Last updated Jan. 5, 2001.
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We seek to estimate the probability that a manual recount of one or several counties, or of the entire state, could have produced a change in the outcome of the election to choose Electors from Florida for the election of a US President in 2000. We have attempted to use the most accurate data to be found as of the revision date above.
We use several models, which make different assumptions about the nature of the changes resulting from a recount. None of them suggest that either candidate could expect to have more than a couple thousand vote lead over the other, so that the race is arguably a statistical dead heat -- certainly denying both candidates the surety that the race was unambiguously won.
In our most detailed model, we find that a recount of the "undervotes" from all 63 counties which have not already had a manual review, carried out with a uniform and somewhat permissive standard, would likely produce a net gain for Bush of up to about six hundred votes. We find the recounts completed in Volusia and Broward counties to be consistent with these standards, and the partially-completed but legally-rejected recount in Dade county to be nearly so. We predict that a recount of the entirety of Dade county would net Gore a couple hundred votes more than the roughly 160 votes already netted by him in the partial recount.
Thus the outcome of the recounting depends on the fate of Palm Beach County. If no recount were allowed there, Bush would win by about 500-700 votes statewide. If the completed but rejected recount were included, as suggested by the Florida State Supreme Court, we still predict a Bush win (by just a few hundred votes). However, we note that the results of the recount which was already conducted fall significantly short of what this same model predicts for the county, giving credence to Democratic claims that the canvassing board failed to count many hundreds of ballots which would have been accepted in other counties. Our best estimate is that a fresh review in that county, conducted according to the same standards as used in other counties, would find nearly a thousand more votes for Gore than Bush. Added to the total for the rest of the state, this would give Gore a lead of several hundred votes.
We note that these estimates are dwarfed by the effects of other election irregularities. Overvotes were much more common in Gore-leaning areas, led again by Palm Beach county, with statewide totals suggesting that these cost Gore perhaps ten thousand more votes than they cost Bush. Likewise ballot irregularities in Palm Beach are suspected of giving Buchanan votes not intended for him, again costing Gore more heavily by perhaps another thousand votes. It is unclear to what extent any of these could be recaptured in a recount; our point is merely to illustrate that both the official lead and any lead resulting from a recount of undervotes only will be very slim compared with other effects of comparable importance.
The thrust of our argument is that a manual recount of votes in Dade County or other counties can be expected to add votes favoring the candidate already leading there (Gore). A statewide recount would nominally be expected to add more votes for both candidates in proportion to their official vote counts statewide (about 48.8% each), but there are variations from county to county which account for the use of voting methods of differing levels of accuracy. In addition, purely random effects could add dissimilar amounts of votes to the two candidates; there is a range of values which would be considered unremarkable. Since the race was very close and, in the view of one side, the counting was incomplete, it is appropriate to ask whether or not there is a reasonable probability that further counting would produce a change in the net lead of one candidate over the other of a magnitude sufficient to change the outcome of the contest. We develop several different models of the voting process to attempt to answer this question.
We present first a simple model (sect. II, III): is the race close enough that we could expect random counting errors alone to turn the race? We estimate that it would be highly unlikely for a lead of 537 votes to be the result of an unbiased, random error of a realistic magnitude. (One model gives odds of about 70 to 1. The odds were much better when the Bush lead was under 300 votes, and when the lead was over 900 votes, this model showed an overturn to be essentially impossible.)
Next (sect. IV and V) we consider the possibility of making the model more realistic by clarifying the actual mechanisms by which errors are introduced. We outline the many refinements which would be necessary. These include many parameters which we have found it difficult to measure (sect. VI). Reasonable choices of these parameters increase the likelihood that an overturn can occur; without more accurate data we cannot assess whether an overturn is more likely or less likely than the preservation of the Bush lead.
Certainly it is true that a random but biased error, of reasonable magnitude, computed only in several heavily-Democratic counties together, could turn the balance. It is difficult to estimate those error rates with any precision, however, making the precise outcome of a Dade County recount unclear. We estimate in section VII that a recount there would yield Gore a net gain of several hundred net votes, if the recounts are ruled to be legal inclusions in the official vote counts, and if the ballots are are accepted as expressing the "will of the voter" with criteria similar to those previously used.
Some preliminary data by precincts have been obtained by major news media; we consider the use of these data in sect. VIII. They indicate that, for reasons not entirely clear, in some large counties the errors likely to be corrected by recounts tend to accumulate in precincts with stronger Democratic support than the rest of a county. Whenever true, this will of course tend to move the county's contribution to a recount towards Gore.
Finally, we attempt to extend the model to the whole state (sect. IX). In view of the fact that Gore's support was concentrated in the large counties most likely to have "lost" votes, a statewide recount could be expected to favor him. However, if we examine separately the three largest counties and then use our statewide model only for the other 64, we find those three counties account for the whole of the Gore advantage; indeed a recount in all the other counties would probably favor Bush slightly in those counties. A few (mostly political) variables can have a deciding effect at this small scale, however, making the final conclusion too close to call. (Compare sect. X).
Here is our best reading of the situation: if a statewide recount were conducted, using a consistent standard for accepting marginally-marked ballots as valid, which made a reasonable effort to determine the intent of each voter who cast a ballot, then we would expect Gore to net enough votes to manage a very slim lead over Bush. The lead would probably grow to the thousands if it should be possible to make a statistical correction for ballots spoiled in certain predictable patterns. On the other hand, a recount which holds to very tight standards regarding voter intent would make a Gore lead less likely. And we find a Gore lead to be improbable if a moderately permissive recount standard were applied statewide except, as decreed by the Florida State Supreme Court, that there be no revisit of the Palm Beach County recount: it is only that county which provides Gore the most realistic cache of as-yet-unrecovered ballots (unless some overvotes are also counted).
Judge Sauls ruled on Dec. 6 that the Democrats' lawyers failed to show there was a reasonable probability that the results of the requested recounts in Dade and Palm Beach could affect the outcome of the election. Whether the plaintiffs proved the point is a legal question we are not qualified to address. Whether such a limited recount is appropriate is an ethical question we choose not to consider. Whether a recount would likely change the outcome is a statistical question which we discover cannot be answered with any certainty. However, we note the judge's ruling stands at odds with our conclusion. From our computations, it seems a hand recount in Dade county, combined with other actions requested by the Gore camp, had a reasonable likelihood to have changed the outcome of the count. The concerns of the U.S. Supreme Court are different: from our calculations, a uniform re-reading of all ballots statewide would give Gore and Bush roughly equally good claims on the plurality of the votes; the statistical expectations alone slightly favor Bush but give neither side an a priori claim that they have a certain lead.
Naturally the political decision about whether or not to proceed with such a recount is then a subjective one: given these odds that the effort could indicate an error in the selection of President of the United States, is it appropriate to invest the necessary time and money in a statewide manual recount? Given the ambiguities inherent in the measurement process, how thin a margin of victory would be considered sufficient for the general public to accept the results? These questions are beyond the reach of mathematics and this paper.
Not to discount the tremendous political cost of recounting the ballots, it is nonetheless very peculiar from a scientific perspective that the sort of analysis we are performing should be necessary. Objectively speaking, we have approximately 6 million pieces of data whose information content we wish to determine -- for example, counting the numbers of ballots marked for Bush, Gore, Other, None, or Unclear. A statistical model is appropriate when forecasting an election, for example, or when the data we wish to consider are very distant in time or space. Why it should be necessary to estimate these numbers when the actual datasets are already available in Florida is, from a scientific point of view, incomprehensible. Indeed, under Florida's liberal freedom-of-information statutes, it is only a matter of time before each of the six million ballots is scanned and made available for inspection on the Internet, at which time all the analysis of this document will become moot.
Disclaimer: This report was prepared by a crack research team consisting of a large number of highly-paid, full-time investigators. Yeah, right -- I wish. I am a mathematician, not a political scientist, nor a politician, nor a resident of Florida. --djr
The election for U.S. President on Nov. 7, 2000 was very close nationwide, both in terms of popular and electoral vote. After several other states completed close races, the selection of President depended on the assignment of Florida's electoral votes. The Florida race was also extremely close; exactly was what the outcome of the Florida vote is now the subject of debate. Several hours after the polls closed, a snapshot of the count showed GOP candidate George Bush with an 1,831-vote lead over Democrat Al Gore, with 2,891,922 and 2,890,091 votes respectively. Many counts have since been reported, including several with some official status. All of them since late Nov. 7 have shown Bush in the lead; all of them have reported this lead to be less than 2,000 votes. The Democrats have claimed that, in fact, the true vote count favors Gore, and that only a set of errors gave Bush an apparent lead, and indeed nearly every effort to review the ballots recovered more votes for Gore than for Bush. This prompted us to consider the closeness of the race from a mathematical perspective. A short version of this paper was circulated beginning Nov. 12; we have attempted to keep the details up to date and to extend our analyses to cover new features of the vote count as they arose. (Timeline)
We will not itemize the many adjustments to the vote count since Election day. Briefly, state law mandated an automatic recount statewide. At candidates' requests, additional machine recounts were performed in Palm Beach and Miami-Dade counties. The Gore camp requested hand recounts in several counties; of those requests which were accepted, only a count in Volusia county was completed before the Nov. 15 filing deadline, and only a count in Broward county was then completed by an extended deadline Nov. 26. A recount in Palm Beach county was nearly completed but not accepted by the state. (It was completed later, but called into question by both parties, for opposite reasons.) A recount was partially completed, then abandoned in Dade county. These recounts are a primary focus of this report. Other adjustments to the vote totals included the final tabulation of overseas absentee ballots on Nov 17, a number of exploratory (sample) hand recounts, and many corrections of clerical errors or reconsiderations of ballots. On Nov 26 an official total was certified by the state but some of these numbers were contested in court. A Dec 8 split court decision mandated recounts of all ballots statewide which had not previously been tallied as a vote for a presidential candidate. A Dec 9 stay by the US Supreme Court froze this recount and the definitive ruling Dec 12 disallowed it (these also being split decisions). Gore conceded the election Dec 13, and on Dec 19 the Electoral College made official the selection of Bush as president-elect. At about the same time several news and political organizations began a review of some of the problematic ballots in several counties; this is ongoing.
The final statewide official vote totals are Bush: 2,912,790; Gore: 2,912,253; Others: 138,067; for a total of 5,963,110. In addition, a number of ballots were disqualified for the Presidential race; our data on these are from Nov. 15, and so are not quite current, but at that time there were 180,110 disqualified ballots across the 67 counties. These numbers, giving Bush a 537-vote lead, include all adjustments listed above, but not the incomplete recounts in Palm Beach and Dade counties. Other reported sizes of the Bush lead range from 200 (CNN at 5am EST on Nov. 8) to 1,784 (official at 6am EST Nov. 8, just before the automatic recount), and include 784 (after the automatic recounts), 300 (after the Volusia hand recount), 930 (after the inclusion of overseas ballots), 537 (the lead certified by Secretary Harris, including the Broward hand recount), and 193 (adjusted total after the Dec. 8 decision, including the Palm Beach hand recount, itself adjusted Dec.9 to a 176-vote Gore net gain).
The critical question, of course, is what is the "true" count of the ballots? Given no other information, our best guess would be that the official numbers are the correct numbers of votes actually cast (that is, intended and actuated) by the voters. We know, however, that machine tabulation introduces a number of systematic errors -- instances in which the tabulated vote count differs from the numbers of votes cast. Therefore, we set ourselves the task of estimating the likelihood that these errors made a large contribution to the Bush lead, including the possibility that the direction of the lead is in fact entirely the result of these accumulated errors.
In particular, it has been suggested that the hand recounts begun but not included in these totals (in Palm Beach and Dade counties) have the potential to correct errors of sufficient magnitude to alter the lead. (The numbers of ballots for Bush, Gore, Other, Undervote (blank), and Overvote (spoiled) are here:)
Dade: 289533, 328808, 7108, 10750, 17851 Palm Beach: 152951, 269732, 10503, 10582, 19120One may similarly speculate whether careful recounts of the ballots in other counties could shed more light on the situation. We wish to assess how likely it is that such a recount could reverse the lead.
What is the fraction of votes which are not recorded correctly? The hand count of 4695 Palm Beach County ("PBC") votes on November 11 2000 found 49 more votes when assessed by sight than were counted by machine. While there are many sources of error in vote tabulation, the one most often mentioned in this race is the incomplete removal of chad; in this case the type of enumeration error would rather uniformly be the non-tabulation of an otherwise acceptable ballot -- that is, real votes are lost. We use this sample to estimate that there is an error rate of approximately 1% in the vote count . (We will argue later for a lower value.) It is important to observe that this phenomenon is most likely a systematic error, meaning it is a flaw in the system which is just as likely to affect votes in favor of either candidate. Therefore, it is in most elections not a significant concern: the net effect on an evenly-divided race is likely to be nil. However, the vote in this particular contest is very close and thus it is appropriate to measure just how likely is this "likelihood to be nil".
The next two sections of our analysis focus on this precise question: at what point is it appropriate to think that an election outcome might be solely the result of a random, unbiased, loss of votes on the order of 1%? Back to top
If six million voters determined their vote by flipping a fair coin, we would expect about half to vote for each candidate. But we wouldn't expect exactly half to vote for each; most of us would recognize that such an outcome is remarkable. It would also be remarkable if 90% or more coin flips came out "heads". We know these outcomes can occur, but we also know they should be rare. Well, what outcome would not be "remarkable"? 75% heads? Ten more than half the total? There must come a range of values around a perfect 50-50 split which we can recognize as being wide enough so that it wouldn't surprise us to get that close, but narrow enough so that we wouldn't be surprised to fall outside that range either. Statistically, we are describing here the difference between the mean (the three million expected votes for each side) and the standard deviation, a measure of how wide that "unremarkable" range is.
It turns out that a fair coin flipped N times will come up heads on exactly half those flips with probability roughly equal to: (4/5) divided by the square root of N. (Assuming N is even, of course.) From this one can show that the deviation from a perfect 50-50 split is about as likely as not to be within, roughly, (1/3)-times-the-square-root-of-N flips. We could take this number as a measurement of "remarkableness". If we want to predict the outcome of this "coin-flip election" we would include a margin of error which is a multiple of the square root of N. (Note that every single deviation from a perfect split actually gives a 2-point increase in the lead of one candidate.)
Here are final tabulations of some statewide votes in the US Presidential election of 2000, and a few other elections, showing the size of the lead L, the percentage of the vote this represents (L/N) and the "remarkableness" (L / square-root-of-N) (roughly, the number of standard deviations from the mean). The "remarkable" races are the ones for which this last measure is far from about 2/3, that is, IF the votes were obtained from coin tosses, we would be surprised if this index was very much less than 2/3 or very much more than 2/3. (Since we have not proposed that coin-tossing is a model for voting behaviour, these numbers may be taken "for entertainment purposes only" :-) )
|1974 NH Senate seat||10||0.005%||0.02|
|New Mexico (Nov14)||17||0.0034%||0.02|
|1984 IN House seat||39||0.017%||0.08|
|1948 TX primary||87||0.01%||0.09|
|New Mexico (final)||366||0.061%||0.47|
|1996 MA House primary||108||0.22%||0.49|
Races which are exactly tied at several hundred votes each are not particularly uncommon. The "largest" tie decided by lot of which we are aware is a 1999 local judge race in New Mexico, tied at 798 votes for each of Republican Jim Blanq and Democrat Lena Milligan. Other ties have been resolved by recounts and legal actions, including several "official" vote counts. An interesting example is Penny Pullen v. Rosemary Mulligan in the 1990 Republican primary for Illinois House which, after a recount, was determined to be a tie at 7387 votes each. Legal action centered on the validity of partially marked chads. Ties seem to be becoming increasingly common; at this writing the US Senate and four statehouses are tied too.
To repeat, it is both the very close and a very uneven outcomes which would be remarkable in a coin toss, so the early New Mexico and the Wyoming results would be the ones most strange to see in such an experiment.. Using a coin-toss model, the Ohio lead (for Bush) looks like a sure win, since it would be remarkable for a coin toss to make two even candidates show up so far apart (even though in ordinary terms Ohio was a close race, with the leader just shy of half the ballots.) By this same analysis, it would not be at all remarkable if this voting method gave one candidate a lead of 1784 votes (Bush's lead over Gore in Florida on Nov. 8). But it would be remarkable for these random coin tosses to keep the lead to within 300 votes -- the odds against this are about ten to one. (Similar conclusions also reported in NYTimes).
Well, no one is suggesting the coin-tossing is a good representation of the method people use to vote. Moreover, most people (other than mathematicians!) want to know the winner of the election, not the remarkableness of the result. To know the winner, we need the actual count of votes for each candidate. This count is affected by several types of variables:
Now, if coin-tossing is not a good model for the balloting, what is? We will show that a standard tool known as the "random walk" can model actual voting results in which the candidates are about equal but there is some unbiased error in tabulating the votes.
Let us first illustrate with an idealized situation which is easier to visualize. Suppose that there were 500,000 votes cast for the two candidates, and that the votes cast were evenly split between the two. One could imagine placing the votes onto a balance, at each moment showing the net amount by which Bush (say) is favored over Gore. The initial reading is zero; the final reading would also be zero in this illustration as the vote is a tie. Indeed, the intermediate vote counts are more likely to show a difference of zero than any other single difference. If indeed the vote is evenly split, it is even possible to arrange the voters deliberately so that the votes are even balanced periodically (say, after every 100 votes).
Now let us consider the effect of random error: assume about one vote out of each hundred -- with equal probability from each side -- will fail to be enumerated. In this case, after each 100 votes, we will NOT show an even balance of votes; rather, one side or the other -- affecting each side randomly -- will lose a vote and the net lead will shift one more vote towards the other side. (In a very small number of sets of 100 votes there can be two or more errors, which will not noticeably affect our analysis.)
We thus obtain a model in which every set of 100 votes can be viewed as a step randomly taken in either direction on the number line. The full set of 500,000 voters amounts to 5000 steps along the line. Thus we are modeling the process as a Random Walk.
It is possible that all 5000 steps would be taken in one direction, resulting in an apparent lead of 5000 votes for one side, despite that fact that the true numbers of votes for the two sides are equal. However, this is exceedingly unlikely. It is also possible that the 5000 steps would be exactly evenly split into errors favoring one side and errors favoring the other. This would result in a vote count which accurately reflects the even balance between the voters. However, it too is quite unlikely: only in about one such election out of ninety would the errors cancel each other exactly.
The odds are just slightly less in favor of a net gain of 2, 4, 6, ... for either party. In particular, the odds are about 50-50 that this process will result in a net lead of about 90 or less for one party; equally well, the odds are about even that one candidate will show a lead of about 90 or more. We might say that the initial data establish a 90-vote threshold: a hand recount performed to find the lost ballots is as likely to change the lead by less than 90 votes as it is to change the lead by more than 90 votes, under this model.
Now, there are other ways to model this situation; they produce somewhat different numerical conclusions. The key point is that it is highly probable that an evenly-split electorate of this size can, with unbiased tabulation errors, produce an apparent lead of several dozens of votes; and it is highly improbable that these errors will produce an apparent lead of several hundreds of votes.
Our numbers are only slightly changed when the initial vote isn't quite even. If one side actually has a lead of 100 votes, say, we could imagine these 100 voters placing their votes into the ballot box first, and then the others voting in groups of 100 as before. The vote lead would then start at 100 and take 4999 random steps from there. We conclude that the final position will be near 100 with a certain probability; the odds are about 50-50 in this example that the final resting position will be within the range of a 10- to 190-vote lead. So in this illustration, a tabulated lead of less than 0 (i.e. a win for the other side) requires an accumulation of errors in that direction which is just a little remarkable -- it can happen but with a probability a little under 50%.
Working backwards, we can see that if there are 500,000 votes and a random 1% error, and the tabulated vote is, say, 120 votes in favor of one of the candidates, then with high probability we can say that the starting position -- the actual lead of that candidate -- is somewhere between 30 and 210 votes. Other starting positions are possible, of course, but they are less likely when we move further from the nominal lead of 120 votes.
This situation holds fairly generally. In particular, it is always possible that the actual winner of the vote could be different from the nominal winner, if the initial lead is less than 1% of the total vote (that's the assumed error rate). Whether or not it should be considered a rare, or remarkable, event that the tabulated vote shows the wrong victor depends on whether or not the tabulated lead is much less than the "remarkable" threshold, something on the order of (2/3)x(square root of number of errors).
In particular, we may apply this to the Florida presidential vote. Assuming that: there is a 1% unbiased random chance that each vote will be lost (i.e. not counted), and that there are six million votes altogether, very nearly evenly split between the two candidates, then there will be sixty thousand votes lost, taken from one side or the other essentially by the flip of a coin. If the coin flips split exactly down the middle, the measured vote lead is accurate. If 30,000 + 268 coin flips come up heads and the other 30,000 - 268 come up tails, the lost votes will incorrectly move the lead by 536 votes in favor of one side. The probability that the number of flips will deviate from a 30,000:30,000 split by no more than 268 flips comes out to about 97.2%, leaving only about a 1.4% chance that it would favor a particular side by 269 or more flips. Thus a measured lead of 537 votes for Bush would mean, with high certainty, that the actual lead is between 537-536=1 and 537+536=1073 votes. The chance that Gore actually held the lead would, by this model, be only a 1.4% chance (about 70 to 1 against this happening). The corresponding odds were about four-to-one against Gore being the actual winner when the lead was around 200, as it was when we began this paper. They dropped to about 14,000-to-1 odds when the Bush lead was 930. In summary: it is highly unlikely, though not inconceivable, that the correct vote count would show a lead for Gore, under this model.
It is important to remember that this conclusion depends on the specific numbers ("6 million", "537", and "1%") and on the assumption that every vote is equally likely to be lost. Changes in the numbers are easy to incorporate into the model; changes in the assumption of purely random loss will require a more sophisticated model for the accumulation of errors in vote counts. We take up this question in the next three sections.
As a general rule, the observations of this section imply that when estimates are listed in following sections, we should not be surprised to find that the effect on the lead could vary by roughly the square root of the number of total ballots involved in the estimates. Back to top
There is considerable confusion about the nature of the "errors" assumed in the model above. In fact there are many types of errors, and their effects on the vote counts are different, so we ought to model them separately -- as best we are able. In this and the next two sections, we would like to itemize these errors and estimate the magnitude of their effects. We certainly do not claim this listing is complete, and welcome improvements (especially if quantifiable information is available!)
But first we need to decide what is the thing that the these errors are a departure from! (It is not clear that such a thing really exists [Essay].) Let us agree that a "vote" is the expression by one person of preference for presidential candidate. A "vote count" is a pair of processes conducted, in practice, by counties (or precincts) and tracked by the state and by news agencies: votes are segregated (conceptually or physically) into piles, and also enumerated by pile:
Next, observe that each vote passes through several stages. Not all votes need pass through all stages (e.g. some, not all, votes have been recounted twice by machine). Not all votes pass through the stages simultaneously (e.g. some 10% or more of the votes are filed as absentee votes and completed before Election Day; the final overseas absentee votes were only counted Nov. 17). Among the stages we might find useful to distinguish:
It probably bears repeating that the word "count" is used slightly differently by partisans in the media. By now, all the ballots have been enumerated: using headcounts by poll-watchers or the tabulations of the ballots, it is probably possible to state with great precision how many votes there were in stages 2 and later. We do not have the most recent corrected data, but it's about 6,142,000. No "new" ballots should be found after stage 2, and machines are quite capable of enumerating the ballots accurately. On the other hand, the ballots are partitioned into the various piles in slightly different ways at different stages, and it is true that most of them have not been examined carefully to assess with certainty just which pile they belong in. For at least 95% of the ballots, there is probably no need to do so -- the machine count is fast and accurate. At issue however is exactly what should be done with the hundreds of thousands of under- and over-punched ballots at any stage. When the Gore camp claims these votes have not been counted, they presumably mean they have not been assessed in this sense. When the Bush camp claims these votes have been counted and recounted, they presumably mean they have been repeatedly enumerated. Both claims are accurate.
For the purposes of this document, we will treat the "true" measure of a voter's preference to be the subset into which it is segregated at the third stage -- the earliest stage from which a concrete record of the vote exists. That is, we choose to ignore in our model all possible errors which occur before that stage, and leave those to the courts. (This author is of the opinion that some of the error types before this stage may represent hundreds or even thousands of errors in this election, but that no reliable way to measure these errors can likely be found.) We understand that the legally binding vote count will be one of the later stages, and so we seek to estimate the errors which occur from the third stage forward.
We must take it as axiomatic that each vote does in fact belong to precisely one of these subsets of votes at the "true" stage. We cannot consider subsequent vote counts to be right or wrong unless there exists a "correct" disposition of each ballot against which to compare. Thus for example if a voter intended to vote for candidate X but only barely filled in the corresponding oval on a ballot card, then by definition the mark either will be dark enough to count as a vote (the vote is in the "candidate" pile), or it will be too light to count and the "true" disposition of the vote is that it is in the "underpunched" pile. Implicit here is the existence of some rubric, however arbitrary, which allows a segregation of the ballots into the piles. (It does not matter for this analysis whether a "punch" requires completely severing all four corners of a chad rectangle, or requires only dimpling the chad, as long as each ballot is assigned to one of the subsets.) Note that for our purposes, the segregation scheme need not even be "fair" (e.g. one may decide that a certain depression of a chad constitutes a vote on one ballot and an identical depression does not count as a vote on another ballot, as with the Texas law). All we really need to begin the analysis is the assumption that each ballot has a correct interpretation.
Now, all this may appear ridiculously belabored, but breaking things down this way allows us to itemize the various errors and other irregularities which have been noted in the voting process. Depending on which vote count we consider the true one, and which errors we consider to be random, we may apply a variant of our previous model. If we can estimate the sizes of various error rates, we will be able to estimate the likelihood that the "true" vote count showed a different outcome, as at the end of the previous section; however, the specific conclusions depend greatly on the numbers which replace the assumed values "1%", "6 million", and "537" in the previous section. Most of these numbers will be extremely difficult to measure with any certainty.
The mathematically-minded can perhaps see the model being proposed here: vote-counting is a stochastic process in which we have identified several states in which a vote may reside. The movement of votes between "stages" should be representable by a transition matrix. We can estimate only a few of the entries of these matrices well, and in general only if we finely divide the states (e.g. states are not labeled "Bush" but rather "Bush in a Punched-Card county", etc.) We do not proposed to complete this analysis in full. Back to top
Now let us consider in turn which mechanisms might move a vote from one subset to another as it passes through each stage. We will ignore errors of communication, transcription, and arithmetic which can occur at all stages but which will likely be quickly fixed in this highly-charged situation. To assist the reader in following the list, we provide a Helpful Illustration:
A. We begin with each voter considering how to vote. As we know, many voters begin in the subset "no vote", i.e. they do not intend to cast a ballot at all. Most others decided to vote for a particular candidate, though some have perhaps decided to vote only for non-Presidential races, or have decided to spoil their ballots deliberately (as protest, say). Now, how can it be that the voter's vote is no longer in the same subset by the time the physical ballot is under his or her control? The following events have been alleged:
B. Once the voter has obtained a ballot, he or she attempts to register his or her preference. How can it be that the voter does not succeed in delivering an adequate record of his or her intentions to the state? The irregularity receiving the most attention is probably the layout of the PBC "butterfly" ballot, which may have contributed to some voters converting a vote for candidate X to a vote for candidate Y or to an overpunched ballot. [Analyses] The same occurs when voters attempt to change their ballot before submission (e.g. crossing off a marked oval), and likewise a voter who forgets to complete the ballot, or circles choices instead of marking them according to instructions, converts a "candidate" vote to an "underpunch". Other irregularities alleged:
C. At this stage, the state is in possession of the ballot and conceptually it belongs to one of the piles, which we have called the "true" disposition of the ballot. Passage to the next stage requires a first vote count. The mechanism here varies by county. Paper ballots are counted by hand, marked ballots are optically scanned, and punched cards are passed through a card reader. (Possibly some absentee ballots are hand-counted even if cast on punched cards or scanned cards.) Depending on the county, the counting may be done within the precinct or at a central county location. Many sorts of errors were discussed in the media in the weeks following the election. [Sample mail]
D. Since the (first) statewide vote count showed a lead for Bush of less than 0.5%, state law requires an automatic recount of the ballots. The intent of the law was perhaps to validate the first count, or to expose and then correct errors. In fact, the recount did both these: the absolute numbers were approximately the same, and whatever variations exist are often explained as the correction of previous errors. Of course, it is to be expected that there is just as good a chance as in the first machine count that the raw enumeration will be incorrect for the same reasons as before.
Significantly, however, the total number of "candidate" ballots increased between first and second machine counts. [Data] The primary explanations seem to be
It must be observed that in some counties, the automatic recount was performed without referral to the original ballots. In effect, the recounts performed in these counties allowed no motion of ballots between the piles except to correct clerical errors. [St.PeteTimes]. Conversely, in other counties officials evidently took it upon themselves to manually inspect some ballots called into question by the machines [Salon], that is, to perform what constitutes in the large counties the now-disputed hand recount.
E. A second machine recount (i.e. a third count) was conducted in Palm Beach county and later in Dade county; additional machine recounts may have been conducted in select precincts elsewhere. Here were expect again the same possible range of enumeration errors, favoring increased tallies for the same reasons as in the previous paragraph. In each of those cases, the additions to the tallies should be smaller than upon the first recount.
As far as we can determine, there are no cases of ballots which underwent four or more machine counts, nor does that seems appropriate to this author. The expected net gain in tallies must be approaching zero, and random statistical variation would begin to dominate. Multiple recounts introduce wear and tear on the ballots, possibly causing more errors and certainly running the risk of destroying what will probably be taken as legal evidence. Finally, it is true that multiple counts give statistical confidence that the machine counts obtained are accurate measures of what the machines are able to count.
F. A hand recount was requested in several counties, was performed on samples of the ballots in some of them, and is in the process of being carried out in just a few. We observe that a hand-count is not "just another recount"; at this stage a team of humans is searching a ballot for evidence of a vote, and they can use clues which the machine cannot sense to determine whether a vote was cast or not. Thus we are not surprised when most of the precincts examined in Dade county showed increases for both candidates beyond what three machine counts found. Nor is it true that a hand recount is "more error-prone". We have itemized above several sources of error in hand counts, and indeed, it is certainly true that human error in enumeration is frequent. However, we observe that the hand recounts are performed by teams and observed by witnesses (presumed hostile!) representing both major parties, as well as the media. This environment seems designed to be the least efficient, but most accurate, method of segregating the ballots into piles and enumerating them.
The greatest differences between "true" and manual-recount tallies are probably the chance for human dishonesty or miscount, and the difficulties of applying an assessment rubric. That is, although we have had to assume in advance that for each vote there is a "correct" disposition, in practice some ballots would be very difficult to judge. Moreover, it is important that those performing the assessment follow the rubric as scrupulously as possible, but it is extremely difficult to write that rubric to capture the sense of what people would agree is correct. (Example: a voter who circles the position of the hole he intended to punch, but does not punch anything at all, would be counted as "underpunched" by any useful rubric, even though the "true" expression of this voter's will would seem reasonably clear.)
Different counties have taken different approaches to the problematic issue of assessing, during a hand count, whether a ballot was a "true" vote for a candidate or was a "true underpunch". In the next section we will attempt to determine to what extent these deliberations have been "disenfranchising voters" or "manufacturing votes". Justifiable caution probably tends to move more votes from "candidate" to "underpunch" in this way than the other way around.
Summary The goal of all the counting ought to be to discover a useful measure of the "true" vote count. At this moment a hotly contested issue is, what would be the result of a hand recount? Our best model for this is: use the known machine counts (stages 4,5,6) to estimate the "true" count (stage 3), and then estimate the hand count which would result from that (stage 7). But our best description of the result of a hand count is that it differs from the true count mostly by dishonesty and we know of no good measure of how much of an effect that has. Therefore, we choose instead only to estimate, as well as possible, the "true" vote counts (stage 3), and let the reader adjust these numbers as he or she finds appropriate to predict the results of a hand recount (stage 7). (Equivalently, one may say we are building a prediction for the outcome of a hand recount by assuming a dishonesty rate of zero.) As hand-recount information becomes available, we use this to improve our estimates for the error rates and other quantifiable parameters above.
So we wish to use the known data in order to estimate as well as possible what the "true" tally of the votes was. All the data we have are the vote counts from the statewide count and recount and the limited hand counts. Some errors have already been introduced, and a portion of those corrected; what we must do is estimate the ones remaining. Back to top
We have indicated in the previous section a few components of the error suite which we believe are most likely to be corrected during recounts. In this section we wish to estimate numerical parameters associated with these processes: the "abstention rate", the "disambiguity rate", and the "drift rate".
A. A very large fraction of the underpunched ballots would be found, upon inspection, to be inarguably blank, that is, these ballots are "underpunched" in the "true" ballot count (stage 3) because the voter apparently intended not to cast a vote for President when submitting his or her ballot. (This is quite surprising to this author, and possibly significant as an indication of the mood of the electorate.) This number will have an important effect in our model, so we will try to estimate it carefully.
Our best estimate so far is that about 0.4% of the voters actually intended not to vote for President, and that this fraction does not vary considerably from county to county.
We may justify this estimate from the reported numbers of undervotes listed by CNN. Not all Florida counties are represented there but the fourteen missing ones are mostly smaller and use punched-card balloting systems. We view the underpunch rates in punched-card counties as inflated, since these numbers are (barring a hand recount) the numbers of ballots not registering a vote on the tabulating machine, which includes both the intentional abstainers and those marking dimpled chad or experiencing other failures. Possibly the numbers reported from optically-scanned counties are inflated too, though we currently envision no mechanism likely to produce large numbers of unregistered ballots from voters who intended to cast a vote in these counties.
CNN shows data for all but four of the 41 counties using an optical scanning method of vote tabulation. (Leon, Escambia, and the smaller Santa Rosa and Holmes are missing.) Among these 37 counties, we find the undervote rates to be 0.3% overall. Only about a third of these counties report undervote rates to be more than 0.5%, and they tend to be smaller counties (where larger percentage variations are less "remarkable" as we showed in an earlier section). Indeed, the only counties among these with an undervote rate above 1% are quite small: Baker (8154 valid votes, 1.1% undervote); Taylor (6808, 1.2%); Liberty (2410, 1.2%); Franklin (4644, 1.5%); Calhoun (5174, 1.5%); and Washington (8025, 3.5%). (This county is one of the ones in the Central Time zone possibly affected by untimely news reports "calling" the election early.) CNN incorrectly reversed the under- and over-vote counts in Columbia county as per a phone call to the County Supervisor.
It is important to stress that there is no clear reason why voters would be more likely to prefer to abstain from a vote for President simply because they live in a punched-card county. For example, when Brevard county switched to an optical system between the 1996 and 2000 elections, they found the undervote rate to drop by an order of magnitude. By no mechanism known to us would it be reasonable for a roughly unchanged populace to change its abstention rate so markedly when no change was evident statewide. We concur with the county Supervisor of Elections that the adoption of the optical system can be assumed to have avoided the loss of a number of votes (possibly accounting for some 453 votes out of Bush's 537-vote lead!).
With such experiences in mind, we choose to view the undervote percentages in these optically-scanned counties as a predictor of deliberate abstention rates statewide. We conclude that abstention rates in the 0.3%-0.5% range are accurate (or slightly inflated). So why are the undervote rates higher in the punched-card counties? (All are between 0.8% and 2.6% in the CNN data.) Given the prevalence of "dimpled ballots" and so on, it seems clear that these higher rates include not only the intentionally-blank ballots but also some ballots erroneously classified as blank. That is, we interpret this difference as an indication of votes which were intended by the voter but erroneously not included in the vote tabulations after stage 6. These amount to about 1% of the statewide vote total (thus more or less accidentally confirming our assumed error rate in earlier models).
We have other corroborating data which we used earlier in an attempt to determine the rate of deliberate undervoting; these provide similar estimates. In conclusion, we feel that an appropriate estimate of the number of ballots intended by the voter to be blank tends to be in the neighborhood of 0.4%.
Of course, the abstention rates may vary somewhat by region and from year to year. For example, the presence of a hotly-contested local race may draw to the polls many voters who have no preference for President, giving a higher abstention rate than another region. (This applies to any region with distinctive region-wide elections, including state, county, and subsets of a county such as city wards during aldermanic elections.)
B. Next we wish to consider the "disambiguity rate" -- the rate at which ambiguous ballots are recognized as actual votes for president. This must be defined carefully. Our remaining models -- and the recount ordered Dec 8 -- consider only the recovery of votes from among those ballots which have already been rejected as undervotes. We will use the estimated abstention rate, above, to account for ballots intended by the voter to be blank; thus the "true" disposition of each of the remaining ballots is that they must have been intended by the voter as a vote for one of the candidates. However, cautious interpretation rubrics inevitably lead to a certain fraction of these ballots rejected as either unclear, ambiguous, or contested -- that is, they remain in the "undervote" pool. We will need to know what this recovery rate is.
No such single number exists. Rather, this rate depends both on the point of view of the ballot canvassers and on the condition of the ballot itself. We will estimate an "average" rate, but recommend that a careful model be formulated which applies different rates to different subsets of the ballots.
(This differential recovery rate is, apparently, a key concern for the U.S. Supreme court in its Dec. 9 ruling. Evidently this is seen as a serious abridgment of the 14th Amendment's "equal protection" clause. But from a modeling perspective, this issue accounts for a much smaller expected error -- and smaller headache -- in measuring vote totals than the obvious differences among the counties resulting from the use of different voting apparatus. Not having any legal training we are unable to determine why a hand recount is deemed to increase the magnitude of the inequities among voters in different counties, when already significant inequity is caused by asking some of them to use a more error-prone voting system!)
Let us derive first an "average" recovery rate from the partial recount in Dade county. Using data in the Dec 8 court decision, and precinct-level data for Dade county provided by the Miami Herald, we see this examination of about one fifth of the county included a review of precincts with 112,304 ballots (17% of the county total), and found 417 more votes for president than the last machine count. Now, these precincts contained 2030 undervotes (leaving 8720 undervotes still unexamined from the original 10750). Among these we expect about 449 votes intended to be blank (the 0.4% abstention rate), leaving an expected total of some 1581 underpunched ballots which, according to voter intent, should have been a vote for president. The 417 ballots actually accepted thus imply about a 26.3% "disambiguity rate" in this portion of the Dade recount.
Now, might this value be acceptable for a general model? We may itemize several reservations.
Our reading of the Dec. 12 US Supreme Court decision suggests that it was precisely this unevenness of the disambiguity rate from county to county (or even within a county) which prompted the Justices to declare the recount proceedings unconstitutional, and indeed it will be clear that a comparatively small variation in this rate between Bush and Gore strongholds would be sufficient to change the overall outcome of the recount.
In the hopes that effects #1 and #3 tend to cancel, we will use a disambiguity rate of 1/3 in our later models. Again, we suggest lowering this, perhaps to 0, for some counties as described in #2 and #4. We will present calculations with this rate set to 1, to which the reader may simply apply whatever rate he or she feels more realistic.
C. Finally we consider the "drift" of the recount numbers towards one of the candidates. That is, we wish to predict the extent to which the additional votes recovered in a recount would tend to favor one of the candidates more or less heavily than was observed in the machine counts for a similar representative group.
Our best estimate for this quantity is: zero. That is, we have no credible mechanism to explain why it should be true that a greater fraction of votes should have been lost from one candidate's pool than from another candidate's pool. We will use this, appropriately qualified, in our models below. However, it has been observed to be false in recounts undertaken so far; for example, the first automatic recount found noticeably more Gore votes than Bush votes statewide despite the fact that Bush had just slightly more votes than Gore among the machine counts up to that point. So we offer some arguments which could account for this "drift" so that we may attempt to incorporate such a drift into future predictions. These arguments depend upon the observation that voting units are not homogeneous.
Indeed, at the state level, we have already accepted the possibility that a recount can alter the essentially even split of the machine vote total. The basic argument -- that Gore has more support in punched-card counties -- is valid since the different voting machines have different error rates and, significantly, are not uniformly distributed across the state.
It is somewhat more difficult to make a similar argument within a county, however, as each Florida county uses (nominally) a single type of voting machine. However, this is not completely true; for example, in Palm Beach both Votomatic and DataPunch machines are used. Nominally identical, they have overall markedly different error rates. It is possible (and by some arguments likely) that the cheaper, more error-prone machines could be deployed in precincts more lively to favor one candidate. [PalmBeachPost,Dec8]
It is also not true that all voters are identical. Plausible arguments can be made suggesting that elderly voters, first-time voters, non-native speakers of English, and so on may have greater difficulty with the process of voting and so may be more likely than other citizens to be unable to execute their vote properly [AP]; thus their votes are over-represented in the pool of undervotes. To the extent that these voters favor one candidate (which is not particularly clear in Florida), that candidate would gain votes at a faster rate in a manual recount than in the rest of the state.
We have access to limited data by precinct (see sect. VIII.); these confirm our expectation: that the results of hand recounts tend to add votes to the parties in roughly the same proportions as the original vote count in small areas, such as precincts. The tendency of county-wide recounts to lean more heavily towards one candidate in recounts appears then to be a consequence of the fact that the higher-error rate precincts tend to coincide with the precincts where that candidate has greater strength. We remain hopeful that as more recounts are conducted and more precinct-level data are released, the precinct-level correlation between machine- and hand-counted ballots will be confirmed.
A classic example may help illustrate how it can be that a set of norm-following units can appear not to follow the norm overall. Consider a university charged with discrimination against women in admissions. Suppose 20 women apply for the Math program and 80 for the Politics program, and that a hundred men apply in the opposite ratios. If Politics admits 20% of its women applicants and 10% of its men, and if Math admits 80% of its women applicants and 60% of its men, only 32 women are admitted but 50 of the men are admitted. Thus neither program has discriminated against women, but the university as a whole appears to do so.
One more mechanism should be mentioned which may affect the distribution of hand-counted votes among the candidates. There is some concern that repeated handling of the ballots would tend to introduce additional stray marks and indentations in the ballot papers, effectively manufacturing votes. Unless some reason is presented to the contrary, we should expect these accidentally-added votes to show indentations equally often at each position on the ballot. When such a 50-50 split of votes is added to an otherwise uneven split, it will have the effect of drawing the distribution of the added ballots closer to even. We expect the incidence of this phenomenon to be small. In particular, we note that such a happenstance would be just as likely to create votes for minor-party candidates as to add votes for either Bush or Gore. In fact, we have observed this to a certain extent in the Palm Beach recount: the four leading candidates increased their totals by less than 0.9% in the recount; the others gained 2.7% or more and in fact Moorehead jumped from 104 to 125 votes in the recount! Now, we have advocated estimating vote gains by precinct, and this method predicts the minor parties' share of the undervote is some 937 votes, as opposed to the 535 from a county-wide extrapolation -- that is, we do indeed predict that the third-party candidates should have done well in the recount -- but the rate of gain of the minor parties is surprising and tends to support the hypothesis that a certain amount of this type of error has been introduced during the hand recount. Yet given the small absolute numbers of votes apportioned to minor candidates, this disparity reflects a small portion of the vote change during a recount, suggesting that this phenomenon is not particularly significant. (We have not performed a detailed analysis but estimate that these data could support a claim that as much as 5% of the votes added in the recount are the result of random markings of the ballots.)
In summary, until by-precinct data are available for the state, we find no reasonable course of action but to duplicate county-wide proportions of votes for Bush and Gore when predicting the disposition of hand-recounted ballots, although it is easy to adjust our calculations to reflect any other assumed proportions. We also find, with this analysis, that there is no a priori evidence of fraud simply because a set of recount ballots does not mirror the county's overall distribution of votes to candidates. Given limited recount data so far, it would not be unreasonable to assume a small drift to Gore in general when comparing hand- to machine-counts.
We will apply these parameters to the models developed in subsequent sections. Back to top
Dade county, largest in the state and leaning towards Gore, was seen by the Democrats as a major source of additional votes during a hand recount. After quite a number of reversals and incomplete attempts, the county reached the end of the post-election season with a portion of its precincts recounted, yielding some additional Gore votes, and nearly nine thousand undervotes still unexamined. We wish to estimate the number of additional votes each candidate could have obtained from a hand recount of the entire county.
On Nov. 14, the Dade County canvassing board manually recounted 5870 ballots (just under 1% of the county total) to determine whether or not to recount the whole county. At that time they found six additional ballots for Gore, and none for Bush. With blatant disregard for statistical validity, we might extrapolate to expect that a complete Dade County recount might yield about 600 net votes for Gore -- our first prediction for the recount. But we hope to develop models in which we might hold a greater measure of confidence.
It is possible to use the discussions of the previous three sections to predict the possible effects of a hand recount in one county. Roughly speaking, we compute the numbers of ballots already counted for each party, and then add back in a fraction of the votes assumed lost through chad or other problems. If we assume as in section II that about 1% of the votes would be recovered in a recount, we can simply increase each party's totals by about 1%. Since this recount is performed in a county which has favored Gore so far, we expect Gore to pick up votes, which we may subtract from the tabulated Bush lead. Simplicity itself!
In Dade county, the official data show 289,533 Bush votes, 328,808 Gore votes, and 7108 votes for other candidates. If we assume these represent a 1% loss of ballots, the original totals must have been 292,458 - 332129 - 7180. Thus a recount would pick up 2925 more votes for Bush, 3321 more for Gore, with a net advance of 396 votes for Gore in a Dade recount -- this is our second prediction.
Note: our precinct-level dataset for Dade county shows 87 fewer ballots total and correspondingly fewer votes in each of the categories above. We cannot explain the discrepancy.
We may even express our confidence in this result using the language of section II. These recovered votes amount to a random walk of 6246 steps on the number line, favoring steps towards Gore in a 3321:2925 ratio; while we expect this to move us 396 steps toward Gore, we ought to accept any result between a 350- and 450-vote net gain for Gore as "unremarkable".
It is easy to criticize this model, too. To begin with, we might ask where these 6318 "recovered" ballots would come from. In the case of Dade county, we have record of 28,601 disqualified ballots -- more than enough to provide ballots which perhaps should not have been disqualified. But county records show that 17,851 of these ballots were over-punched; a correction of the "hanging chad" problems would not transform any of these into legal ballots. Only 10,750 of the ballots registered as blank after all three machine recounts; it is among these which we must find the votes we claim would be "recovered". While this is still compatible with the given numbers here, it is clear, after perusing data from other counties, that similar reasoning cannot apply in general (that is, there is a logical flaw here): the number of undervotes is in some counties already less than 1% of the total, leaving no source for the votes we claim we would recover.
So let us now refine the model by considering the progress of ballots through the "stages" in section IV. The county vote totals given above refer to the standings after the first (automatic) recount. These numbers reflect an increase of 108 votes from the original machine count, commonly held to be the result of loose chad falling free.
On Nov 19, the ballots were passed again through tabulating machines; much chad was again freed. In some cases, this may represents the passage of a ballot from "candidate" to "overpunch" piles; more likely, it represents either the passage from "underpunch" to "candidate", or else no movement at all. (Presumably about equal fractions of the new chad are falling from punches in the senatorial and other races. Unless thousands of new chad fell Nov 19, we expect fewer than 100 ballots to have been recovered from the "underpunched" pool at that time.) It is also likely that the number of ballots changing piles is smaller than during the automatic recount -- at some point whatever was loose would be freed.
Thus we still have perhaps 10,700 ballots in the "undervote" pile. What should we expect to happen to these in a manual recount?
Certainly a fraction of the ballots can be expected to be blank; we have estimated in the previous section that there is a small number of deliberate abstentions in the elections -- we estimate this at 0.4% of the total number of ballots cast (654050, including overvotes and undervotes). In the case of Dade county this abstention rate would translate to about 2616 votes -- certainly substantially fewer than the 10,750 underpunched ballots. To repeat: barring a situation in Dade county which is substantially different from the situation in other states, we would expect about 2600 votes to be intentionally blank and some 8000 of the ballots rejected by the tabulation machine should be expected to reflect some intented vote. A split in which many more of the ballots are deemed to be "votes" would be an indication of "manufacturing votes" as claimed by the GOP; a split in which many more of the ballots are deemed to be "no-votes" would be an indication of "disenfranchisement" as claimed by the Democrats. In either case it may be appropriate to readjust the threshold of "dimpling" at which the canvassers decide that a vote has taken place. Even if we were to assume a higher 1% rate of ballots which actually began (in stage 3) as "underpunched", we believe there should be several thousand ballots in Miami-Dade County which have erroneously been classified as "underpunched" by the tabulating machine. Lacking suggested mechanisms to the contrary, we expect the pile of "undervotes" includes about 8000 ballots intended by the voters to have counted as a vote for President.
Next, we recall from the previous section that it is unlikely that even as many as a third of the marginally-marked ballots would be found, under scrutiny, to have a sufficiently clear of voter intent to warrant the inclusion of these votes in the legal count. Using the 26.3% rate established in the first portion of the county, we estimate approximately 2139 votes would actually be accepted as legal among the undervotes in Dade; the higher 33% rate we have suggested would yield 2711 votes to be found. (Note that in sections II-III we postulated a loss rate of 1%, which would mean 6,541 votes in Dade county. In fact the total of 8,134 lost votes in the previous paragraph is higher than this, but if we assume only 2,139 of the votes will be held to be legal, this is equivalent to a much lower loss rate of about 1/3 of 1% to apply in the arguments of sections II-III.)
Finally, we ask for the expected distribution of those ballots. How should we expect these 2139 (or 2711) additional votes to be divided? We could justify several different answers. First, the county-wide proportions would divide these votes 1125 to Gore, 990 to Bush, and 24 to others, giving Gore a net gain of 135 (assuming no additional overvotes are found). On the other hand, in the partial manual recount of 135 precincts, the distribution was 287 Gore and 130 Bush. This net Gore gain (157) already exceeds what we have just predicted for the whole county! Using this division rate of the recovered votes, we would find 1472 of the votes for Gore, 667 for Bush, none for others, and thus a net gain of 805 for Gore.
(Note: Our 157-vote gain is also what was reported when on Nov 22 the recount was halted, However, a 168-vote advance for Gore is quoted in the Dec 8 Florida Supreme Court decision. We cannot explain the discrepancy.)
But these data suggest these first 135 precincts recounted must have been Gore strongholds (and indeed, we can compute that they originally favored Gore by a 3:1 ratio !), and our suspicion is that undervotes recovered in a recount do not necessarily mirror the county distribution of votes but rather are more likely to mirror their precinct's distribution of votes. (Taken as a group, the 135 precinct's pattern of support for Bush versus Gore would split a pool of 417 votes into 312 for Gore and 99 for Bush -- not too far from the actual numbers of 287 and 130.)
Therefore, we could consider a finer model in which we partition the county totals into individual precincts, subtract 0.4% intended abstentions in each, and use the precinct distributions of the ballots to determine the expected breakdown of the votes. Setting aside the 98 precincts (with 13,548 ballots) with an undervote rate less than 0.4%, we may perform these computations (allowing for fractional vote gains within a precinct) to find the county-wide distribution of the undervotes would be about 3295 "intended" for Bush, 4786 for Gore, and 77 for Other (and leaving a slightly low 2592 Abstentions). Staying with the 0.263 disambiguation rate, we now have a slightly higher total of 2146 recovered ballots: 867 Bush, 1259 Gore, 20 Other, and thus a net gain of 392 for Gore. (More precisely, the model predicts gains of 89, 321, and 5 respectively for the first 135 precincts, and 778, 938, and 15 for the remaining 479 precincts and 176 absentee precincts. So we may add a projected 160 net votes for Gore in these other precincts either to our projected 232-vote gain in the first 135 precincts or to the measured 157-vote gain. That is, this model offers a prediction of either 392 net votes gained for the county or 317 votes.) Clearly the process of ascribing the undervotes according to precinct patterns rather than county-wide patterns suggests a "drift" toward Gore on the order of a couple hundred votes.
One may adjust the figures easily to account for different assumed disambiguation rates such as those used in Broward or Palm Beach counties. For example, we have suggested a 33.3% rate, which would have the effect of adjusting all the foregoing figures upwards by a factor of 33.3/26.3 = 1.27, for example giving projections of 203 + (either 157 or 294) for the county.
For comparison, we point to a different model of the Dade county recount by Bruce Hansen (UW/Economics). His model extrapolates the by-precinct data from the partial recount to the rest of the county, and estimates 254 votes gained by Gore. Arguably the variation among the estimated vote gains is a reflection of the difficulty of making models accurate to such a small fraction of the overall vote. It is also true that his model does not account specifically for intended-blank votes; it is possible that this could adjust the conclusions.
So, various models have suggested predictions of 137, 157, 254, 317, 360, 392, 396, 497, 600, and 805 net votes for Gore upon performing a county-wide recount in Dade county, each making various assumptions (such as assuming a criterion for disambiguating marginal ballots which continues the pattern established in their partial recount). We believe the two smallest and the two largest estimates have the least supporting arguments; the others suggest a net gain of several hundred votes for Gore.
In summary: it is roughly consistent with previous experience to expect a recount in Dade county to cut the Bush lead by perhaps another hundred or two votes beyond the 157 or so already found. (We regret to disappoint the reader who was under the assumption that all models would lead to the same answer!)
Operative maxim: "All models are wrong. Some models are useful." Back to top
We will illustrate this in this section by examining the cases of counties for which additional information about spoiled ballots has been obtained precinct-by-precinct. Since voting patterns seem to vary widely within some counties, our expectation is that this approach will allow us better to fine-tune our predictions. This procedure reveals some interesting observations, and in particular accounts for the "drift" of county results (usually towards Gore) when recounts are performed.
We and several other groups of researchers have collected data in various formats from several counties. Bruce Hansen has kindly agreed to collect and serve these data; see http://www.ssc.wisc.edu/~bhansen/vote/data.html. As of this writing, they exist (sometimes incompletely) for the following counties: Brevard, Broward, Collier, Dade, Duval, Escambia, Hillsboro, Lee, Leon, Marian, Monroe, Palm Beach, Pasco, Pinellas, Polk, Sarasota.
Here is the typical situation: in each county, we know the number of votes cast for Bush, Gore, and Other, as well as the number of disqualified ballots. What will happen during a recount? Our basic premise is that none of the votes counted in the first three categories will change, but that some of the disqualified ballots will be found to move to the other three piles. But how shall we play with the numbers? How do we expect each 100 disqualified ballots to be split into piles, say? We have already discussed the number we expect to be deliberate abstentions, and we have considered what fraction of the remaining undervote can be recovered. But we may split those into piles in different ways: considering each disqualified ballot to be representative of the county as a whole, we will split the ballots in one proportion, while considering it as a representative of its precinct, we might split the ballots differently. We will encounter other problems in counties for which we only know how to distinguish over-votes from under-votes at the county level, and in one case we cannot distinguish them at all.
We have carried out this process for Dade in some detail above and noted a "drift" toward Gore when the votes are tallied by precinct. (We can measure this drift as the number of percentage points of increase in Gore's fraction of the vote, comparing the initial counts to the added votes in a recount. In the case of Dade county the drift is about 3 percentage points.) We will also carry this out for Duval county below, which is large and has been held to be of significance, and in which the drift is somewhat larger (about 8 percentage points). Unfortunately we do not have the data to carry this out for all counties, so we will not include these numbers for any of the remaining counties, but we may report them here.
In the ideal case we simply compute for each precinct in a county the amount by which the undervote there exceeds .004 of the ballot total there, and divvy up the ballots according to the candidates' strength there. We have to the data to do this for Brevard, Hillsboro, Marion, Pinellas, and Sarasota.
For Brevard county, there is nothing to report, since for the county as a whole, and for nearly every precinct, the number of undervotes is less than 0.4% of the total ballots cast.
For Hillsborough county, the projections for Bush, Gore, and net lead are
county-wide model: 2033 1907 126 net for Bush precinct-wide model: 1962 2033 71 net for GoreThese are totals of all non-abstaining ballots, which must be reduced by a disambiguation rate to make a prediction for the result of a recount. With a rate of 1/3, this means we would predict about 24 votes for Gore with this model rather than 42 for Bush -- not much of a change, but indication of this "drift", in this case of about 3 percentage points. (Note that in all the models of this section, we are not incorporating the disambiguity rate and invite the reader to do so.) Note On Jan 4 Salon magazine reported the results of an independent recount in Hillsborough. Using the most liberal standard of several offered by the Tampa Tribune, the paper indicated 1,878 ballots to be recoverable (corresponding, in our model, to a 46% "disambiguation rate"). Our by-precinct model would split these votes approximately 894 for Bush and 927 for Gore, a lead of 33 for Gore. The Tribune's count gives Bush 879 and Gore 999, suggesting a drift in Gore's direction just a little larger than our precinct model predicts.
For Marion, the corresponding numbers are
county-wide model: 1086 879 206 net for Bush precinct-wide model: 1105 862 243 net for Bushmeaning a precinct-wide model would predict about 81, rather than 69, net votes for Bush -- a drift in Bush's direction by a half a point.
Pinellas also shows a net improvement for Bush when carried out by precinct:
county-wide model: 1205 1308 103 net for Gore precinct-wide model: 1292 1296 3 net for Goreabout a one-point drift toward Bush.
For Sarasota county we lack the precinct-by-precinct breakdown of the 991 overvotes, but these would have only a tiny effect on the outcome of our models, so we proceed as in the other cases.
county-wide model: 598 524 74 net for Bush precinct-wide model: 617 530 87 net for Bushagain about a half-point drift toward Bush.
So for all of these counties together, we see that the effect of a computation at the precinct level is nearly nil but favors Bush slightly.
We have been able to obtain similar precinct-level data for Lee, Collier, and Pasco counties, unfortunately not separating undervotes from overvotes. Since we have the county-wide total for over-votes and under-votes in each case, we might simply split the disqualified ballots by precinct in the same ratios to get a presumed distribution of the undervotes by precinct. Here are the numbers which result:
Collier county-wide model: 1134 561 573 net for Bush Collier precinct-wide model: 1119 576 543 net for Bush Lee county-wide model: 735 509 226 net for Bush Lee precinct-wide model: 698 552 146 net for Bush Pasco county-wide model: 570 578 8 net for Gore Pasco precinct-wide model: 539 613 74 net for Gore Monroe county-wide model: 0 0 0 net Monroe precinct-wide model: 7 10 3 net for Gore Leon county-wide model: 0 0 0 net Leon precinct-wide model: 9 14 5 net for Gorewhich is to say it looks like a more careful model would reduce the expected Bush gain by about two dozen votes in each of Lee and Pasco counties, and only about 10 votes in Collier.
We caution however that this model assumes that a disqualified ballot in one precinct is just as likely to be an overvote as a disqualified ballot in any other precinct in that county (or at least, that the propensity for a precinct's disqualified ballots to be overvotes rather than undervotes not be correlated with Bush or Gore support). Unfortunately this is not supported by the other data we have considered. In the Hillsboro, Marion, and Pinellas examples above, we see that the ratio of undervotes to overvotes in each precinct can vary widely within a county. Moreover, this can have an effect on the quality of our predictions. If in those cases we were to combine over- and undervotes and treat those counties as we did the others in the preceding paragraph, we would get markedly different results with the precinct-wide models! For example, the Pinellas models yield
county-wide model: 1205 1308 103 net for Gore true precinct-wide model: 1292 1296 3 net for Gore this precinct-wide model: 1056 1491 436 net for Goresince in Pinellas county the disqualified ballots in Gore-leaning precincts are much more likely to be overvotes (as opposed to undervotes) than in the Bush leaning counties. This new precinct-wide model, in which we are allowing the same fraction of all precincts' disqualified ballots to count, would give Gore over 300 supporters whose ballot is in fact spoiled rather than undercounted. Likewise in Marion county the same test shows this model would over-estimate Gore support and underestimate the Bush lead:
county-wide model: 1086 879 206 net for Bush true precinct-wide model: 1105 862 243 net for Bush this precinct-wide model: 1064 902 162 net for Bushand in Hillsborough:
county-wide model: 2033 1907 126 net for Bush true precinct-wide model: 1962 2033 71 net for Gore this precinct-wide model: 1727 2246 519 net for GoreThus we suspect that this precinct-wide model, which ignores the distinction between under- and over-votes, tends to exaggerate the extent of Gore support which would result from a recount which considers only undervotes. Indeed, with these examples in mind, we believe it quite plausible that a true precinct-wide model in the counties of the previous paragraph would should stronger Bush support than in the county-wide model.
The situation in Pinellas, at least, suggests a better model may be made by segregating at least the absentee and the in-precinct votes, but we have not attempted to do so.
There is one more county for which we have been able to obtain data by precinct, but in this example we not only have no information by precinct as to the disposition of disqualified ballots -- we also lack that information county-wide. In Escambia County we can follow the same pattern as above but partition all the disqualified ballots:
Escambia county-wide model: 2431 1363 1068 net for Bush Escambia precinct-wide model: 1801 1997 196 net for Gorewhich suggests a very large change in the prediction based on the choice of model, but we must treat this with considerable caution. First, these numbers include both over- and under-votes, and we have no idea what fraction of them are indeed undervotes -- indeed, these numbers are surprisingly large for a number of reasons, as we will see below, so that no way of deducing an undervote rate here seems good! Second, we have the problem as in the preceding paragraphs, that we do not know whether the Gore precincts tend to overvote while the Bush precincts tend to undervote, or vice versa. This county shows a marked correlation between the strength of Gore support and the rate of ballot disqualification, but only if Gore supporters tend to undervote much more often than they overvote is this likely to help Gore in a recount limited to undervotes. Finally, of course, even if every one of the disqualified ballots were indeed an undervote, we would have to discount these numbers by the usual disambiguity rate to compute an actual number of votes to be recovered -- for whichever candidate -- in a recount.
Our conclusion, then, is that in some very large punched-card counties, there can be a significant difference between the results predicted by county-wide extrapolation and the results predicted by extrapolation by precinct. But this effect is thus far limited to just a few counties, notably Dade and Duval; it appears to be true from the other examples studied that there is likely to be little change in our predictions in all other counties taken together whether we use precinct- or county-models. We will choose the latter in the next section so that we can have a uniform way to deal with all but the largest four counties. For now we will merely note below by [Pct] the counties for which precinct-level analyses are included here.
As other counties provide precinct-level data we will analyze them similarly. Back to top
Prompted in part by the state Supreme Court's Dec 8 decision, we attempted to carry out an analysis similar to that of the previous two sections, for all 67 counties. We will provide an analysis by county, so the reader may piece together predictions for the effect of recounts conducted in smaller numbers of counties, e.g. the Gore request for recounts only in Dade, PBC, Broward, and Volusia.
The methodology is simple: for each county, use the numbers of undervotes or disqualified ballots to estimate the number of true ballots which have been erroneously recorded as blank by the tabulating machine. (We have assumed that overvotes are unlikely to be counted as legal votes in any recount, though this is apparently not always true [Gadsden].) Then use the county-wide split of the votes to estimate in some way how many votes each candidate would recover. Then total statewide. The outcome, in short, is simply that Gore could be expected to gain many hundreds of votes, simply because the counties with the most "lost" votes tended to vote Democratic. But it is primarily the three or four largest counties which make this difference; apart from them, the situation favors Bush slightly. Since some of these counties have already carried out recounts, Gore's potential to recover more ballots is greatly diminished. We will see how these numbers play out.
For a comparable analysis, we recommend an article in the Orlando Sentinel. Note however that their estimate of a "true" Gore lead statewide (about 1700 votes) assumes first that a recount is conducted statewide, and second, that the error rates in the punched-card counties are more or less the same in all such counties. This is definitely not likely to be true, since the cards in Palm Beach and Miami-Dade have been passed through the counting machines an additional time, thereby knocking out some more loose chad. It can only be assumed, then, that the numbers actually counted by these machines already show a partial correction of precisely the error to be recovered by the hand count. Taken together, the two- or three-thousand vote gain predicted by the Orlando Sentinel must be reduced, perhaps to one-third of their estimate. Thus the effect of a recount in the chosen counties can only expected to be in the range of several hundred to a little over a thousand. This range includes the 537-vote tabulated Bush lead, effectively making it impossible for this author to use their analysis to predict the final outcome of the recount at this time. (Reuters also evidently decided the prospect of a Gore win was possible but chose not to make a prediction.)
It seems clear that the estimates by the Sentinel greatly overestimate the votes likely to be recovered. Indeed, these predictions look little like the additional votes recovered as the recounts progressed. This is due in part to conservative judgements used when deciding whether or not to count a marginal ballot. (It is also affected by the practice of setting aside contested ballots until the end of the counting; these are precisely the ballots for which the hand count is expected to offer a more refined assessment than the machine count!) Naturally, the outcome of the recount then depends on the expected disposition of these ballots. If they are all rejected, the vote changes would be minimal; if they are all accepted, then the Gore gain is likely to be much larger. (For example, during the PBC recount, there was little change in the ballot totals until the very last days, when Gore picked up about 200 votes during the examination of the set-aside marginal and contested ballots; he would gain even more if dimpled ballots were accepted as legally binding.)
Let us begin our own statewide analysis.
Our analysis is made more complex because we have had to work with various datasets which are incomplete and occasionally inconsistent. We have some CNN data for some counties (apparently containing some errors) and AP data (which also have some errors ). The final county data we used are available here.
We should probably remark that this methodology could be refined simply by continuing to divide the votes into smaller groups such as precincts, where actual data probably exist but are not available to us. Taken to its logical extreme, of course, we propose a perfect model, in which the statewide election results are broken into some six million voting units; we request that the state make available an accurate database indicating for each such unit the the fraction of the votes in that unit which lean towards Bush or Gore :-)
We will proceed through the analysis by viewing the election as a collection of 67 more or less independent races. Because, in part, of the limitations of our data, we must partition the races into several sets. Here first are all the data necessary to see the pattern of our analysis; details will then follow. There are:
A. 4 counties using optical scanning equipment, not reporting undervoting rate Bush leads 153,364 to 117,349; there are 5,057 disqualified ballots B. 1 hand-count county, not reporting reasons for disqualifications Bush leads 2,332 to 1,407; there are 258 disqualified ballots C. 8 counties using punched cards and not reporting separate undervoting rate Bush leads 56,092 to 62,013; there are 3,713 disqualified ballots D. 37 counties using optical scanning equipment and reporting undervotes Bush leads 1,080,595 to 911,749; there are 6,672 undervotes E. 13 small-error counties using punched cards and reporting undervotes Bush leads 978,240 to 835,491; there are 27,426 undervotes F. 1 large county with significant over-voting and separate undervote Bush leads 152,098 to 107,864; there are 4,967 undervotes G. 3 large counties with Democratic majorities and large, reported undervote Gore leads 620,386 to 986,243; there are 28,018 undervotes
We separate the first three groups from the others since we will have to estimate and discard the numbers of ballots likely to be irretrievably disqualified e.g. for over-voting. We separate the hand-, optical-, and punchcard-counties since both over- and under-voting numbers reported seem to vary with system type. Finally, we segregate the largest counties since they account for so many recoverable ballots and we wish to analyze them separately.
Roughly speaking: from the numbers of rejected votes, obtain what would be reported as undervotes by subtracting some of the total vote count when overvotes are not explicitly segregated; 1%, 6%, and 4% might be appropriate in the first three cases, respectively. Then from the tabulated or estimated undervotes, subtract about 0.4% of the total ballots cast to account for deliberately blank ballots. The leaves "recoverable" votes. Partition to the candidates. Using the numbers just given to measure candidates' strength in the groups of counties, we can get a preliminary sense of what the detailed data should show:
Type Rej'd Underv Recov +Bush +Gore +Bush(net) A 5057 2350 1267 718 549 169 B 258 33 18 11 7 4 C 3713 0 0 0 0 0 D 6672 0 0 0 0 E 27426 20171 10879 9292 1587 F 4967 3927 2298 1629 669 G 28018 21591 8337 13254 -4917
This table assumes 100% of the "recoverable" votes are accepted, and that those votes precisely mirror the aggregate splits to Gore and Bush; The reader is invited to substitute the parameters which he or she finds most likely to represent reality. We will also look at individual counties, not groups of counties, to improve our predictions. But it should be clear from these analyses why the attention has been showered on the three counties in the last group -- Dade, Palm Beach, and Broward Co. Of the groups of counties here demarcated, only this last group offers Gore any real hope of recovering votes, but in this group they can be expected to exist in abundance -- on the face of it, more than enough to overturn a 537-vote Bush lead and even to be able, in addition, to balance a Bush boost from a statewide recount.
Now let us pursue the analysis of these groups of counties in detail. This will change the specific numbers in the table above, but not the general tenor of the conclusion.
A. The counties not reporting a separate undervote rate in the CNN table include the four optically-scanned counties (Leon, Escambia, Santa Rosa and Holmes) mentioned in section VI.A, tiny Union county, which uses hand-counted ballots, plus eight punched card counties. We will discuss these thirteen counties first; the lack of appropriate data makes for a more complicated discussion and less reliable projections, but fortunately the overall numbers are comparatively small. In section A we consider the four optically-scanned counties.
Leon county [Pct] reported only 181 votes disqualified for any reason, out of 103,305 ballots, which is well below our threshold for deliberate no-votes; therefore our models certainly predict no votes to be added in a hand recount there. (Their web page claims overvotes are impossible with their voting system, although their description of the disposition of ambiguous absentee ballots suggests these ballots have already been subjected to the sort of scrutiny which the Supreme Court found objectionable!)
To assess the nature of the invalidated ballots in the other three counties we consider the mechanisms for voting there. Escambia, Santa Rosa, and Holmes counties use the "Election Systems & Software, Inc., OPTECH" optically-scanned ballots, which are centrally-tabulated, meaning that a ballot would not be disqualified with the voter still standing by to make corrections. The system is also used in large Orange county, mid-sized St.Johns, Bay, and Clay counties, and small Washington and Baker counties. Many of these counties are in the Florida panhandle (from West to East: Escambia, Santa Rosa, Okaloosa, Walton, Holmes, Washington, Bay, ...), and thus possibly affected by the previously-noted phenomenon of the premature news broadcasts. (About 2% of the voting day was left, possibly with a somewhat greater fraction of the votes cast during that time.) The other counties are in the northeast around Duval county (Baker, Clay, St.Johns) except for Orange county, further south (containing Orlando). The counties which report a breakdown show between 0.1% and 0.6% of the ballots to be overvotes, with about 0.5% perhaps most representative. (For comparison, there are 16 counties listed by the state as using the same manufacturer's "Model 115" and "Model 315" optical systems, and tabulating the ballots centrally. The over-vote rates among these counties are around 6%. The only significant outlier is large Polk county, with over-vote rate only 0.4%. -- which CNN reports to use precinct-tabulation.)
Escambia county [Pct], fairly large with 121,020 ballots cast Nov. 7, verified for us on Dec. 1 that there were indeed 4,372 invalid votes (3.6%), but does not report a breakdown the reasons the ballots were rejected. This number of errors has been reported since at least Nov. 15. We are unable to determine why this county should have a markedly higher number of disqualified ballots than the other optical-method counties. While we expect 484 deliberate undervotes (0.4%), and perhaps about 605 ballots overvotes (assuming 0.5% as the over-vote rate with this voting system), and no accidental undervotes (when deriving the 0.4% abstention rate we assumed optical-card systems had a rate of zero erroneously-undercounted votes), this still leaves some 3283 additional disqualified ballots unaccounted for. Let us pause to consider why there might be so many.
In what way is Escambia different? It is of unknown significance that counties in this part of the state went more heavily towards Bush than any others. Escambia, with a large naval base in Pensacola, was recipient of a major portion of the overseas absentee ballots. It is possible that Escambia experienced a higher-than-normal (0.4%) rate of voters choosing only to vote in other races; in addition to reasons already mentioned, we note that they show very high rates of participation in normally low-yield races such as sheriff (a narrow win for one candidate with 50.9% of the vote in a 2-person race) which registered a very high 116,263 votes -- just a few hundred fewer than voted for President! Thus it is conceivable that a greater than customary number of voters showed up to vote without a preference in the Presidential race. (Compare neighboring Santa Rosa county, in which no other race drew as much as 90% of the Presidential race).
To repeat, this county is exceptional among the optical-scan method counties. It either must have more than the usual number of abstentions, or of overvotes, or of erroneously-ignored ballots. Not wishing to be accused either of "inventing votes" nor "disenfranchising voters", we might simply split the unexplained excess number (3283) of disqualified ballots evenly among the three categories; that is, we will treat Escambia has having an unusually large number of abstentions (now 1094+484=1578), an unusually large number of overvotes (now 1094+605=1699) and an unusually high number of (recoverable) undervotes (1095). We recognize this split is largely guesswork but see no alternative at present.
But now we may estimate the votes which could be recovered in a recount. Using the proportions of the rest of the county to partition the 1095 erroneously-undercounted ballots between Bush and Gore, we could argue that as many as 685 additional votes for Bush and 384 more votes for Gore could be expected among these, for a net gain of 301 more votes intended for Bush, assuming no drift in the Bush/Gore ratio from the rest of the county.
In the previous section we noted that a by-precinct accounting of all disqualified ballots would change the ratio noticeably; 1095 such ballots would be split about 507 to Bush, 562 to Gore, for a net gain of 55 more votes intended for Gore. See section VIII for our reasons to be somewhat skeptical of this result; but lacking better data we cannot dismiss it entirely. So the net effect of a recount in Escambia county seems difficult to predict at present. Taking into account our assumed disambiguity rates, we might expect anywhere from an 18 vote gain for Gore to a 100 vote gain for Bush!
Next-door Santa Rosa county reported to us Dec. 1 that (contrary to the CNN data) the Optiscan system they use is tabulated in precinct, that is, a ballot which contains an over- or under-vote in any race is rejected on the spot. Indeed, they report 1,532 ballots declared spoiled and then replaced by the voter. Presumably, then, the bulk of the 365 ballots listed by the AP as 'disqualified' are ballots on which the voter chose to under- or over-vote after having been notified that there was a problem. It seems unlikely, then, that any of the ballots would, upon hand-examination, show a clear vote for any candidate. Indeed, after our assumed 0.4% abstention rate is subtracted from their 0.7% overall disqualification rate, the remaining 0.3% is within the range of normal for the over-vote rates of other counties using this system (indeed it is a little low). We thus expect that the net effect of a recount here would be: no votes recovered.
Holmes county shows a 1.7% overall rejection rate. As in the last two paragraphs we may accept about 0.4% intended abstentions and perhaps half a percent as over-votes, leaving some 71 ballots. Assuming, as for Escambia, that this county has slightly higher rates of all of abstention, overvoting, and erroneous-undervoting, this means about 24 ballots were intended to be votes but ignored. Given Holmes's split among the candidates, this would mean that Bush would recover 16 and Gore 7 (assuming as usual a split along the county's ratios), giving Bush +9 toward the lead.
We summarize our best guesses for these counties (Undervotes = Rejected ballots minus estimated overvotes):
County Rej'd Underv Recov +Bush +Gore +Bush(net) Leon 181 0 0 0 0 0 Escambia 4372 2681 1107 685? 384? 301? Santa Rosa 365 0 0 0 0 0 Holmes 139 135 24 16 7 9 Total 5057 2816 1131 701? 391? 310?Assuming a disambiguation rate of 1/3, this means these four counties could deliver as many as 103 net votes for Bush in an actual hand recount. It is unfortunate that we have no further information about Escambia county; its presence renders this estimate highly speculative.
B. Union county votes with paper ballots. They have already disqualified 258 ballots for some reason, a rather high 6% of the total. Only 16 deliberately-blank ballots would be expected with our 0.4% standard. We have no way to judge the likely incidence of over-voting with this system, but presumably the expected 242 non-blank ballots would be found, upon a second hand examination, to be disqualified for the same reasons as they were when first counted. (Should they all, instead, be found to truly be votes cast, the county's roughly 5:3 split toward Bush would yield Bush 148, Gore 89, and hence a net shift of 59 votes towards Bush's lead.)
County Rej'd Underv Recov +Bush +Gore +Bush(net) Union 258 0 0 0 0 0
C. Next we turn to the punched-card counties not reporting undervotes (Martin, with 62570 ballots cast, and seven others with at most nine thousand votes each).
The only moderately-large county of this type is Martin county, which reported only 0.9% disqualified ballots altogether. Setting aside the usual 0.4% abstainers, the remaining 0.5% is easily explained as likely to represent over-votes. Indeed, this is the over-vote rate among the best optical systems, and well below the over-vote rates reported in most other punched-card counties. (Martin is the only county listed as using the Fidlar & Chambers "Election Management System" - DataVote type.) If it were true that this last 0.5% represented only erroneously ignored ballots, the county split would give Bush 168 votes, Gore 131, for a net Bush boost of 37.
Glades county uses the Sequoia Pacific System, Corp. "TeamWork Election Management System" punched cards (Datavote type, central tabulation), shared only by Nassau county. Nassau reports about 5.5% overvoting (and 0.7% undervotes), while Glades reports a 9.6% overall disqualification rate. It may perhaps be reasonable to subtract the usual expected 0.4% abstainers and an identical 5.5% overvoting rate for this type of machine; that leaves 3.7% of the vote unaccountably disqualified. Fortunately, Glades is small, so that this represents only 137 votes, which would be expected on county lines to split 75 - 59 for Bush and Gore, giving Bush about 16 votes net.
The remaining counties without explicit breakdowns of undervotes use the Triad Governmental Systems, Inc. "ElecTab Ballot Tabulation System" (Datavote type, central tabulation). This is used by Hardee county, which reported 4.9% overvotes, as well as Wakulla, Desoto, Madison, Jefferson, Gilchrist, and Dixie. All these are small counties. If we accept a 4.9% overvote rate as typical for this kind of system, and add the 0.4% expected intended-blank, we obtain figures as shown in the table below.
Again, it is unfortunate that we do not have explicit undervote data for these counties, and must therefore first estimate overvote rates from flimsy data, but fortunately the expectation is that these counties would produce few net votes for either candidate in a recount anyway. Here are our estimates:
County Rej'd Overv+Abst. Recov +Bush +Gore +Bush(net) Martin 577 577+ 0 0 0 0 Glades 357 205 152 75 59 16 Wakulla 422 477 0 0 0 0 Desoto 701 451 250 136 106 30 Madison 480 352 128 63 63 0 Jefferson 571 329 242 106 130 -24 Gilchrist 293 301 0 0 0 0 Dixie 332 265 67 39 26 13 Total 3713 839 419 384 35Actual recovered votes in a manual recount must be discounted as usual; Bush's lead would go up by 12.
D. Now we examine the many counties for which we have a reported number of undervotes. These are ballots not showing registering any choice for president, and consist (only) of those actually intended to be blank and those erroneously not registering. We have estimated this first group to be 0.4% in every county; however, one might legitimately argue that it is only the punched-card systems which are likely to experience this to any appreciable degree -- possibly all the undervotes in optical counties are deliberate abstentions. So we consider the optical- and punched-card counties separately.
Obviously if we assume the reported undervotes on the optical cards to be intended undervotes, we see there would be no net change in the vote totals if a hand recount were held. If instead we assume a fixed 0.4% intended-blank rate, then 16 of the 37 counties would show erroneously-blank ballots. Taking the usual step of partitioning these according to the county-wide rates, we obtain the following changes:
County +Bush +Gore Net(Bush) St. Lucie 99 118 -19 Walton 39 18 21 Calhoun 32 24 8 St. Johns 118 58 60 Bay 191 93 98 Washington 161 90 71 Baker 42 18 24 Jackson 13 10 3 Gadsden 18 36 -18 Okeechobe 21 19 2 Bradford 1 1 0 Hendry 2 1 1 Taylor 31 20 11 Gulf 13 9 4 Franklin 26 22 4 Liberty 10 8 2 Total 817 545 272This total suggests there were 272 more votes for Bush than Gore among the erroneously-undercounted ballots in this group; in a recount, we would expect an increase in Bush's lead by 91 votes.
Note that Volusia county is included in this pool. Our model predicts no votes to be recovered in a recount; this conclusion would have to be forced anyway since a hand recount has already been conducted, and further hand recounts are unlikely to register any significant change. (It is also our understanding that the Florida Supreme Court's Dec. 8 ruling did not contemplate a repeated hand recount here or elsewhere.)
Also included in this group is Lake county, with a scant 245 undervotes. A review by the Orlando Sentinel discovered among these six votes which they thought clearly intended for Bush, and another six intended for Gore. (They also reviewed the 3114 overvotes and found 376 doubly -- but invalidly -- marked for Gore and 246 so marked for Bush; if allowed in a recount this would contribute 120 net Gore votes from a Bush-leaning county! But no overvote data are included in our models above or in the sequel.)
E. As for the 17 punched-card counties, it is again true, but in this case more clearly irrelevant, that if we accept the stated undervote counts as intended blanks, then a recount would produce no net change. Taking instead the same 0.4% rate of intended blanks as always, the same process shows these changes from converting "erroneous undercounts" to candidate votes; here we segregate out the four counties with the highest disqualification totals:
County +Bush +Gore Net(Bush) Hillsboro 2033 1907 126 [Pct] Pinellas 1205 1308 -103 [Pct] Marion 1082 876 206 [Pct] Collier 1116 552 564 [Pct] Lee 726 503 223 [Pct] Pasco 571 579 -8 [Pct] Sarasota 596 522 74 [Pct] Indian Riv 492 339 153 Sumter 273 217 56 Osceola 195 209 -14 Highlands 198 139 59 Nassau 65 27 38 Hardee 35 22 13 Total 8587 7200 1387that is, we expect over 16,000 votes intended by the voters in these 13 counties were erroneously counted as "undervotes", including 451 for minor candidates and a net 1,387 more for Bush than Gore. A recount in these counties might thus be expected to add 462 Bush votes.
F. Next is one county with a very large total of disqualified ballots: Duval County. Using the same algorithm as above, we compute these numbers of recoverable ballots:
County +Bush +Gore Net(Bush) Duval 2185 1549 636With our usual conventions, this means Bush could expect a boost to his lead of about 212 votes from a recount in Duval. However, in this and the other large counties we have more data and have been able to perform a more refined analysis. We will discuss this county in some detail, in part as an illustration of the "drift" we have mentioned earlier. We will see that the Bush gain in Duval is likely to be only a fraction of what our coarse model has suggested; indeed conceivably this county could produce votes for Gore.
Duval county is interesting and, given our analysis, has been under-discussed in the media, compared with Palm Beach County. As in Palm Beach, an unusual ballot structure seems to have led to a very high number of disqualified ballots, most due to over-punching. Also as in PBC the expected type of multiple punching should make it possible to surmise, with reasonable accuracy, what was intended by the voter on most of those ballots; but as with PBC this would likely not be allowed by the courts. A New York Times article reports 4,927 underpunched ballots; our records show 4967. Of these, about 1166 would be "intended blank" by our 0.4% standard, leaving 3,801 votes to be recovered -- about half what should have been recovered in PBC by these standards. As shown above, this could be expected to represent hundreds more Bush votes than Gore votes erroneously lost, a story comparable to the loss of Gore votes in Palm Beach. (Of course, the actual manual recount in the larger and more partisan PBC only netted Gore only about this many votes, due in part to the conservative interpretation of which votes could be counted. A similar standard applied to Duval would limit its addition to Bush's lead to less than 100 votes.)
However, as noted in the New York Times article, the disqualified votes in Duval county were concentrated in Democrat-leaning precincts. We have been able to obtain some of these precinct-wide numbers. Using precisely our county-wide model on individual precincts, and keeping our 0.4% abstention rate assumption (1159 ballots), we find only three precincts with no recovered votes; the total gains for the other precincts amount to 1886 recovered votes for Bush and 1857 for Gore, a net gain of only 29 Bush votes.
This concentration of undervotes in Democratic-leaning precincts therefore accounts for a dramatic drift of the vote towards Gore, when the manual-recount recovered ballots are compared to the machine-counted ballots. This does not represent an extraordinary change in candidate strengths (we refer to a change from a 58.5%-41.5% ratio to a 50.5%-49.5% ratio as an "8 percentage point drift); it is of course because the recount operates at the margins of the vote totals that the effect seems so large.
G. Finally we turn to the three last counties: Palm Beach, Dade, and Broward. We can of course apply the same model we have previously used to these counties:
County +Bush +Gore Net(Bush) Palm Beach 3082 5436 -2354 Broward 1338 2917 -1579 MiamiDade 3765 4276 -511That is, these are potentially large reservoirs of Gore votes, which, presumably, is why the Democrats requested recounts here. These numbers suggest that 4444 net Gore votes were intended among these tens of thousands of erroneously-undercounted ballots. Applying our standard assumed disambiguation rate, we still recover 1481 Gore votes from these three counties.
However, as with Duval it is both possible and appropriate to consider these counties in detail.
We have discussed Miami-Dade county in section VII. Of the models proposed there, we might consider the estimate of 497 net votes for Gore to be most in keeping with the models we used in the other counties, and the estimate of 360 to be perhaps the best guess of the effect of what would have happened had the recounts of Dec.8 been taken to completion. (Note that both those estimates are of actual votes to be recovered, that is, unlike the previous subsections, these estimates take into account the disambiguation rate.)
Our best analysis of Broward county is simply: there are no net ballots to be found. Since a manual recount has already been performed, it is unlikely that an additional manual recount would recover more ballots (for either candidate), unless a more generous interpretation of "uncontested" ballots is used. It is our understanding that the Florida court did not intend there to be a repeat hand-recount. We do note, however, that our model's prediction of the effect of a recount (1579/3 = 526 net gain for Gore) is surprisingly close to the 567 actually observed in their recount. We consider this to be an accident :-) We have obtained data by precinct for this county, lacking votes for third-party candidates. These predict a slightly higher gain for Gore: 1304 Bush votes, 3179 Gore votes, a net gain of 1865 votes, and thus about an actual net gain of 622 votes for Gore assuming a disambiguation rate of 1/3. (The observed rate in Broward was about 0.40 -- higher than in the other counties with manual recounts.)
Finally we turn to Palm Beach County. Our model shows a 2354-vote predicted "recovery" of net Gore votes, which by our assumptions would result in 785 net votes for Gore actually being counted. We have also run a by-precinct analysis with the same model; it shows 2753 votes expected for Bush, 5522 for Gore, with a significant increase in votes for third-party candidates; this 2769-vote net gain for Gore translates into 923 net votes for Gore with our assumptions. (The precinct model suggests 10582 undervotes, not 10310 as we used in our data, which accounts for most of the difference between these two models.)
On the other hand, we observe that a conservative interpretation of the marginal votes was already used in the hand recount and resulted in a 176-vote gain for Gore. Democratic partisans have objected that some 800 net Gore votes were clearly noted but rejected by the canvassers' rubric, consistent with our model. For consistency's sake, we might continue to claim that the effect of a statewide manual recount, performed under more or less consistent conditions, would allow the remainder of the 923 predicted net Gore advance.
However, when the Florida Supreme Court reviewed Judge Sauls' decisions, it let stand his ruling that there should be no re-examination of a set of some 3300 Palm Beach County ballots which elicited Democratic concern during the recount. Therefore, the effect of a then-legal recount would be to place Palm Beach County in the same camp as Broward County: net change = zero, beyond what was actually measured. The hand recount added 176 net votes for Gore beyond what PBC reported in the official certified results (the ones giving Bush a 537-vote lead). (This number was reported as +215 but corrected to +176 for Gore on Dec. 9. [CNN]. We cannot explain the discrepancy.)
Early versions of this document noted that Gore stood to gain many votes in these three counties in a statewide recount -- roughly enough to balance the Bush gains previously and elsewhere in the state. With the disallowance of more Palm Beach votes, this balance is greatly upset.
It is interesting to note that the effect of these three counties is then: Broward,0; PBC, 176; Dade 360. That is, had the recount taken place as envisioned on Dec. 8, these three counties are predicted to have contributed approximately 536 votes toward the Gore lead -- at a time when the Bush lead was 537 !
Summary. Here then is our best statewide analysis, attempting to use uniform methods, as appropriate, for all counties, while considering the distinct influences in certain groups of them: allowing a constant 0.4% of the vote as intended by the voter to be blank, adjusting as well as possible for unmeasured over-votes, and applying the counties' existing Bush/Gore ratios to the (assumed) erroneously-blank vote, we have computed the votes intended by the voters which were not, in fact, tabulated; scaling by 1/3 to account for the numbers of them likely to be accepted legally, we have:
Net addition to Bush lead: A. 103? Optical counties without explicit undervote data (i.e., Escambia) B. 0 Paper ballot county C. 12 Punched-card counties without explicit undervote data D. 91 Counties using optical methods : unknown mechanisms for undercounting E. 462 Counties (not next 4) using punched cards : partial chad etc. F. 9 Duval G. -536 Dade, Broward and Palm Beach (Fl.S.C. Dec. 8 standard)Here lines A-F add to 677 votes for Bush; adding this to line G makes the total change be: 141 net votes for Bush, beyond his official 537-vote lead.
These numbers reflect a disambiguation rate of 1/3, comparable to the Miami and Broward rates. Any other figure may be used instead by simply scaling all totals. In particular, if the Palm Beach criteria --- under which only about 7% of the expected lost votes were recovered -- were used in the 64 counties listed in A-F, then these counties could be expected to yield only some 129 additional net votes for Bush.
So what do these numbers really mean? If the Democrats had succeeded in obtaining the recounts they requested in Dade and Palm Beach (only), our models have predicted they would have gained 497 and 923 votes, respectively, making line G read '1420', certainly sufficient to overturn the official Bush lead by our models. If in addition the court had ordered the recounts of the other 63 counties (A-F), our models there predict a statewide Gore net gain of 1420-677=743, again a very good chance to claim a solid lead. If the court had allowed only the completion of the Dade recount and the completed Palm Beach recount, Gore could reasonably have expected about 536 net votes gain, certainly putting the final balance much too close to trust our models to predict who the actual victor would be. Even if the comparatively generous recounts in Broward and Dade were allowed and completed, but the rest of the state were recounted according to the more restrictive criteria used in PBC, the Bush lead could be projected to be (537 official base)+(129 statewide)-(176 PBC)-(360 Dade) = 130 votes -- still, in our opinion, far smaller than the likely margin of error in our models, giving Gore a reasonable hope to come out ahead.
But in fact none of these scenarios stood. As we understand the intent of the court, the recount was to include the 63 un-recounted counties and the incomplete portion of Dade county manual recount, but not the contested portion of the completed PBC hand recount. Given the models so far, this appears to give Gore virtually no chance of success and, as noted above, even an expected loss of some 141 votes. Even some justifiable tinkering with the hypotheses -- assuming, for example, that hand counts of all optically-scanned ballots find no erroneously undercounted ballots, and that Escambia county came out a wash -- seem insufficient to produce a Gore victory; Bush stands to gain considerably in the counties in group (E), and Gore's chances of recovering votes in group (G) have been explicitly limited by the courts.
We concede, however, that our model has a potential flaw already shown in the Duval example: recovered undercount votes lean more heavily toward the Democrats. We can easily adjust the computations to account for a shift of votes, although we have little solid evidence of how large such a shift should be. If we assume, as happened in our Duval precinct model, that 8% of the votes expected above to be Republican are actually found to be votes for the Democrat in categories A-E, we may adjust the totals at the end of each subsection above (remembering that only one out of three missed ballots is accepted in our model); then the recount numbers would instead show
Net addition to Bush lead: A. 66 Optical counties without explicit undervote data B. 0 Paper ballot county C. -11 Punched-card counties without explicit undervote data D. 47 Counties using optical methods : unknown mechanisms for undercounting E. 4 Counties (not next 4) using punched cards : partial chad etc. F. 9 DuvalThese add up to just a 115-vote net gain for Bush, meaning that the net effect of a recount in all the other 64 counties is nearly nil; then once again it would be clear that the main question is, exactly how do we carry out the recount in the three remaining counties -- Broward, Dade, and Palm Beach. The completed Broward recount, the complete but disputed PBC recount (+176 Gore), and the incomplete Dade recount (+157 Gore after 20% of precincts) together could still be sufficient to overturn a Bush lead estimated to be 537+115 votes.
Our data to compute this drift are still limited, but it seems unlikely that a drift of this magnitude will appear. One can compute the effect of a smaller drift, since this effect is linear: the Bush gain drops by about 70 with each percentage point of recovered votes which are found to support Gore instead of the expected support of Bush.
Again, we must emphasize that the total numbers of ballots under consideration is over six million, and that we are considering changes in the two separate vote totals on the order of several tens of thousands of votes; yet the net effect on the size of the lead is at most in the low thousands (and the lead itself is even smaller). This means the margin of error in our analysis is quite large; our conclusions can easily be upset by comparatively small changes in any other large component of the vote totals. For example, we have simply guessed which fraction of the "unexpected" undervotes in Escambia county are in fact recoverable votes -- a 100-point change in either direction would hardly be surprising. Similarly, not only the absentee ballot cases in Seminole and Martin counties (affecting tens of thousands of votes) but even proposed challenges to the overseas absentee ballots (affecting perhaps one thousand ballots) take on great significance as their magnitude easily outweighs the effects we are considering here. In fact, issues which would ordinarily be of little consequence can have an effect of magnitude comparable to the changes being considered here: for example, the question of how to handle the roundoff of fractional votes in our by-precinct models can change the vote totals by hundreds of votes per county, and as in sect. II we discover that it would not be remarkable for this effect alone to alter the statewide net lead by a hundred votes or more!
Apart from actually examining the ballots, we are somewhat at a loss to develop better methods of predicting the outcome from the data which have been collected. Certainly it would be interesting to examine the precincts of some more counties in group (E). It is probably also appropriate to compare undervoting and other patterns in relation to demographic data in order to determine how it can be that undervoting is so much more common in some precincts than others using the same voting apparatus. We do not feel competent to undertake these analyses. Back to top
Our overall conclusion, then, is that the court decisions since Dec. 4 have made it rather less likely that Gore would be able to end up with a comfortable lead in Florida. On the other hand, as the news media begin their independent reviews of the ballots, it will be possible to re-examine the Palm Beach ballots in particular, making it possible for Gore to manage a statewide lead of a few hundred votes. Thus we retain the title of this section, chosen earlier in the post-election season.
So there we have it: the election is too close to call. In a state of some 12 million citizens of voting age, we are reduced to the half of them who voted, looking only at the 1% of those ballots which have not already been found to be a clear expression of a vote, retaining only about half of them as likely to have been intended to be a vote, and only a third of those as being clear enough to escape protest by one side or the other, and finally calculating that most of these are cancelled by such a vote for the other candidate. The tiny remainder could give a net change in the lead in Gore's favor by a vote total in the mid- to high-hundreds, thus reasonably allowing for the possibility of a reversal of the official lead. Unfortunately, while these fractions might be reasonable estimates of the mean values which arise in other recounts, the standard deviations of these numbers appear very large -- there is great variation even between adjacent counties. In other words, while a lead change of more than 537 votes may be more likely than a lead change of less than 537, given all our assumed values of the parameters, we find the differences between these probabilities to be low; each candidate still has excellent ground on which to claim he had a greater number of Florida votes.
Again, we stress that the conclusions we have drawn are only valid to the extent the assumptions are met: for example, that overvotes are not included in any recount; that the "dimpled" and other marginal ballots are included in the totals whenever there is reason to think the voter intended to vote; that the recovered ballots favor the Democrats at least in proportion to their standing in the rest of the county; and that there are no other adjustments to the vote totals (such as a late inclusion of more overseas absentee ballots than originally accepted). We find the lead to be so remarkably close that small changes in any of these variables would invalidate our conclusions.
The main topic in this paper has been the comparison of the tabulated Bush lead (537 votes) to the expected size of the small variation in the vote due to a particular sort of error. Meanwhile, the conclusion of this total vote of 6 million ballots is affected by many other vote tallies of a considerably larger size. For reference we mention:
|5,963,110||Valid votes for President in Florida|
|180,110||Florida votes with invalid or no vote for President|
|97,422||Florida votes for Ralph Nader|
|29,793||Threshold for vote lead necessary to trigger automatic recount|
|10,311||Votes ignored in one county (Palm Beach) for having no vote for President detected by machine|
|5,870||Votes sampled in one county (Dade - 1% sample) in order to decide whether or not to proceed with a recount|
|2,800||Votes received in Palm Beach County by Buchanan but estimated to be miscast  (approx)|
|2,202||Uncontested overseas absentee ballots (statewide total)|
|614||Precincts in Dade county alone|
|558||Votes cast in Florida for 10th place finisher James Harris (Socialist Workers Party)|
|547||Nationwide votes for independent Louie Youngkeit of Utah for President|
|538||Size of United States Electoral College|
|384||Votes added to Bush column in one county (Seminole) in automatic recount|
|250||Maximum reasonable amount of erroneous lead which can be attributed to unbiased random 1% ballot loss (est.)|
|213||People in Florida named A. Gore, G. Bush, R. Cheney, or J. Lieberman [http://people.yahoo.com]|
|537||Official Bush lead in Florida|
Clearly this was an election fought at the margins. This election is an excellent example of what should be called a statistical dead heat! Whether it is politically useful to attempt to determine a winner from the existing ballots is not at all clear to this author; perhaps there would have been less acrimony if the election were simply determined by a coin toss.
It is possible that clicking here will allow you to mail this URL to a friend (no guarantees!)
Links to further technical analyses of the Florida vote count may be found at http://madison.hss.cmu.edu/
Prof. David Rusin Director of Undergraduate Studies Department of Mathematical Sciences Northern Illinois University DeKalb, Illinois, 60115 USA Email email@example.com Web http://www.math.niu.edu/~rusin/ Telephone 1-815-753-6739 Fax 1-815-753-1112
 For the purposes of this discussion, we assume that a ballot clearly marked for Buchanan, for example, was intended to be a vote for Buchanan. It has been alleged that this was frequently untrue [http://madison.hss.cmu.edu/], but we are not able to assess such allegations with this analysis. The claim that the "butterfly" ballot used in Palm Beach County was confusing seems to this author to be difficult to believe, but a close look at the 1% sample of the ballots cast nonetheless provides anecdotal support. See, for example, the distribution of overvotes.
 The difference is assessed by comparing the machine count of ballots to a visual inspection of the ballots. We are in this document assuming that the result deduced from visual inspection is indeed the vote intended by the voter. This is somewhat problematic as the human interpretation of validity of a vote is subjective. It has been alleged that this form of error is at least as common as machine-tabulation error. [http://www.sptimes.com/News/111400/Election2000/Punch_cards_fraught_w.shtml, http://www.sptimes.com/News/111300/Election2000/Is_hand_counting_bett.shtml] For the purposes of this document we shall assume that there exists some mechanism to determine the true intended vote; we are assessing here whether or not the spread of the votes makes it reasonable to undertake that determination.
 This source of error tends to diminish on a second count as bits of chad are loosened and freed; the effect would be similar but lessened on a third count. Exactly these tendencies have usually been observed in the recounts, some multiple, done in all the Florida counties, about half of which use the punched-card system of balloting. Other sorts of mistakes could reasonably be expected to follow the same error pattern, e.g. users of marked-ovals systems who chose to circle, rather than fill in, the ovals corresponding to their choices. These mistakes would not diminish on multiple machine counts.
 For comparison we mention the lower 0.5% rate which triggers an automatic recount and the higher 2%-5% quoted by elections professionals as commonly accepted for unbiased errors in non-controversial elections. http://www.gopbi.com/partners/pbpost/epaper/editions/sunday/news_16.html http://www.gopbi.com/partners/pbpost/epaper/editions/monday/news_13.html
 One may allege that errors of these or other types of miscounts tend to strike votes for one candidate preferentially. It is difficult to assess these claims from the limited data available, but certainly a differential error rate would affect the true vote totals to be inferred from the measured data.
 Anecdotal reports suggest quite a few coins were tossed in voting booths. Historical data show that some 7% of voters decide on election day for whom they will vote.
 In 1996, some 2500 of these votes gave Dole a 15% advantage over Clinton at a time when Clinton carried the state. Pundits have explained this to be a consequence of the high number of military personnel included in this tally. On the other hand, there are also many Florida expatriates in Israel, and it has been suggested that this group may vote in large numbers and with a high preference for the Democrats this year. It is difficult to surmise the "true" contribution of this group since so many of these votes were disqualified on technical grounds.
 When the number in the last column is small, we can interpret it as follows: the probability of a random coin toss coming in this close or closer is about 0.8 times that number. When the number in the last column is about 2/3, that is, there were about (square-root-of-N)/3 more heads than tails (say), then we are not at all surprised, since we expect to end up with either more than this lead or less than this lead with about equal odds.
For those who just have to know: the vote spread after N tosses will be an even integer L; the probability that a particular integer L results is C(N,(N+L)/2)/2^N, suitably adjusted for odd N. This can be estimated as being about sqrt(2/Pi/N) exp(-L^2/2/N).
Interested readers may wish to consult a statistics book. For the next section of this document, we suggest a reading of Bayes' Theorem as the relevant tool for "going backwards" from observed to initial vote counts.
 Here is the reasoning we used before the CNN data were available to us. First, we deduce from press reports of statistician Hengartner's testimony that this number is about 1.2% statewide, but he kindly corrected this for me in email: he finds the undervote rate in optically-scanned counties is about 0.3% and in punched-card counties is about 1.5%; and that PBC's rate is about 2.2%. The Orlando Sentinel shows 59,845 blank votes statewide, just under 1% of the total vote, which is consistent with his findings (the 25 punched-card counties account for 61% of the vote; 41 counties use optical systems and contribute 39% of the vote. Tiny Union county uses paper ballots). (Nationwide, about 2% of voters fail to record a vote for president [Fed.ElectionsComm.] for whatever reason.)
The record for the optical counties seems indeed to be much less than 1%. The NYTimes reports, for example, a 0.3% no-vote rate in Orange county (which is both large and Democrat-leaning), compared to a 0.9% overall rejection rate there. We note that sixteen Florida counties are listed at the Florida state elections site as using the Global Election System's "Election System 2000" voting method, described as "Marksense; precinct and/or central tabulation". Here "precinct tabulation" implies, we believe, that ballots with multiple candidates marked per ballot are rejected in the voter's presence; therefore the fractions of ballots not counting a vote for president listed at SunSentinel are, for these counties, only those which are underpunched. For eleven of these counties, the no-vote rate is between 0.18% and 0.82%, with an average of 0.4%. The other listed rates are higher, but these may include ballots not counted for other reasons; indeed in a phone call to Columbia county we learned that these ballots were not tabulated in precinct, and indeed there were 617 overvotes and only 76 undervotes, the latter representing, again, 0.4% of the vote. Likewise Manatee reports 111 blank ballots, 0.1% of the total. (The remaining counties are Okaloosa, 1.1%; Walton, 1.2%; Calhoun, 1.5%; we do not know whether these counties use precinct ballot-checks.)
We continue to attempt to estimate the fraction of intentionally blank ballots. Comparative data from other counties must be adjusted if, for example, those others allow straight-ticket voting (as Jeff. County, AL); or if they have hotly-contested local races for which people may be more inclined to vote than for President; or if they have characteristics which might suggest a more careful electorate (county size, voter experience, party affiliation,...). Comparative data from other elections might not reflect purported high degrees of ambivalence during this Presidential race. So far, at least, we have no good data which suggests that it would be reasonable for more than 1% of the ballots to contain no attempt to express a preference for President.
 Note that these numbers are somewhat dated; for some counties recounts have already reduced the numbers of disqualified ballots, although this is of measurable size only for the 4 large counties undergoing manual recounts. Note also that these numbers do not include the disqualified overseas absentee ballots (about 1200(?)) which could reasonably expected to favor Bush by a margin similar to the accepted overseas ballots (about 2:1).
Finally, note that there are some typos in that table. Based on the numbers of disqualified ballots in AP's first column, we find that the disqualified percentages for Gadsden and Franklin counties are reversed. The numbers for Duval county are inconsistent; but another source shows the total rejection rate there is actually 26,909 votes, indeed 9.23% of the total. This brings the statewide total of disqualified ballots to a total of 180,110.
The Dec 8 Florida State Supreme Court decision gives the statewide total of rejected ballots as 177,655. Exactly which ballots are included in this total we cannot say; for example it is unclear to this author whether this includes the 434 net recount votes added to Gore's total by that court ruling.
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http://www.math.niu.edu/~rusin/uses-math/recount/index.htmlLast modified 2001/01/05 by Dave Rusin, firstname.lastname@example.org