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.

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