Poll numbers

Reuters today reported: "Democratic challenger John Kerry expanded his slight lead over President Bush to three points in a tight race for the White House, according to a Reuters/Zogby poll released on Monday." For the next few weeks, we will be bombarded with poll numbers reporting each candidate's position in advance of the election. A few things must be kept in mind whenever one reads poll numbers.

First off, all polls should report some margin of error. For the poll referenced above, the error is 2.9 percentage points. This number is derived by simply taking the square root of the number of participants in the poll, divided by the number of participants. There were 1,214 participants, so the error is (√1214)/1214 or 0.029 which is 2.9 percentage points. This calculation is valid on the assumption that the number of participants is large and the sample is truly random. Therefore, when the poll reports Kerry at 47%, they really mean 47±2.9%.

What does the error bar mean? The 47±3 (rounding for simplicity) means his placement is somewhere between 44% and 50%. Really, it means there is a 67% confidence that, if the election were held today, Kerry would poll somewhere in this range. It also means there is a 33% chance he would poll outside the range. Doubling the error bar to 6% means there is a 75% chance Kerry would poll somewhere between 41% and 53%.

Now, of greater interest to both voters and candidates than Kerry's number or Bush's number is the difference between them. This poll reports Kerry ahead by three points. Above, we've discussed how the 3% error bar on each number is derived. What is the error bar on the difference? That is found by taking √2 times the error bar on each number, for a result of 4.1%.

So, the poll is telling us that, if the election were held today, there is a 67% confidence Kerry would place somewhere between 1.1% behind Bush and 7.1% ahead of Bush and a 75% confidence he would place somewhere between 5.2% behind and 11.2% ahead of Bush.