There is a huge political battle raging to Wisconsin, but you probably know this already. The drive to recall Governor Scott Walker has gotten plenty of media attention. Some 540,000 signatures are needed, and the Democrats turned in well over one million signatures to be verified. Walker supporters are hard at work trying to identify false signatures, to get as many petition thrown out as they can. That means about half of all signatures could be bad and the Democrats would still have enough to force a recall election. Given this rather daunting situation, how hard should Walker's supporters try? Do they really need to check 500,000 signature, a million signatures? How many is enough to be confident of a reliable result?
Regardless of your opinion about Scott Walker and the recall, some simple statistical sampling can help answer the question, and it requires checking far fewer than 500,000 signatures. I'm going to assume that each petition form contains 20 signatures, and 50,000 petition forms to total one million signatures. Essential to this process is a simple random sample**, where we can select a sample of petitions so that each form has an equal chance of being selected. There are fancier schemes, but this is the easiest way get an unbiased sample, and for me to explain.
Starting with a generous assumption that half of all signatures are fake, and that this number varies with a standard deviation of 2.6 bad signatures per petition, meaning most petitions have between 5 and 15 bad signatures (also generous). For a sample of size n petitions that gives a total count of x bad signatures the formula for the percentage bad is x/(20 * n) [x divided by (20 times n)] with a standard error of 2.6/sqrt(n) [that's 2.6 divided by the square-root of n]. Based on the mathematical law of statistical averages from a random sample, we can say that the actual number of bad signatures is close to x/(20 * n), where close means it is within about 2 standard errors in 95% of all such samples performed in this way. For a sample of size n = 1000 petitions, our assumptions and statistics say we should observe 50% plus or minus about 3.2%, or between 46.8% and 53.2% bad signatures - IF the assumptions are correct. We can say there is a confidence level of about 95% (19 times out of 20) that the interval generated this way will capture the true rate of false signatures.
Now the good news: If the actual percentage is more or less than 50%, the standard error should be a little smaller either way, meaning the estimate will be a little more accurate. More good news: this setting is what statisticians call a "finite population sample", which means this estimate will be a little more accurate yet, because the recount is sampling a significant fraction of the total population.
Long story short, if you want to verify recall petitions, take a sample of about 1000 petitions, check them carefully, and calculate the percentage of bad signatures. If that percentage is less than 45% or so, then it is time to stop counting and start campaigning.
** In practice random samples are not always "simple", but this is what statisticians call it.
Regardless of your opinion about Scott Walker and the recall, some simple statistical sampling can help answer the question, and it requires checking far fewer than 500,000 signatures. I'm going to assume that each petition form contains 20 signatures, and 50,000 petition forms to total one million signatures. Essential to this process is a simple random sample**, where we can select a sample of petitions so that each form has an equal chance of being selected. There are fancier schemes, but this is the easiest way get an unbiased sample, and for me to explain.
Starting with a generous assumption that half of all signatures are fake, and that this number varies with a standard deviation of 2.6 bad signatures per petition, meaning most petitions have between 5 and 15 bad signatures (also generous). For a sample of size n petitions that gives a total count of x bad signatures the formula for the percentage bad is x/(20 * n) [x divided by (20 times n)] with a standard error of 2.6/sqrt(n) [that's 2.6 divided by the square-root of n]. Based on the mathematical law of statistical averages from a random sample, we can say that the actual number of bad signatures is close to x/(20 * n), where close means it is within about 2 standard errors in 95% of all such samples performed in this way. For a sample of size n = 1000 petitions, our assumptions and statistics say we should observe 50% plus or minus about 3.2%, or between 46.8% and 53.2% bad signatures - IF the assumptions are correct. We can say there is a confidence level of about 95% (19 times out of 20) that the interval generated this way will capture the true rate of false signatures.
Now the good news: If the actual percentage is more or less than 50%, the standard error should be a little smaller either way, meaning the estimate will be a little more accurate. More good news: this setting is what statisticians call a "finite population sample", which means this estimate will be a little more accurate yet, because the recount is sampling a significant fraction of the total population.
Long story short, if you want to verify recall petitions, take a sample of about 1000 petitions, check them carefully, and calculate the percentage of bad signatures. If that percentage is less than 45% or so, then it is time to stop counting and start campaigning.
** In practice random samples are not always "simple", but this is what statisticians call it.
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