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

, by Nigrini, Mark; Wells, Joseph T.**Note:**Supplemental materials are not guaranteed with Rental or Used book purchases.

- ISBN: 9781118152850 | 1118152859
- Cover: Hardcover
- Copyright: 4/24/2012

This book will provide an overview of the applications of Benford's Law to summarize its current uses. It will also set out a series of Benford's Law tests that can be run on any data set and show how to run these tests on any one of ten software programs. It will show Benford's Law can be used as a tool for auditors and data managers to insure integrity of large sets of data. There will be a companion site for the book with sample data sets readers can practice tests on and other interesting items.

**MARK J. NIGRINI**, PhD, is a professor at The College of New Jersey where he teaches forensic accounting courses. His research involves advanced theoretical work on Benford's Law and the legal process surrounding fraud convictions. Nigrini is also the author of

*Forensic Analytics*(Wiley), which describes tests to detect fraud, errors, estimates, and biases in financial data. He has been published in national media including the

*Wall Street Journal*and has published papers on Benford's Law and accounting in academic and professional journals. Nigrini regularly presents professional seminars for accountants and auditors in North America, Europe, and Asia with recent events in Singapore, Switzerland, and New Zealand.

Foreword | p. xi |

Preface | p. xiii |

About the Author | p. xix |

Introduction and Mathematical Foundations | p. 1 |

Benford's Expected Digit Frequencies | p. 5 |

Defining the First and First-Two Digits | p. 6 |

Digit Patterns of U.S. Census Data | p. 8 |

Logging on to Benford's Law | p. 10 |

General Significant Digit Law | p. 13 |

Log and Behold, the Census Data | p. 13 |

Love at First Sight | p. 15 |

Mantissa Test and Census Data | p. 19 |

Number of Records and Benford's Law Tests | p. 20 |

When Should Data Conform to Benford's Law? | p. 21 |

Conclusions | p. 23 |

Theorems, Truisms, and a Little Trivia | p. 25 |

Digits of Corporate Payments Data | p. 26 |

Digits of Lake Data | p. 28 |

Scale Invariance Theorem | p. 31 |

Mean Absolute Deviation | p. 34 |

Scale Invariance and Census Data | p. 34 |

Scale Invariance and Corporate Payments | p. -35 |

Scale Invariance and Lake Data | p. 36 |

A Level Playing Field Becomes Benford | p. 38 |

Multiplication by 1/X | p. 42 |

All Distributions Lead to Benford | p. 43 |

Getting a Gripf on Benford and Zipf | p. 46 |

Conclusions | p. 50 |

More Formulas and Facts, and a Little Fibonacci | p. 51 |

Fibonacci Numbers | p. 51 |

Lucas Numbers | p. 53 |

Back to Square One | p. 55 |

3n+1 Problem | p. 58 |

Ultimate Uniform Distribution | p. 60 |

Benford Embraces Other Bases | p. 62 |

Nigrini's Second Last Theorem | p. 65 |

Conclusions | p. 69 |

Primary Benford's Law Tests | p. 71 |

Corporate Payments Data | p. 72 |

Data Profile | p. 72 |

First Come, First Served | p. 74 |

Playing Second Fiddle | p. 7^ |

First-Two Digits Test | p. 78 |

Running the Digit Tests in Excel | p. 80 |

Running the Digit Tests in Access | p. 83 |

Conclusions | p. 87 |

Advanced Benford's Law Tests | p. 89 |

Summation Test | p. 90 |

Running the Summation Test in Excel | p. 94 |

Running the Summation Test in Access | p. 95 |

Second-Order Test | p. 97 |

An Analysis of Payments Data | p. 102 |

An Analysis of Journal Entry Data | p. 104 |

An Analysis of Census Data | p. 107 |

Running the Second-Order Test in Excel | p. 108 |

Excel, Thanks a Million(s) | p. 110 |

Scale Invariance and the Second-Order Test | p. 113 |

Conclusions | p. 114 |

Associated Benford's Law Tests | p. 117 |

Number Duplication Test | p. 117 |

An Analysis of Payments Data | p. 119 |

An Analysis of Census Data | p. 121 |

Running the Number Duplication Test in Excel | p. 122 |

Running the Number Duplication Test in Access | p. 126 |

Last-Two Digits Test | p. 129 |

An Analysis of Payments Data | p. 130 |

An Analysis of Census Data | p. 131 |

An Analysis of Election Results | p. 132 |

Running the Last-Two Digits Test in Excel | p. 135 |

Running the Last-Two Digits Test in Access | p. 136 |

Distortion Factor Model | p. 138 |

Distortion and the Census Data | p. 145 |

Conclusions | p. 146 |

Assessing Conformity to Benford's Law | p. 149 |

Z-Statistic | p. 150 |

Chi-Square Test | p. 153 |

Kolmogorov-Smirnoff Test | p. 157 |

Mean Absolute Deviation Test | p. 158 |

The Logarithmic Basis of Benford's Law | p. 160 |

Creating a Perfect Synthetic Benford Set | p. 163 |

Mantissa Arc Test | p. 165 |

Conclusions | p. 169 |

Examples of Fraudulent Data | p. 171 |

The Inside Story | p. 172 |

The Vendor Who Never Was | p. 174 |

Not Paying Attention | p. 175 |

Funny Money | p. 177 |

The Heart of the Matter | p. 181 |

Going the Extra Mile | p. 182 |

Laugh All the Way to the Bank | p. 184 |

Culture Shock | p. 187 |

Having a Bad Hair Day | p. 189 |

An Unclean Bill of Health | p. 191 |

Turning the Table on Tax Evasion | p. 193 |

Conclusions | p. 196 |

Fraudulent Financial Statements, Part I | p. 199 |

Number Crunching | p. 200 |

Wrong Numbers | p. 205 |

A Look at Enron's and AIG's Numbers | p. 207 |

Figuring Out the Controllers | p. 210 |

Conclusions | p. 213 |

Fraudulent Financial Statements, Part II | p. 215 |

Digital Yoga by Absaroka | p. 216 |

Can't See the Forest for the Trees | p. 218 |

Digit a Little Deeper into Papua New Guinea | p. 221 |

The Good, the Bad, and the Ugly | p. 227 |

Dig a Little Deeper | p. 233 |

There Are More Questions than Answers | p. 237 |

Conclusions | p. 244 |

Madoff and Other Ponzi Schemes | p. 247 |

The Madoff Claims | p. 248 |

Don't Bank on Kaupthing | p. 249 |

The Whole Ball of Waxenberg | p. 252 |

General Motors Demoted to Private | p. 253 |

Chrysler Unable to Dodge Bankruptcy | p. 255 |

Discussion of the Claims Results | p. 258 |

A Review of the Madoff Returns | p. 259 |

Apple, Dell, Berkshire, and Benford | p. 262 |

Discussion of the Returns Results | p. 265 |

Earth Science and Income Tax Applications | p. 267 |

Still Waters Run Deep | p. 268 |

The Lay of the Lake | p. 274 |

For a Few Dollars Less | p. 281 |

A Clean Bill of Clinton | p. 286 |

Conclusions | p. 290 |

Future Directions and Conclusions | p. 293 |

My Law | p. 295 |

Insights into Number Invention | p. 300 |

Lehman's Charitable Gifts | p. 306 |

The Bottom Line | p. 310 |

Glossary of Selected Terms | p. 315 |

References | p. 323 |

Index | p. 327 |

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