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BENFORDS LAW
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BENFORDS LAW
History
What is Benfords Law
Types of Data That ConformUses in Fraud Investigations
Examples
Other uses of Benfords LawCautions 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 from1 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 eachposition 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 1/9that the first digit will be 1 - 9
This would mean digits are equally likely tooccur, but this is not the case
According to Benfords Law the probability
of obtaining a 1 in the first digit position is30.1%
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Expected Frequencies Based on
Benfords 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.
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Benfords Law
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http://testingbenfordslaw.com/rails-core-line-count
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Logic Behind Benfords Law
If a data entry begins with the digit 1 it hasto double in size (100%)before it beginswith the digit 2
If a data entry begins with the digit 9 it onlyhas to be increased by 11% in order forthe first digit to be a 1
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Types of Data That ConformWhen Benford Analysis Is
Likely Used
Examples
Sets of numbers that result from
mathematical combination of
numbersResult comes
from two distributions
Accounts receivable (number sold * price),
Accounts payable (number bought * price)
Transaction-level dataNo
need to sampleDisbursements, sales, expenses
On large data setsThe more
observations, the betterFull years transactions
Accounts that appear to conformWhen 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
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Types of Data That Do NotConform
When Benford Analysis IsNot
Likely UsedExamples
Data set is comprised of assigned
numbers
Check numbers, invoice numbers, zip
codes
Numbers that are influenced byhuman 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
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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 pricesamong locations
New combinations of selling prices
Customer refunds
Duplicate payments
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Uses in Fraud Investigations
Invented or altered numbers are not likelyto follow Benfords Law
Human choices are not random
1993, State of Arizona v.Wayne JamesNelson
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Examples of Benfords Law
Benfords Law was used to analyze the
first 2 digits of accounts payable data for aNASDAQ-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-storefamily-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 BenfordsLaw to analyze the stores disbursement data
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Small Business Owner Example(Source: Rose, 2003)
First 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
Rate Benford
Sample
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Small Business Owner Example
Second Digits Test
00.05
0.1
0.15
0.2
0.250.3
0.35
0 1 2 3 4 5 6 7 8 9
Digit
Rate Benford
Sample
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Small Business Owner Example First 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
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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 expectedOwner 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 populationstatistics & 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 detectedusing 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/Benfords%20Law.html, accessed 9/21/2004. Browne, Malcolm W. Following Benfords 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 Benfords Lawto Assist in Detecting Fraud in Accounting Data, Journal of Forensic Accounting1524-
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 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.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 MillenniumMathematics Project, University of Cambridge, Sept. 1999).
http://www.intuitor.com/statistics/Benford's%20Law.htmlhttp://www.rexswain.com/benford.htmlhttp://www.nigrini.com/images/ForensicAccountantReport.htmlhttp://www.euromoneyplc.com/http://www.aicpa.org/pubs/jofa/may1999/nigrini.htmlhttp://web9.epnet.com.ezproxy.udayton.edu/http://web9.epnet.com.ezproxy.udayton.edu/http://pass.maths.org.uk/issue9/features/benfordhttp://pass.maths.org.uk/issue9/features/benfordhttp://web9.epnet.com.ezproxy.udayton.edu/http://www.aicpa.org/pubs/jofa/may1999/nigrini.htmlhttp://www.euromoneyplc.com/http://www.nigrini.com/images/ForensicAccountantReport.htmlhttp://www.rexswain.com/benford.htmlhttp://www.intuitor.com/statistics/Benford's%20Law.htmlTop Related