Ballot errors are regrettable, and they should be minimized. But ballot errors are much less worrisome when they affect all candidates roughly in proportion to their support in the voting population. If this happens, the relative positions of the candidates in the voting are unchanged. Clearly this was not true for Buchanan who benefitted from these mistakes, but Buchanan was not a contender in Florida, so the additional vote he received did not change his circumstances. Al Gore and George Bush, however, are both contenders for Florida's electoral votes and relatively small numbers of voters could affect their final status. Thus, it is very important to establish whether those voters who mistakenly voted for Buchanan were Bush or Gore supporters. Whether the mistakes were made by Bush or Gore supporters can be answered by considering county data from all Florida counties other than Palm Beach County and precinct level data from Palm Beach, Broward, and Miami-Dade counties. The basic argument is that if the "extra" Buchanan voters are Bush supporters who made a mistake, then we would expect a quite different pattern in the data than what we would get if the "extra" Buchanan voters are Gore supporters. This argument was first elaborated in my paper "What Happened in Palm Beach County?" which was posted on the web on Thursday, November 9, 2000. This paper makes some simple assumptions and uses standard statistical methods to conclude that at least 2000 of the Buchanan votes in Palm Beach County were cast by Gore supporters who mistakenly punched Buchanan's name.
The logic of the analysis that concludes it was Gore and not Bush voters who made mistakes can be explained through a hypothetical example. We begin with Figure 3 which depicts what happened in the rest of the state of Florida outside of Palm Beach County by considering 10,000 voters. For simplicity, the example puts aside all third party candidates except Buchanan. In the hypothetical example, Bush and Gore obtain the same number of votes, 4,990. Buchanan obtains 20 votes or .2% of the total.
Assume that we can split this hypothetical state into three equal-sized precincts--one with 33% Bush support, another with 50% Bush support, and a third with 67% Bush support. Because Buchanan was once a Republican and because he still appeals to Republicans, we find that he does better in precincts with more Bush voters. In our example, he gets about four votes in the precinct with only 33% Bush supporters, 7 in the precinct with 50% Bush supporters, and 9 in the precinct with 67% Bush supporters. Thus, Buchanan's support goes up as the percentage of Bush supporters goes up.
This is exactly the pattern we observe in scatterplots of Buchanan vote percentage versus Bush vote percentage for each precinct in Broward (Figure 4) and Miami-Dade (Figure 5) counties. Scatterplots like these are used by all statisticians because of their utility and simplicity. Indeed, plotting scatterplots like this is now a standard grade school exercise. In Figures 4 and 5, Buchanan support in Broward and Miami-Dade counties increases as we go from precincts with lower to higher Bush support. The same relationship also exists across the counties in the state of Florida. Statistical tests decisively confirm that the lines in Figure 4 and 5 slope upwards.
The lines that are drawn are Ordinary Least Squares's (OLS) fits to the data. OLS is, perhaps, the most commonly used statistical method, and it is the workhorse of statistical analysis. It was invented almost 200 years ago for solving problems in astronomy and physics where scientists wanted to describe the line that best described a scatterplot of points. This method is now widely used in all of the sciences. OLS determines a straight line through a set of points by finding that line which minimizes the sum of the square of the "residuals" which are defined as the difference between the line and the observed data. Figure 6 shows what would happen if all of the mistakes came from people who would otherwise be Bush supporters. That is, it shows what would happen if the ballot form had caused Bush supporters to mistakenly vote for Buchanan. In this example, I assume that the chance for Bush supporters to make mistakes is the same across all Bush supporters. This assumption seems valid because the mistake is due to the ballot form. Therefore, it is the characteristics of the form and not the characteristics of Bush supporters that lead to errors. As shown in Figure 6, if it is Bush supporters making mistakes, then a scatterplot of Buchanan voters in each precinct against the level of Bush support would still be upwardly sloped.
Figure 7 shows what would happen if all the mistakes came from people who would otherwise be Gore supporters. That is, it shows what would happen if the ballot form had caused Gore supporters to mistakenly vote for Buchanan. In this case, precincts with more Bush supporters obviously have fewer Gore supporters who are available to make mistakes. Now, if the rate at which Gore supporters make mistakes is high enough, then the "true" Buchanan supporters who vote for Buchanan (shown in Figure 3) are swamped by "mistaken" Buchanan voters who come from the ranks of Gore supporters. The result is a line that slopes downwards.
In effect, each precinct is made up of some true Buchanan supporters and some mistaken Buchanan voters who come from among Gore supporters. The first group gets bigger with Bush support. The second group gets smaller. If there are enough Gore supporters who make mistakes, then they cause the slope of the line to go down. Thus, Figure 6 which assumes that Bush supporters have made errors and Figure 7 which assumes that Gore supporters have made errors are strikingly different. In Figure 6 the slope of the line is positive - it goes up. In Figure 7, the slope of the line is negative - it goes down. These results provide a clear-cut way to distinguish between a situation where it is all Bush or all Gore voters making mistakes.
Figure 8 shows what happens when we look at the relationship of Buchanan vote to Bush vote in Palm Beach County. Whereas these lines slope upwards in scatterplots for Broward and Miami-Dade counties, the line for Palm Beach County slopes downwards. Moreover, the slope is highly significantly different from zero.
These graphs show, with a very high degree of scientific certainty, that there are significant numbers of Gore voters who mistakenly voted for Buchanan. Using the method in my paper cited above along with multiple sets of data, it is possible to get highly refined estimates of the fraction of Gore supporters who mistakenly voted for Buchanan and the fraction of Bush supporters who mistakenly voted for Buchanan.
My estimates of these models leads me to say with a very high degree of scientific certainty that essentially none (at most perhaps a handful) of the Bush supporters mistakenly supported Buchanan while at least 2000 Gore supporters mistakenly supported Buchanan.