A novel Benford-based approach for detecting statistical anomalies in precinct-level election data

We introduce a novel extension of Benford’s law that shifts attention from leading digits to the joint distribution of final digit pairs in numerical data. Traditional Benford-based tests rely on first-digit frequencies and require observations to span several orders of magnitude, a condition frequently violated in precinct-level election returns. To overcome this limitation, we develop a formal probabilistic framework for analyzing the behaviour of the last two digits under Benfordian assumptions, which we denote L2-BL 10. Within this framework, we derive closed-form expressions for digit-pair probabilities and establish convergence bounds toward their limiting uniform distribution, providing new theoretical results that extend the statistical foundations of Benford analysis beyond its classical first-digit form. We then show that precinct-level vote counts are well approximated by a discrete Weibull distribution, linking this theoretical development to realistic data-generating processes encountered in elections. Monte Carlo simulations and empirical illustrations using data from nine states in the 2016 and 2020 US presidential elections demonstrate how the L2-BL 10 framework can be operationalized as a practical diagnostic for identifying anomalous digit behaviour. The result is a method that is both theoretically grounded and directly applicable to election forensics in settings where standard Benford tests are not appropriate.