Using Benford’s Law for Fraud Detection & Auditing Referred to as the First-Digit Law, Benford’s Law is a mathematical theory conceived over 70 years ago that has aided numerous anti-fraud professionals in solving embezzlement, insurance claims and money laundering cases. Benford's Law gives the expected patterns of the digits in unaltered data, and explains there is a large bias towards the lower digits, so much so that nearly one-half of all numbers are expected to start with the digits 1 or 2. In this webinar, we will explain the theory behind the law and how it can be used to find potential fraud and errors to help turn your internal audit or fraud investigation into a revenue generating center. In this session, you will learn: • How to apply Benford’s law analysis to find outliers in processes such as cash disbursement, general ledger, insurance claims, tax assessments, etc. • The types of data that do and do not conform to Benford’s Law • A practical guide to apply Benford’s tests using IDEA software (1st digit, 2nd digit testing, advanced analytics – fuzzy logic, etc.)

1. Using Benford’s Law
for Fraud Detection & Auditing
Rohit Kundu, CAATs Expert
July 2014

2. • AuditNet® features:
• Over 2,000 Reusable Templates, Audit Programs,
Questionnaires and Control Matrices
• Networking Groups & Online Forums through LinkedIn, Google
and Yahoo
• Audit Guides, Manuals, and Books on Audit Basics
CaseWare Analytics (IDEA) users receive full access to AuditNet templates

3. • Founded in 1988
• An industry leader in providing technology solutions for
finance, accounting, governance, risk and audit professionals
• Over 400,000 users of our technologies across 130 countries
and 16 languages
• Customers include Fortune 500 and Global 500 companies
• Microsoft Gold Certified Partner
CaseWare International

4. International Acceptance

5. • What is Benford’s Law?
• Conforming/Non-Conforming Data Types
• Practical Applications of Benford’s Law
• Major Digit Tests
• Demo
• Q&A
Agenda

6. Timeline
1881- Simon Newcomb
1938 – Frank Benford
1961 - Roger Pinkham
1992 - Mark Nigrini
From Theory to Application
Simon Newcomb’s Theory:
Frequency of Use of the Different Digits in Natural Numbers
“A multi-digit number is more likely to begin with ‘1’ than any other number.”
Pg. 40. American Journal of Mathematics,
The Johns Hopkins University Press

7. Timeline
1881- Simon Newcomb
1938 – Frank Benford
1961 - Roger Pinkham
1992 - Mark Nigrini
From Theory to Application
Frank Benford:
• Analyzed 20,229 sets of numbers, including, areas of rivers, baseball
averages, atomic weights of atoms, electricity bills, etc.
Conclusion
Multi digit numbers beginning with 1, 2 or 3 appear more frequently than
multi digit numbers beginning with 4, 5, 6, etc.

8. Timeline
1881- Simon Newcomb
1938 – Frank Benford
1961 - Roger Pinkham
1992 - Mark Nigrini
From Theory to Application
Data First Digit 1 First Digit 2 First Digit 3
Populations 33.9 20.4 14.2
Batting Averages 32.7 17.6 12.6
Atomic Weight 47.2 18.7 10.4
X-Ray Volts 27.917 15.7
Average 30.6% 18.5% 12.4%

9. Timeline
1881- Simon Newcomb
1938 – Frank Benford
1961 - Roger Pinkham
1992 - Mark Nigrini
From Theory to Application
Roger Pinkham:
Research conducted revealed that Benford’s probabilities are scale invariant.
Dr. Mark Nigrini:
Published a thesis noting that Benford’s Law could be used to detect fraud
because human choices are not random; invented numbers are unlikely to
follow Benford’s Law.

10. The number 1 occurs as the
leading digit 30.1% of the
time, while larger numbers
occur in the first digit less
frequently.
For example, the number 3879
 3 - first digit
 8 - second digit
 7 - third digit
 9 – fourth digit
Benford’s Law

11. Benford’s Law Key Facts
 For naturally occurring numbers, the leading digit(s) is (are)
distributed in a specific, non-uniform way.
 While one might think that the number 1 would appear as
the first digit 11 percent of the time, it actually appears
about 30 percent of the time.
 Therefore the number 1 predominates most progressions.
 Scale invariant – works with numbers denominated as
dollars, yen, euros, pesos, rubles, etc.
 Not all data sets are suitable for analysis.

12. Benford’s Law Defined

13. Conforming Data Types
• Data set should describe similar data (e.g. town populations)
• Large Data Sets
• Data that has a wide variety in the number of figures e.g.
plenty of values in the hundreds, thousands, tens of
thousands, etc.
• No built-in maximum or minimum values
Some common characteristics of accounting data…

14. Conforming Data Types - Examples
• Accounts payable transactions
• Credit card transactions
• Customer balances and refunds
• Disbursements
• Inventory prices
• Journal entries
• Loan data
• Purchase orders
• Stock prices, T&E expenses, etc.

15. Non-Conforming Data Types
• Data where pre-arranged, artificial limits or nos. influenced
by human thought exist i.e. built-in maximum or minimum
values
– Zip codes, telephone nos., YYMM#### as insurance policy no.
– Prices sets at thresholds ($1.99, ATM withdrawals, etc.)
– Airline passenger counts per plane
• Aggregated data
• Data sets with 500 or few transactions
• No transaction recorded
– Theft, kickback, skimming, contract rigging, etc.

16. Usage of Benford’s Law
• Within a comprehensive Anti-Fraud Program
COSO Framework
Risk
Assessment
Control
Environment
Control
Activities
Information and
Communication
Specify
organizational
objectives
Monitoring

17. High- Level Usage of Benford’s Law
• Risk-Based Audits
– Planning Phase
 Early warning sign that past data patterns have changed
or abnormal activity
Data Set X represents the first
digit frequency of 10,000 vendor
invoices.

18. High- Level Usage of Benford’s Law
• Forensic Audits
– Check fraud, bypassing permission limits, improper
payments
• Audit of Financial Statements
– Manipulation of checks, cash on hand, etc.
• Corporate Finance/Company Evaluation
– Examine cash-flow-forecasts for profit centers

19. Major Digit Tests (using IDEA)
• 1st Digit Test
• 2nd Digit Test
• First two digits
• First three digits
• Last two digits
• Second Order Test

20. 1st & 2nd Digit Tests
1st Digit Test
• High Level Test
• Will only identify the blinding glimpse of the obvious
• Should not be used to select audit samples, as the sample
size will be too large
2nd Digit Test
• Also a high level test
• Used to identify conformity
• Should not be used to select audit samples

21. First Two Digits Test
• More focused and examines the frequency of the numerical
combinations 10 through 99 on the first two digits of a series
of numbers
• Can be used to select audit targets for preliminary review
Example:
10,000 invoices -- > 2600 invoices
-- > (1.78% + 1.69%) x 10,000
-- > (178 + 169) = 347 invoices
Only examine invoices beginning with the
first two digits 31 and 33.
Source: Using Benford’s Law to Detect Fraud , ACFE

22. First Three Digits Test
• Highly Focused
• Used to select audit samples
• Tends to identify number duplication

23. Last Two Digits Test
• Used to identify invented (overused) and rounded numbers
• It is expected that the right-side two digits be distributed
evenly. With 100 possible last two digits numbers (00, 01,
02...., 98, 99), each should occur approximately 1% of the
time.
Source: Fraud and Fraud Detection: A Data Analytics
Approach, John Wiley & Sons, Inc., Hoboken, New
Jersey

24. Second Order Test
• Based on the 1st two digits in the data.
• A numeric field is sorted from the smallest to largest
(ordered) and the value differences between each pair of
consecutive records should follow the digit frequencies of
Benford’s Law.
Source: Fraud and Fraud Detection: A Data Analytics
Approach, John Wiley & Sons, Inc., Hoboken, New
Jersey

25. Continuous Monitoring Framework
• Automated & Repeatable Analysis
• Input New Analytics with Ease
• Remediation Workflow & Resolution Guidelines
• KPIs (Root Cause Analysis)

26. Continuous Monitoring Framework
Turn-key Solutions
• P2P
• Purchasing Cards and T&E Monitoring
– Identify transaction policy violations
– Spend, Expense & Vendor profiling
– Identify card issuance processing errors
– Evaluate trends for operational/process improvements

27. Conclusion
Benford’s Law
• One person invents all the numbers
• Lots of different people have an incentive to manipulate
numbers in the same way
• Useful first step to give us a better understanding of our data
• Need to use Benford’s Law together with other drill down
tests
• Technology enables this faster and easier to produce results

28. Rohit Kundu
Rohit.kundu@caseware.com
Sunder Gee
Sunder.Gee@ZapConsulting.ca
IDEA Inquiries
salesidea@caseware.com
Q & A