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There's a lot of confusion about MOE and polling.  People using words like 'tied' when the difference is within the MOE, or saying the difference is only significant if it's bigger than the MOE, or bigger than two MOE, or all sorts of things.  But, what we're really interested in is not the MOE.  It's who's going to win, and, in non-winner-take-all races (like the primaries) by how much.  

We can estimate this for any combination of percentages and sample sizes.

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When we're estimating two proportions, it's called a binomial, and if it's more than two proportions, a multinomial.  If we look at, say, Obama and Clinton, then all the polls have an 'undecided'; some have other, as well. We can make this three categories: Obama, Clinton, Other. A recent Gallup poll had Obama 45, Clinton 44, leaving 11 for other.  I don't see the sample size, but let's guess it was 500.  So, let's simulate 100,000 replications of a multinomial of 500 people with those proportions.

What do we care about?  Who wins.  Well, Obama is ahead of Clinton 58.95% of the time; Clinton is ahead of Obama 39.58% and they get exactly equal numbers 1.83% of the time.  Since, in the real election, there are so many voters that a tie is almost impossible, lets assign those 1.8% in proportion, and find that Obama wins 59.8% of the time and Clinton 40.2% of the time.

Now, how does sample size affect this? Well, suppose the same results were based on samples of 300.  Then we simulate Obama winning 56.36% of the time, Clinton 41.24%, and a tie 2.40%.  Let's do some others.  Remember, all these are for results of Obama 45, Clinton 44, other 11

                      Obama wins         Clinton wins        tie
100                      52.06%              44.05%            3.89%
300                      56.36               41.24            2.40
500                      58.95               39.58            1.83
1000                     62.48               36.26            1.25

We also care about the margin of victory.  Let's say we're curious about whether either would win with a margin of 5% or more.

                     Obama + 5 or more     Clinton +5 or more
100                        32.5%                     25.4%
300                        22.5                      13.0
500                        16.9                       7.7
1000                        8.8                       2.1

Notice how, with larger samples, the chances of being way off decline.

I can give you these for any combination of sample size and polling results.  

Of course, that all assumes the polls are perfectly done.  (Yeah, like that's gonna happen!

Originally posted to plf515 on Wed Feb 13, 2008 at 05:11 PM PST.

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