Statistics for Digits
I show how election results may be used to calibrate a test that compares the second digits of a set of precinct-level 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.
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