10. Second Digits and Beyond — The Fuller Law

The Expansion of Statistical Resolution

The examination of numerical distributions is not confined to the leading digit alone, as the underlying logic of the law extends to every subsequent decimal position. Walter R. Mebane, Jr. demonstrated in 2006 that the second-digit test, known as the 2BL test, provides an additional layer of scrutiny for analyzing data sets. By applying this method to precinct vote counts, researchers can identify patterns that might otherwise remain obscured when observing only the primary digit of a number. However, this increased granularity does not come without a cost, as it introduces a higher frequency of false positives that require careful and nuanced interpretation. In his 2006 study, Mebane noted that the 2BL test frequently flagged jurisdictions that were actually experiencing normal political phenomena rather than genuine instances of electoral fraud. This illustrates the fundamental tension between achieving higher statistical resolution and maintaining the practical utility of an analytical tool in real-world environments.

Comparing the Power of Multi-Digit Tests

Dieter W. Joenssen conducted a comprehensive Monte Carlo comparison in 2013 to evaluate how effectively various tests perform across a range of different sample sizes. His research measured performance across sample sizes from 25 to 10,000, utilizing 1,000,000 replications per cell to ensure the robustness of the statistical conclusions drawn. The testing of the joint first-two-digit distribution gained between 1 and 3 percentage points of power with the Kolmogorov-Smirnov and Freedman-Watson tests against alternative distributions. Despite these gains, the method is not universally superior, as Joenssen found that for the chi-squared test, one-digit testing remained favored at larger sample sizes. The complexity of selecting the correct test-and-digit-count combination means that analysts must be wary of assuming that more digits will always yield a clearer result. Ultimately, the trade-off between statistical power and the risk of misidentifying innocent data remains a central challenge for those applying these sophisticated numerical tests.

The Consensus on Complexity

There is no serious dispute among experts that incorporating more digits into an analysis adds information that can be valuable for detecting specific anomalies. The literature currently disagrees on which specific test-and-digit-count combination performs best across the diverse variety of real-world scenarios encountered by investigators. This lack of uniformity suggests that the selection of a testing protocol must be tailored to the specific nature and size of the dataset being examined.

Example: Relying on the second digit to confirm a finding is like using a magnifying glass to check a signature that already appeared blurry to the naked eye.

Verdict. Adding digits increases the sensitivity of the test but necessitates a higher threshold for evidence to avoid convicting innocent parties of fraudulent behavior.

Overturned if: A replication of the 2013 Joenssen study demonstrates that the chi-squared test consistently outperforms multi-digit tests regardless of the sample size provided.

Sources. 2006 · Walter R. Mebane, Jr., "Election Forensics: The Second-digit Benford's Law Test and Recent American Presidential Elections" — PDF · 2013 · Dieter W. Joenssen, "Two Digit Testing for Benford's Law" — PDF