08. The Election Error — Where the Law Was Abused

The Seduction of Simple Statistics

Benford’s Law describes how the leading digits of datasets spanning several orders of magnitude often follow a specific, predictable logarithmic distribution. In 2020, this mathematical curiosity was repurposed as a tool to allege widespread electoral fraud within the United States precinct-level vote counts. Walter R. Mebane, Jr., a prominent expert in election forensics, investigated these claims by applying his own rigorous statistical models to specific jurisdictions. He examined the 2020 Milwaukee and Allegheny County precinct-level vote data to see if the alleged anomalies held any real mathematical weight. His analysis yielded an estimated probability of "no fraud" equal to .9797 for Milwaukee and .9839 for Allegheny County, refuting the fraud narrative. The application of the law had become a slogan used to manufacture a false sense of certainty for an audience unfamiliar with its limits.

The Failure of the First-Digit Test

The primary error in these election-related claims lies in the naive application of first-digit tests to data that does not meet the necessary criteria. Theodore P. Hill explained in 2020 that there is a widespread error in the use of Benford’s Law when the data lacks sufficient orders of magnitude. Mebane clarified that the first digit in precinct-level vote counts is largely determined by the number of voters registered or present in each distinct precinct. This structural feature of election data has nothing whatsoever to do with any kind of election fraud, despite what the popular, simplified narratives suggested. Because the precinct sizes are restricted by geography and population density, they fail to span the wide numerical range required for the law to apply. Mebane’s own 2020 research found essentially no evidence that election frauds occurred in the specific jurisdictions he analyzed using his forensic models.

A Dispute of Methodology

The debate surrounding these findings is not a clash between mainstream science and fringe critics, but rather a rejection by the field's own methodologists. Walter R. Mebane, Jr. explicitly contested the validity of the naive first-digit applications that were being circulated to characterize the 2020 election results. Theodore P. Hill similarly contested the statistical soundness of applying these tests to election data that lacks the required orders of magnitude for validity.

Example: Using a ruler meant for inches to measure the weight of an object and claiming the resulting error proves the object is missing mass.

Verdict. The application of first-digit Benford tests to precinct-level vote counts is scientifically invalid and fails to provide evidence of electoral manipulation.

Overturned if: A study demonstrates that precinct-level vote counts naturally span multiple orders of magnitude across all jurisdictions, satisfying the fundamental requirement for Benford's Law.

Sources. 2020 · Walter R. Mebane, Jr., "Inappropriate Applications of Benford's Law Regularities to Some Data from the 2020 Presidential Election in the United States" — PDF · 2020 · Theodore P. Hill, "A Widespread Error in the Use of Benford's Law to Detect Election and Other Fraud" — arXiv:2011.13015