
1 
Paper

Statistics for Digits
Mebane, Walter

Uploaded 
07172007

Keywords 
election forensics 2BL test Benford's Law vote counts outliers anomalies election fraud

Abstract 
I show how election results may be used to calibrate a test that compares the second digits of a set of precinctlevel vote counts to the frequencies expected according to Benford's law. For the votes cast for two competing candidates, the calibration is accomplished by tuning a simulation mechanism that mixes normal and negative binomial distributions so that the first two moments of the simulated distribution match the moments observed in a set of precincts. I illustrate the method using data from the counties that had the ten largest values of the digit test statistic for the major party candidates in the 2000 and 2004 U.S. presidential election. Calibration suggests that the peculiar features of the joint distribution of candidate support and precinct sizes explain several of the large test statistic values. I show that artificial manipulations can significantly increase the test statistic's value even relative to the increased distribution the tuned mechanism is producing. So the test can sometimes detect systematic distortions in vote counts even when the baseline mechanism does not produce counts that have digits that are distributed as specified by Benford's law. 

2 
Paper

Election Fraud or Strategic Voting? Can Seconddigit Tests Tell the Difference?
Mebane, Walter

Uploaded 
07062010

Keywords 
election fraud strategic voting gerrymander Benford's Law 2BL American elections turnout presidential House state House state Senate

Abstract 
I simulate a mixture process that generates individual preferences that, when aggregated into precincts, have counts whose second significant digits approximately satisfy Benford's Law. By deriving sincere, strategic, gerrymandered and coerced votes from these preferences under a plurality voting rule, I find that tests based on the second digits of the precinct counts are sensitive to differences in how the counts are derived. The tests can sometimes distinguish coercion from strategic voting and gerrymanders. The tests may be able to distinguish strategic voting according to a party balancing logic from strategic voting due purely to wastedvote logic, and strategic from nonstrategic voting. These simulation findings are supported by data from federal and state elections in the United States during the 1980s and 2000s. 

