By Michael T. Brett
Department of Civil and Environmental Engineering
University of Washington
One of the most troublesome aspects of the November 7th presidential election is the possibility that the results of the popular vote in Florida and the Electoral College vote nationally may have been tainted by a poorly designed ballot in Palm Beach county. In this typically heavily Democratic, and Jewish retiree and African American immigrant dominated county, Reform Party presidential candidate Pat Buchanan received 3407 votes. Democrat operatives have argued this result is highly unlikely because many Jewish and African American voters consider Buchananís views to be anti-Israel and anti-minority, respectively. Republican operatives have countered that it is conceivable that this county could have voted heavily for Buchanan. The claims of Democratic and Republican party activists can be easily tested using simple statistical analyses.
Using the total votes cast as well as the proportion of Democratic leaning voters (Gore supporters) by county it is possible to predict the expected number of Buchanan votes in each county of the State of Florida using the presidential election results. [I obtained Florida State county wide voting results from the CNN News web page]. I used a two variable multivariate regression analysis, and obtained a final result of Log(Buchanan Votes) = -0.9845 + Log(Total Votes)*0.7461 - (%Gore Votes)*0.007013. The predicted votes from this model have an uncertainty (in statistical terms "error") of ± 0.1869 (± 1 residual error) in Log10 units. This statistical analysis had an r2 value of 0.87 meaning the fitted "model" can explain 87% of the observed outcome in the actual election results, and was statistically significant at the highest level reported in any computer statistical package (P > 0.0001 or less than 1 chance in 10,000 that that these results could have occurred due to chance). The Log transformed number of votes in each county explained 85% of the total results, while the %Gore voters only explained 2% of the Buchanan voting outcome; however both factors were statistically significant. Because Palm Beach county had 431,836 voters of which 65.28% voted for Gore, this model predicts Buchanan should have obtained 607 votes when in fact he actually obtained 3,407, or a difference of 2,800 votes. The difference between the expected votes and the actual votes was significant (student t-test) at the 1 in 10,000 level. Furthermore, the results of these analyses show the 99% confidence interval for the "expected" Buchanan vote in Palm Beach County was 193 to 1908 votes, meaning at the 99% confidence level, Buchanan received 1499 votes more than could be reasonably expected due to random chance. This confidence interval was calculated as log(expected Buchanan votes) ± 0.1869*2.66 = 2.783 ± 0.4971. The value 2.66 is the student t-value at the two-tailed 0.01 confidence level for 60 degrees of freedom. To obtain values in non-Log10 units, the anti-Log of these values is taken accordingly: the lower 99% confidence interval equals 102.2862 = 193 votes, the upper confidence interval equals 103.2805 = 1908 votes.
|Results of Statistical Analayses|
|Count:||R:||R-squared:||Adj. R-squared:||RMS Residual:|
|Analysis of Variance Table|
|Source||DF:||Sum Squares:||Mean Square:||F-test:||Probability:|
|Beta Coefficient Table|
|Variable:||Coefficient:||Std. Err.:||Std. Coeff.:||t-Value:||Probability:|
This statistical analysis contains no partisan slight-of-hand. Any person capable of passing a university sophomore level statistics course and who follows these classic steps would obtain exactly the same result. This analysis shows that while Palm Beach county is the second most strongly Democratic large county in Florida, it is not its Democratic electorate that makes the strong vote for Buchanan exceedingly unlikely. Since the results of the expected Buchanan votes statistical model are strongly dependent on the total number of voters in each county, in a probabilistic sense the fact that Buchanan only polled 0.3% statewide and never obtained as much as 1/3 as many votes in any other county points to the exceedingly unlikely result for Palm Beach County. This analysis also strongly suggests that an electoral mistake on the order of 1500 votes occurred in this county. As has been pointed out repeatedly in the media, this over-vote for Buchanan may be due to the confusing structure of the presidential ballot used in Palm Beach county. This margin is more than large enough to turn the popular vote in the State of Florida as well as the Electoral College vote for the entire election.
This analysis did not consider the 19,000 Palm Beach county ballots that were discarded because voters selected two candidates for president. It would be very simple and informative to conduct an analogous statistical analysis of the proportion of spoiled ballots selecting Buchanan and Gore in Palm Beach county, and compare that to the numbers of ballots selecting both Buchanan and Gore in other counties in Florida.