Differences between choices in polling

Main Question or Discussion Point

I'm a bit of a polling junkie, and in general, I'm pretty good at math, but I can't figure this one out (at least, not yet).

As a concrete example, I'm going to use the latest Gallup survey:

Obama: 49%.
Romney: 44%.
Margin of error (95% confidence) = 2%.

As I've found out, the margin of error applies to each person individually, so Obama's share will be between 47% and 51% in 95 out of 100 cases, and Romney's share will be between 42% and 46% in 95 out of 100 cases.

I'm trying to find a formula that will give me the probability that one candidate (say Obama) is ahead of the other candidate. Each candidate's share of the vote is the peak of a normal distribution.

Now, I believe the following represents the probability that each candidate, individually, has at least x% of the vote:

$P = \frac{1}{2} (1-erf (\frac{x-\mu}{\frac{MOE}{2} \sqrt{2}}))$

With
$\mu =$ The percentage in the poll for the candidate.
MOE = Stated margin of error.

I don't know if that helps, but that's what I have so far.

Basically, I'm trying to find a function that describes the probability that one candidate is ahead of the other by any amount as a function of (the difference between the two candidates) and (the margin of error).

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As soon as I posted, I thought of a new way that seems to give somewhat reasonable answers, although they seem a bit off.

Because I'm not totally familiar with using Latex on this forum, here's a link to WolframAlpha Online to what I'm thinking: here.

D = Difference
M = Margin of error

[Disclaimer: I got into the same trouble recently in that horse racing thread. I have a bad habit of pointing out the flaws in these conceptual models rather than saying anything useful about the math. So stop reading now if you only want to read comments about the math.]

National polls are worse than useless. This is the most misleading thing you can possibly do if you're trying to figure out how the US presidential election is going to go.

The outcome is determined by electoral votes; and popular vote means nothing. The election is only being contested in a handful of battleground states. I live in California and I rarely see any ads. Obama and Romney both show up from time to time to hold private fundraisers for well-heeled contributors. Then they fly back to actually campaign in states that matter. My vote doesn't count. I can vote for one or the other or a third party or I can stay home. It simply makes no difference. California goes to Obama no matter what I do.

So if 100% of Californians vote for Obama, your national poll gets incorrectly skewed.

Ok so much for politics. I will make one moderately mathematical remark. A lot of people hate the electoral system but actually it's a brilliant smoothing mechanism that reveals national consensus in ways the popular vote doesn't. In 1992 Bill Clinton handily won the electoral vote while getting only 43% of the popular vote. [Third-party candidate Ross Perot took 19 million votes, mostly from Bush] The electoral vote revealed that Clinton was the most popular candidate across a wide geographical spectrum of the country.

The popular vote is susceptible to geographical anomolies: an extremist candidate hugely popular in only one part of the country. People who favor the abolition of the electoral college should really give this some more thought. The electoral system is brilliant and prevents a lot of extremist disasters (in both directions).

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I disagree that national polls are worse than useless. They are very useful. National polls can tell you how states will poll and state polls will tell you how the national polls will go. It's virtually impossible to win the electoral college this year in any realistic way if you lose the popular vote by 5%. It's extremely difficult even when you lose the popular vote by 2%. The two are highly correlated.

Regardless, the question remains the same for state polls and Senate races.