mean digits are equally likely to occur, but this is not the caseoccur, but this is not the case

According to Benford’s Law the probability According to Benford’s Law the probability of obtaining a 1 in the first digit position is of obtaining a 1 in the first digit position is 30.1%30.1%

Expected Frequencies Based on Expected Frequencies Based on Benford’s Law Benford’s Law

Digit 1st Place 2nd Place 3rd Place 4th Place

0   0.11968 0.10178 0.10018

1 0.30103 0.11389 0.10138 0.10014

2 0.17609 0.19882 0.10097 0.1001

3 0.12494 0.10433 0.10057 0.10006

4 0.09691 0.10031 0.10018 0.10002

5 0.07918 0.09668 0.09979 0.09998

6 0.06695 0.09337 0.0994 0.09994

7 0.05799 0.0935 0.09902 0.0999

8 0.05115 0.08757 0.09864 0.09986

9 0.04576 0.085 0.09827 0.09982

Source: Nigrini, 1996.

Logic Behind Benford’s LawLogic Behind Benford’s Law

If a data entry begins with the digit 1 it has If a data entry begins with the digit 1 it has to double in size (100%)before it begins to double in size (100%)before it begins with the digit 2with the digit 2

If a data entry begins with the digit 9 it only If a data entry begins with the digit 9 it only has to be increased by 11% in order for has to be increased by 11% in order for the first digit to be a 1the first digit to be a 1

Types of Data That ConformTypes of Data That ConformWhen Benford Analysis Is

Likely UsedExamples

Sets of numbers that result from mathematical combination of numbers – Result comes from two distributions

Accounts receivable (number sold * price),

Accounts payable (number bought * price)

Transaction-level data – No need to sample

Disbursements, sales, expenses

On large data sets – The more observations, the better

Full year’s transactions

Accounts that appear to conform – When the mean of a set of numbers is greater than the median and the skewness is positive

Source: Durtschi, 2004, 24.

Most sets of accounting numbers

Types of Data That Do Not Types of Data That Do Not ConformConform

When Benford Analysis Is Not Likely Used

Examples

Data set is comprised of assigned numbers

Check numbers, invoice numbers, zip codes

Numbers that are influenced by human thought

Prices set at psychological thresholds ($1.99, ATM withdrawals

Accounts with a large number of firm-specific numbers

An account specifically set up to record $100 refunds

Accounts with a built in minimum or maximum

Set of assets that must meet a threshold to be recorded

Where no transaction is recorded

Source: Durtschi, 2004, 24.

Thefts, kickbacks, contract rigging

Types of Data That ConformTypes of Data That Conform

Accounts payable dataAccounts payable data Accounts receivable dataAccounts receivable data Estimations in the general ledgerEstimations in the general ledger Relative size of inventory unit prices Relative size of inventory unit prices

among locationsamong locations New combinations of selling pricesNew combinations of selling prices Customer refundsCustomer refunds Duplicate paymentsDuplicate payments

Uses in Fraud InvestigationsUses in Fraud Investigations

Invented or altered numbers are not likely Invented or altered numbers are not likely to follow Benford’s Lawto follow Benford’s Law Human choices are not randomHuman choices are not random

1993, State of Arizona v.Wayne James 1993, State of Arizona v.Wayne James NelsonNelson

Examples of Benford’s LawExamples of Benford’s Law

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

Examples of Benford’s LawExamples of Benford’s Law

Bank audit Bank audit

Small Business Owner ExampleSmall Business Owner Example

Small business owner expanded his one-store Small business owner expanded his one-store family-owned business into a four-store chainfamily-owned business into a four-store chain

Had to relinquish some hands-on control with Had to relinquish some hands-on control with the expansionthe expansion

Concerned about bookkeeping errors or Concerned about bookkeeping errors or possibility of fraudpossibility of fraud

Owner used Excel program based on Benford’s Owner used Excel program based on Benford’s Law to analyze the store’s disbursement dataLaw to analyze the store’s disbursement data

Small Business Owner ExampleSmall Business Owner Example(Source: Rose, 2003)(Source: Rose, 2003)

First Digits TestFirst Digits Test

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9

Digit

Ra

te Benford

Sample

Small Business Owner ExampleSmall Business Owner Example

Second Digits TestSecond Digits Test

00.05

0.10.15

0.20.25

0.30.35

0 1 2 3 4 5 6 7 8 9

Digit

Rat

e Benford

Sample

Small Business Owner ExampleSmall Business Owner Example First 2 Digits TestFirst 2 Digits Test

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

10

14

18

22

26

30

34

38

42

46

50

54

58

62

66

70

74

78

82

86

90

94

98

Two Digit Pair

Rate Benford

Sample

Small Business Owner ExampleSmall Business Owner Example

Benford’s Law AnalysisBenford’s Law Analysis First Digits TestFirst Digits Test

Digits 5, 6, & 7 appear much more than Digits 5, 6, & 7 appear much more than expected, while the digit 1 appeared much expected, while the digit 1 appeared much less than expectedless than expected

Second Digits TestSecond Digits Test Again, the digits 6 & 7 appear much more Again, the digits 6 & 7 appear much more

often than expected, and 0 did not occur at alloften than expected, and 0 did not occur at all

Small Business Owner ExampleSmall Business Owner Example

Benford’s Law Analysis cont.Benford’s Law Analysis cont. First Two-Digits TestFirst Two-Digits Test

56 & 67 are the two digit combinations that 56 & 67 are the two digit combinations that appear more frequently than expectedappear more frequently than expected

Owner pulled sample of disbursements Owner pulled sample of disbursements starting with the 56 & 67 sequencesstarting with the 56 & 67 sequences Discovered pymts to unfamiliar vendorDiscovered pymts to unfamiliar vendor Addtl invest revealed vendor did not exist – Addtl invest revealed vendor did not exist –

pymts going to personal acctpymts going to personal acct

Other Uses of Benford’s LawOther Uses of Benford’s Law

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

Other Uses of Benford’s LawOther Uses of Benford’s Law

Demographic models of population Demographic models of population statistics & vital statistics statistics & vital statistics

Custom’s officials Custom’s officials ballot fraud in Ukraine’s republic electionsballot fraud in Ukraine’s republic elections

CautionsCautions

Not necessarily fraudNot necessarily fraud False positivesFalse positives Certain types of fraud will not be detected Certain types of fraud will not be detected

using Benford’s Law analysisusing Benford’s Law analysis

ConclusionConclusion

Useful toolUseful tool Digital analysisDigital analysis

Suggested ReadingSuggested Reading Works CitedWorks Cited ““Amazing Applications of Probability and Statistics”, Amazing Applications of Probability and Statistics”,

http://www.intuitor.com/statistics/Benford’s%20Law.html, accessed 9/21/2004.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”, Browne, Malcolm W. “Following Benford’s Law, or Looking Out for No. 1”,

http://www.rexswain.com/benford.html, accessed 9/21/2004 (From http://www.rexswain.com/benford.html, accessed 9/21/2004 (From The New York TimesThe New York Times, , Tuesday, August 4th, 1998). Tuesday, August 4th, 1998).

Durtschi, Cindy and William Hillison and Carl Pachini. “The Effective Use of Benford’s Law Durtschi, Cindy and William Hillison and Carl Pachini. “The Effective Use of Benford’s Law to Assist in Detecting Fraud in Accounting Data”, to Assist in Detecting Fraud in Accounting Data”, Journal of Forensic Accounting Journal of Forensic Accounting 1524-5586/Vol.V(2004): 17-34.1524-5586/Vol.V(2004): 17-34.

““Findings and Observations”, http://www.nigrini.com/images/ForensicAccountantReport.html, Findings and Observations”, http://www.nigrini.com/images/ForensicAccountantReport.html, accessed 9/22/2004.accessed 9/22/2004.

Matthews, Robert. “Benford Bend”, http://www.euromoneyplc.com, accessed 12/1/2004 (From Matthews, Robert. “Benford Bend”, http://www.euromoneyplc.com, accessed 12/1/2004 (From World LinkWorld Link; May/June 2000, 10-12). ; May/June 2000, 10-12).

Nigrini, Mark. “Taxpayer Compliance Application of Benford’s Law”, Nigrini, Mark. “Taxpayer Compliance Application of Benford’s Law”, Journal of the American Journal of the American Taxation Association.Taxation Association. 18(1), 1996: 72-92. 18(1), 1996: 72-92.

Nigrini, Mark. “I’ve Got Your Number”, http://www.aicpa.org/pubs/jofa/may1999/nigrini.html, Nigrini, Mark. “I’ve Got Your Number”, http://www.aicpa.org/pubs/jofa/may1999/nigrini.html, accessed 9/22/04 (From accessed 9/22/04 (From Journal of AccountancyJournal of Accountancy, May 1999)., May 1999).

Rose, Anna and Jacob Rose. “Turn Excel Into a Financial Sleuth”, Rose, Anna and Jacob Rose. “Turn Excel Into a Financial Sleuth”, http://web9.epnet.com.ezproxy.udayton.edu, accessed 12/1/2004 (http://web9.epnet.com.ezproxy.udayton.edu, accessed 12/1/2004 (From Journal of AccountancyFrom Journal of Accountancy, , August 2003, Vol 196 Issue 2, 58-61). August 2003, Vol 196 Issue 2, 58-61).

Walthoe, Jon and Robert Hunt and Mike Pearson. “Looking Out For Number One”, 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 http://pass.maths.org.uk/issue9/features/benford, accessed 11/24/2004 (From Millennium Mathematics Project, University of Cambridge, Sept. 1999).Mathematics Project, University of Cambridge, Sept. 1999).