Skip to main content

Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection

Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection

Mark J. Nigrini, Joseph T. Wells (Foreword by)

ISBN: 978-1-118-15285-0 April 2012 352 Pages


In Stock

CAD $96.00


Applications for Forensic Accounting, Auditing, and Fraud Detection

There are hidden patterns in the chaos that we know as data. In the 1930s, the physicist Frank Benford found that there were predictable patterns to the digits in the numbers in tabulated data. For many years, this little secret was known to only a few people, made up mainly of mathematicians and the Benford family. In the 1990s, the accountant Mark Nigrini first advocated the use of Benford's Law as a test for fraud and of data integrity. With 250 tables and figures dealing with 50 data sets revealed over 13 chapters, Nigrini takes us on a pioneering journey in Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection. Our adventure starts with the original 1938 paper on the topic and then moves through the leading-edge mathematical discoveries in the early days, finishing with recent examples from the GM, Chrysler, and Lehman bankruptcy filings, as well as an analysis of the numbers invented by students in an experiment designed to trick the researcher.

The applications include:

  • U.S. census population numbers
  • The Fibonacci numbers
  • Corporate payments
  • Enron and AIG's financial statement numbers
  • General Motors and Chrysler payments data
  • Madoff's monthly returns
  • Streamflow and lake data
  • Taxpayer interest and dividend income
  • The tax returns of Bill and Hillary Clinton
  • Lehman's charitable gifts

Benford's Law is written to be understood and enjoyed by auditors, forensic accountants, fraud investigators, financial analysts, and others concerned with data integrity and data authenticity. The book is both informative and entertaining, and is aimed at professionals with a basic knowledge of statistics and a love for numbers and figures. Nigrini's passion for the topic is clear, from the first page, where we are reminded that numbers play an integral part in our daily lives, to the final chapter, which includes an evaluation of where we are at the moment with respect to Benford's Law and where we need to go in the future. The inclusion of Excel, Access, and IDEA screenshots means that you can start working on your own data right away.

A companion website has links to many of the data sets used in the book, together with PowerPoint slides for college instructors and conference presentations. The website includes the Nigrini Cycle Excel template and several Access databases with their Benford's Law queries. The tests in the book will focus your attention on the abnormal duplications, biases, and anomalies in your data.

Related Resources


Request an Evaluation Copy for this title

Foreword xi

Preface xiii

About the Author xix

Chapter 1: Introduction and Mathematical Foundations 1

Benford's Expected Digit Frequencies 5

Defining the First and First-Two Digits 6

Digit Patterns of U.S. Census Data 8

Logging on to Benford's Law 10

General Significant Digit Law 13

Log and Behold, the Census Data 13

Love at First Sight 15

Mantissa Test and Census Data 19

Number of Records and Benford's Law Tests 20

When Should Data Conform to Benford's Law? 21

Conclusions 23

Chapter 2: Theorems, Truisms, and a Little Trivia 25

Digits of Corporate Payments Data 26

Digits of Lake Data 28

Scale Invariance Theorem 31

Mean Absolute Deviation 34

Scale Invariance and Census Data 34

Scale Invariance and Corporate Payments 35

Scale Invariance and Lake Data 36

A Level Playing Field Becomes Benford 38

Multiplication by 1/X 42

All Distributions Lead to Benford 43

Getting a Gripf on Benford and Zipf 46

Conclusions 50

Chapter 3: More Formulas and Facts, and a Little Fibonacci 51

Fibonacci Numbers 51

Lucas Numbers 53

Back to Square One 55

3nþ1 Problem 58

Ultimate Uniform Distribution 60

Benford Embraces Other Bases 62

Nigrini's Second Last Theorem 65

Conclusions 69

Chapter 4: Primary Benford’s Law Tests 71

Corporate Payments Data 72

Data Profile 72

First Come, First Served 74

Playing Second Fiddle 75

First-Two Digits Test 78

Running the Digit Tests in Excel 80

Running the Digit Tests in Access 83

Conclusions 87

Chapter 5: Advanced Benford’s Law Tests 89

Summation Test 90

Running the Summation Test in Excel 94

Running the Summation Test in Access 95

Second-Order Test 97

An Analysis of Payments Data 102

An Analysis of Journal Entry Data 104

An Analysis of Census Data 107

Running the Second-Order Test in Excel 108

Excel, Thanks a Million(s) 110

Scale Invariance and the Second-Order Test 113

Conclusions 114

Chapter 6: Associated Benford’s Law Tests 117

Number Duplication Test 117

An Analysis of Payments Data 119

An Analysis of Census Data 121

Running the Number Duplication Test in Excel 122

Running the Number Duplication Test in Access 126

Last-Two Digits Test 129

An Analysis of Payments Data 130

An Analysis of Census Data 131

An Analysis of Election Results 132

Running the Last-Two Digits Test in Excel 135

Running the Last-Two Digits Test in Access 136

Distortion Factor Model 138

Distortion and the Census Data 145

Conclusions 146

Chapter 7: Assessing Conformity to Benford's Law 149

Z-Statistic 150

Chi-Square Test 153

Kolmogorov-Smirnoff Test 157

Mean Absolute Deviation Test 158

The Logarithmic Basis of Benford's Law 160

Creating a Perfect Synthetic Benford Set 163

Mantissa Arc Test 165

Conclusions 169

Chapter 8: Examples of Fraudulent Data 171

The Inside Story 172

The Vendor Who Never Was 174

Not Paying Attention 175

Funny Money 177

The Heart of the Matter 181

Going the Extra Mile 182

Laugh All the Way to the Bank 184

Culture Shock 187

Having a Bad Hair Day 189

An Unclean Bill of Health 191

Turning the Table on Tax Evasion 193

Conclusions 196

Chapter 9: Fraudulent Financial Statements, Part I 199

Number Crunching 200

Wrong Numbers 205

A Look at Enron's and AIG’s Numbers 207

Figuring Out the Controllers 210

Conclusions 213

Chapter 10: Fraudulent Financial Statements, Part II 215

Digital Yoga by Absaroka 216

Can’t See the Forest for the Trees 218

Digit a Little Deeper into Papua New Guinea 221

The Good, the Bad, and the Ugly 227

Dig a Little Deeper 233

There Are More Questions than Answers 237

Conclusions 244

Chapter 11: Madoff and Other Ponzi Schemes 247

The Madoff Claims 248

Don't Bank on Kaupthing 249

The Whole Ball of Waxenberg 252

General Motors Demoted to Private 253

Chrysler Unable to Dodge Bankruptcy 255

Discussion of the Claims Results 258

A Review of the Madoff Returns 259

Apple, Dell, Berkshire, and Benford 262

Discussion of the Returns Results 265

Chapter 12: Earth Science and Income Tax Applications 267

Still Waters Run Deep 268

The Lay of the Lake 274

For a Few Dollars Less 281

A Clean Bill of Clinton 286

Conclusions 290

Chapter 13: Future Directions and Conclusions 293

My Law 295

Insights into Number Invention 300

Lehman's Charitable Gifts 306

The Bottom Line 310

Glossary of Selected Terms 315

References 323

Index 327