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On the standards to use when hand counting ballots

In this article, I advance a probabilistic argument that the goal of accuracy is best achieved by using a lenient standard when hand counting ballots.
I argue that the basic criteria in judging a vote counting mechanism is that it correctly chooses the winner, in the sense of reaching the same conclusion that would be reached if all the voter's intentions were perfectly measured.

Mathematically, the goal is to get the size (actually, just the sign) of the "differential" correct -- the value of VOTES_GORE - VOTES_BUSH.

In particular, this criteria dictates that purely random errors are much less important than non-random errors. Purely random errors will, on average, cancel out and have 0 effect on the differential. Non-random errors, in contrast, will not cancel out -- they will have an impact on the differential.

So, consider how one should interpret an "undercount" for which there is some doubt. Depending on whether one uses a stringent or an inclusive standard, one will make one of two errors:

  1. stringent standard -- you will toss out votes that were "sloppily" made
  2. inclusive standard -- you will include votes that were "accidental', that were not meant to be cast

Case a is an example of a non-random error. Assuming that these sloppy voters were distributed the same as the population of voters in the underlying precinct, then the net effect of tossing them out will be to void more Gore then Bush votes (or more bush then gore votes, depending on the precinct).

Case b is an example of a random error. Assuming that these "accidental" votes are randomly cast (i.e.; when one drops a stylus, it can land anywhere), then they will just as likely be for gore as bush, hence will cancel out.

Thus, adoption of the stringent standard will tend to have a greater negative impact on the "correctly measure the differential" criteria. Hence, the inclusive standard should be adopted.

There is a 2nd order concern -- each recount will tend to yield a different number. So, there is a random chance that a loser will be measured as a winner, should you recount enough times. In particular, the net effect of including even randomly distributed "accidental votes" may not be 0.

However, this chance can be measured empirically:

  1. assume 50% chance of an accidenal vote being counted as gore, 50% chance for Bush
  2. This yields a binomial distribution with mean=n*0.5 and a variance of n*0.5*0.5 (where n is the number of accidental votes counted)
  3. the standard error of the mean will be sqrt(n*0.5*0.5)
  4. the 95% confidence interval (CI) of the mean will be approximately 2*sqrt(n*0.5*0.5)
Suppose that n=1000. Then, the 95% CI will be 32.

In other words, 95% of the time the impact on the differential will be + or - 32. In contrast, if the gore vote was 52%, then tossing out 1000 "sloppy" votes will yield an average loss of 40 votes.

In other words, it is highly unlikely that the error from incorrectly counting accidental votes (i.e.; 32 vote) will be greater then the error from neglecting to count sloppy votes (i.e.; 40 votes).

[note: to be precise, I should compute the CI of the 40 votes, and compute an overlap of the confidence intervals.]

Conclusion: if accuracy is your goal, you should err on the side of counting a ballot as a vote.

First written in November 2000. Minor editing, March 2004.