Modern Benford’s Law

Daniel J. McCarville is a recognized audit analytics professional, researcher, and speaker with more than a decade of experience developing advanced tools for auditors and compliance teams. He currently leads the audit analytics team at an S&P 500 insurance firm. His statistical expertise spans Benford’s Law, machine learning, and simulation, which he applies to detect anomalies and strengthen decision-making.His peer-reviewed research has introduced novel applications of Benford’s Law and new methods for identifying irregularities in standardized test data. He has published in Emerald Open Research and received the National Legislative Program Evaluation Society’s Impact Award multiple times.Daniel has presented at major professional conferences, including IIA events, and frequently speaks at universities to inspire the next generation of data professionals. He holds a master’s degree in political science from the University of Kansas, as well as Data Science from Eastern University. He also holds CIA and CRMA designations from the Institute of Internal Auditors and DASM certification from the Project Management Institute.

• Preface. Fundamentals: Non-Statistical Approaches to Benford's Law. 1.0 The Story of Benford's Law. 1.1 Setting Expectations. 1.2 What Does it Mean to "Pass" or "Fail" Benford's Law?. 2.0 The Basics of Benford's Law. 2.1 The First Significant Digit. 2.2 Benford's Law. 2.3 Testing Data. 2.4 Exceptions to Benford's Law. 3.0 Common Results - with Graphs. 3.1 The Classic Example: Avoiding a Limit. 3.2 Missing Data. 3.3 Spikes & Potential Duplicate Transactions. 3.4 Imprecise or Estimated Data. 3.5 Data with Limits. 3.6 Not Enough Data. 4.0 Intermediate Topics. 4.1 Two-Digit Test. 4.2 Example Application: Greenhouse Gas Emissions. 4.3 Second Order Test. 4.4 Example Application: Fleet Mileage. 4.5 Changing Scales. 4.6 Data Transformation. 4.7 Example Application: Accounts Payable Amounts. 4.8 Generating Random Benford's Law-compliant Data. Advanced Techniques: Statistical Approaches to Benford's Law. 5.0 Goodness-of-Fit Scores. 5.1 The D-(Distance) Score. 5.2 MAD (Mean Absolute Deviation). 5.3 Honorable Mentions: Chi-Squared and Other Scores from the Math Department. 5.4 Last Place: The Z-Score. 6.0 Benford-like Distribution. 6.1 The Generalized Benford's Law. 6.2 Example Application: Inventory Adjustments. 7.0 Finding High-Risk Groups. 7.1 Approach to Identifying High Risk Groups. 7.2 Example Application: FinCEN Files. 8.0 Time Series Applications. 8.1 Binned Data. 8.2 Example Application: Journal Entries. 8.3 Windowed Data. 8.4 Example Application: Law Enforcement Gun Usage. 9.0 Benford's Law and Machine Learning. 9.1 Data Quality Monitoring. 9.2 Benford's Law as a Feature. 9.3 Benford's Law as an Outcome Variable. 9.4 Benford's Law for Evaluating ML/AI Models. 10.0 Beyond Base 10. 10.1 The Benford's Law Formula in Other Bases. 10.2 The Concept of Base Invariance. 10.3 Example Application - Determining the Base of a Dataset. 10.1 Example of Base Invariance. Wrapping Up. 11.0 Professional Practice and Benford's Law. 11.1 Benford's Law Throughout the Audit Process. 11.2 Professional Standards. 11.3 Special Notes on Fraud. 12.0 Case Study: IT Incident Management. 12.1 Audit Background. 12.2 Data Preparation. 12.3 Implementing Benford's Law. 12.4 Analysis and Results. 12.5 Wrapping Up. 13.0 Advanced Case Study. 13.1 Audit Background. 13.2 Optimizing Benford's Law. 13.3 Benford's Law in Power BI. 13.4 Finding High Risk Groups. 13.5 Time Series & Continuous Monitoring. 13.6 Wrapping Up. Appendices: Deriving Benford's Law. A.1 Algebraic Approach. A.2 Visual Approach.