More Margin of Error buffoonery

It's not that this really makes any difference, but since we're obsessing over how to interpret a poll's margin of error, we may as well analyze this passage from a report on a new poll of Missouri's Senate race:

Among 800 likely voters polled last week, 49 percent supported state Auditor Claire McCaskill, a Democrat, in her quest to replace U.S. Sen. Jim Talent, R-Mo. That compared to 43 percent who favored retaining Talent, who has been in office since 2003. The remaining 8 percent were undecided.

McCaskill's edge falls within the poll's margin of error of 3.5 percentage points, which means that any individual number could be that much higher or lower.

Set aside the already expressed quibble over the idea that a margin of error "means that any individual number could be that much higher or lower." It doesn't quite mean that, it gives a sense of the range by which you are likely to be off.

The beef I have now is the idea that the fact that the two margins overlap means that they could actually be tied. And that isn't really wrong, but it's the wrong way to examine it. The wikipedia entry on margins of error gives a handy table of the probability that the poll correctly identified the true leading candidate given a particular margin of error and a particular spread between the candidates. The probability that McCaskill is really ahead based on this poll is somewhere between 93% and 97.5%. The probability that Talent was really ahead in the poll the Star reported was somewhere between 94.9% and 89%. Put those together, and the probability favors McCaskill's lead, though only slightly. The Star's poll that put Talent ahead uses a methodology that is questionable, and the trend has favored McCaskill, but in the end, that race is too close to call.

I will say that McCaskill has been a class act, asking people not to respond in kind to Republican smear tactics. Drop by her website if you like and show her that nice gals do finish first.