Some statistical musings on polls and the FPTP system

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Let's say (hypothetically, of course), in a fictitious country S, that Candidates A and B are in the midst of a really tight electoral race run on the First Past The Post (FPTP) system. The news announces that 60% of the votes have already been counted, and Candidate B leads A by a margin of 2%. Total electorate size = 2 million, meaning that 1.2 million of the votes have already been counted.

 

There are two other Candidates C and D, who collectively are only going to garner something like 30% of the votes, with 25% going to C and 5% going to D.

 

So, at the point of the news announcement, the breakdown can be presumed to be something very like:

 

A: 34% (408,000)

 

B: 36% (432,000)

 

C: 25% (300,000)

 

D: 5% (60,000)

 

Total: 100% (1.2 million)

 

Let's focus on A and B. Now, with 1.2 million votes already counted, basic statistics will allow a very precise estimate of the ultimate victor (for example, by the techniques detailed in Section 2 of the attachment (I can't link it, for some reason). Cutting out the math (I can show it if desired):

 

The standard error for the difference in proportions between B and A is about 0.074%. That means that the "true" difference between the proportional difference between B and A lies between 1.85% and 2.14% with a 95% probability. This is called a 95% confidence interval. That means that it's 95% probable that B would've beaten A by a margin of at least 1.85% even if an "infinite" number of votes were polled.

 

In fact, the 99.9% confidence interval (1.76% - 2.24%) is still very convincing for a victory for B over A. That means there's less than a one-in-a-thousand chance that B's victory margin over A is going to be below 1.76%.

 

It can really only be called a "wash" (unpredictable result) if the confidence interval of the difference between the polled proportions encompasses zero. That would take more than 26 standard errors. Want to know the probability of that? There are too many zeroes to bother with - to all intents and purposes, it's a statistical "impossibility".

 

In other words, at the point where 60% of the 2 million votes had been counted, and leading by a 2% (in fact, a "2-plus percent") margin, B should've been celebrating with a very strong assurance of ultimate victory. Yet, it was the *reverse* that actually happened in our hypothetical example, and A won.

 

The only way in which such an "upset" could've legitimately happened is if the initial sample of 1.2 million (!) was a biased one (B over A). In other words, if the voting preference of the remaining 40% of the electorate was strongly biased the other way (A over B). This is possible. But considering that this was a national election, and the candidates were widely regarded as "non-partisan", this is hardly probable.

 

I'll leave you to work out the implications of all this on our little hypothetical example. As it turns out, Candidate D in our little story was, coincidentally, a statistician. And Candidate A held a doctorate in Applied Math. I wonder if they found something privately humourous in all this.

 

It later emerged that the margin of victory was quite a bit less than the votes declared invalid. Assuming (naturally) that our hypothetical vote was conducted with the utmost attention to propriety, it seems a little strange that one could declare a "clear victor" with such a narrow margin, such that even minor perturbations in the subjective decision process of deciding which votes were valid could've reversed the result. If nothing else, this should serve as a cautionary tale against keeping the FPTP system, and this is why some other democracies have done away with such a system and adopted something fairer.

 

MOEFranklin.pdfUnavailable

Edited September 4, 2011 by Turboflat4

[Lethalstrike]

Interesting.

 

But again, we never know if some of the votes for B has been purposely declared as spoilt votes after the recount, which ultimately makes A the final winner by a margin of less than 1%. Such circumstances and outcomes are not a surprise given the control of the Prime Minister's Office over electoral departments.

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  On 9/4/2011 at 4:43 AM, Lethalstrike said:

Interesting.

 

But again, we never know if some of the votes for B has been purposely declared as spoilt votes after the recount, which ultimately makes A the final winner by a margin of less than 1%. Such circumstances and outcomes are not a surprise given the control of the Prime Minister's Office over electoral departments.

 

For me, the line in the sand is the fair conduct of the actual election process. Be as dirty as you want pre-election, but make sure the election itself is carried out with absolute rectitude and secrecy. Once that line is crossed, god help us all, because we're on the slippery slope to becoming an all-out dictatorship.

 

Of course, I'm still talking hypothetically, about our fictitious little country "S".

Edited September 4, 2011 by Turboflat4