Paulos: Statistical Ties and Coin Flips

The Florida election is essentially a statistical tie.

As I’ve written in a previous column, the number of votes in dispute in this election is many times greater than the difference in vote totals between the candidates and so the election might reasonably be settled by a coin flip.

Since such a decisive coin flip is unlikely to occur, we should ideally avoid the appearance of dredging for votes in Democratic counties (although they are the source of almost all the reported irregularities) and hand-count all the counties.

Still, given the closeness of the election and the margins of error and interpretation involved, any recount, even a careful manual one, of the entire state would be more or less tantamount to flipping a coin anyway.

The Buchanan Factor

To illustrate the iffy nature of the outcome, let me examine — after a short statistical detour — one of the continuing points of contention in the post-election campaign: the Buchanan vote in Palm Beach County.

There is a standard approach that statisticians use to understand the relationship between two variables. Take, for example, the heights and weights of people in one group or other. For each person in the group one plots a point on a graph indicating his or her weight (on the vertical axis, say) and height (on the horizontal axis).

Using mathematical techniques that go by the name of regression analysis, one can find and draw the best-fitting straight line through these points. As common sense suggests, we would note that there is a positive relationship between the weight and height of people: the taller someone is, the heavier he or she generally is. There will, of course, be some “outliers,” very tall, light people or short, heavy ones, but these exceptions are unlikely to be extreme.

How is this relevant to the election? Since the vote totals for the candidates in each of state’s 67 counties are readily available, we can examine the relationship between the number of votes Reform Party candidate Patrick Buchanan received in a county and the number that Gov. George W. Bush received in that county by following the same procedure.

For each county in Florida we plot a point on a graph indicating the Buchanan vote (on the vertical axis) and the Bush vote (on the horizontal axis).

Applying the tools of regression analysis, we find and draw a line of best fit through the data and note that there is exactly one extreme outlier: Palm Beach County. It is so far away from the general drift of the data that it’s somewhat analogous to finding a 700-pound person who is 5 feet 6 inches tall in a group of 67 people.

We can also find the regression lines for the Buchanan vote vs. the Gore vote or for the Buchanan vote vs. the total vote, and again we would find that Palm Beach is the only extreme outlier. If our assumptions are correct and Buchanan’s vote in the other 66 Florida counties is any guide, his vote total in Palm Beach is statistically quite extraordinary.

Furthermore, we can estimate with confidence that his total there was approximately 2,000 to 3,000 votes more than it should have been and deprived Gore of enough votes to throw the election to Bush.

Two Kinds of Butterflies

The reason for the excessive Buchanan totals is no doubt the confusing “butterfly” ballot. Amusingly, it gives us a new illustration of the appropriately termed “butterfly effect” in chaos theory.

The term refers to the way a butterfly flapping its wings in Central America can lead to a snowstorm in New York a few months later. More generally, any small change occurring in some quantity can cascade toward a hugely disproportionate consequence down the road. In this case the consequence is the identity of the next president of the United States.

Rather than examining other disputed vote totals mathematically, I’ll end by recalling a classic problem in the history of probability that is relevant to the problem of what to do in indeterminate situations.

Proceeding Under Uncertainty

Consider this scenario: Two men, A and B, bet on a series of coin flips. They agree that the first one to win 6 such flips will be awarded $64,000. The game, however, is interrupted after only 8 flips with A leading B 5 to 3. The question is how should the $64,000 pot be divided?

One argument might be that A should be awarded the full $64,000 since the bet ought to be all or nothing and he was leading. Note the non-mathematical “ought” here.

Another tack might be to maintain instead that A should receive 5/8 of the pot, $40,000, and B the remaining 3/8, $24,000, since the score was 5 to 3 when the coin flips ended. Proportional division has a long tradition.

One could also reason that neither A nor B won the game so the pot ought to be split evenly with each receiving $32,000. Another “ought” with a long history.

Yet another approach involves the probability that A or B would have gone on to win. The likelihood that B would have won had the flipping continued is 1/8 since he would have needed to win three consecutive flips, and the probability of this is 1/8. Thus B should receive 1/8 of the pot or $8,000. And since the likelihood that A would have won is 7/8, he should receive the remaining $56,000.

Settling on the Rules

Other divisions are possible as well. (The above problem would be further complicated if, like the role of Florida in the presidential election, the coin-flipping were part of another larger game that would result in a complete victory for B if he received any of the $64,000 pot.)

The point is the criteria for these divisions of the $64,000 pot are not mathematical!

Unfortunately in both the Florida election and this little example, the criteria for deciding the issue are not at all obvious. Flipping a coin, dividing things equally or proportionally, hiring small armies of attorneys, organizing a foot race or perhaps a vocabulary competition between the candidates — none of these options is specified in election law, which is, by turns, silent, clear, irrelevant, and contradictory.

Mathematics may help determine the consequences of our assumptions and values. But we, not some mathematical divinity, are the source of these assumptions and values.

Professor of mathematics at Temple University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on appears on the first day of every month.