Q: Was Brexit’s 50% threshold too low?

In the Montenegrin independence referendum, 2006 the “Leave” option had to win at least 55% to win and leave the Serbia & Montenegro federation. Was such non-50% threshold good option for Brexit as well?

Believe it or not, this is a statistical question. It’s also a position which comically, a die hard, right wing Brexiteer petitioned the government on before the vote, on the assumption that it would go Remain’s way and wanted it to be declared null and void because it didn’t meet a threshold. I don’t see him being declared a numpty or a hero by Brexiteers, though technically, from a statistical perspective he’s on to something.

Meet the Leave campaigner who set up the petition for a second EU referendum

“William Oliver Healey says his campaign has now been ‘hijacked’ by remain voters”

Genuinely couldn’t make it up :-D

Anyway, I’m going to concentrate on the stats reasoning, as others can do the Brexit politics stuff without me ranting about it.

Statistical Confidence Intervals

Take a look at this graph.

This is the graph of 10 coins being flipped and the combinations (not permutations) being graphed. For example, at 3, it shows any 3 heads out of the ten coins, anywhere.

For example, HHTHTTTTTT is 3 heads and TTHHHTTTTT is also 3 heads (there are more). They count as two items in the height of the graph.

Now, the peak at 5 tells us that flipping 10 coins will most likely result in 5 of them being heads (and 5 being tales).

However, it won’t always be like that. Sometimes, you’ll get 4 heads, other times, you’ll get 4 tails and less of the times. Also, as you reduce the number of heads you can expect, the probability tails off, which is also what the above graph tells you. Hence, you’ll get only 2 heads less than 5% of the time and 5 heads, 25% of the time.

Now, confidence intervals aim to help statisticians understand whether a given result could have been expected, or another way of putting it, is whether the result is significant or not. So flipping 10 coins and getting 4 heads is relatively speaking, still quite likely (20% compared to the 25% of 5 coins) but at the same time, the more you go to the extremes, the lower the probability becomes and the less confident you can be that the result is the same as chance. Hence, the effect would not be regarded as significant (i.e. it isn’t an outlier) if it falls within the confidence interval chosen.

There are formula to choose confidence intervals, but usually most researchers worth their salt will set them at 90%, 95% or 99%. The aim is to find significant differences in the vote to categorically state that the result is definitely what people want “The will of the people”. Without it, you basically have a vote which is no better than chance. That is why other countries choose 55% or 60% as the referendum thresholds. and that is being kind. It should approach the 2 out of 10 position to be declared in any way definitive, but we accept 60% and that is considered democratically reliable but it’s up to the country to decide the level.


With the Brexit referendum, this was a simple in, or out, question. So all it had to do was beat a flip of a coin’s confidence interval. That means it had to reach the 2 out of 10 head or tail position at a 90% (that is 80% either way) or even if we accept the 55% level.

Even as you expand the number of heads Individual people being asked in or out without full understanding of the facts will accumulate like head or tail decisions — each choosing a head (Leave) or a tail (Remain)

(number of heads on a 100 coin flip)

I mean think about it. Round 52% you get a 50:50. It doesn’t even register on the 10 coin graph as a difference and even if you use the 100 coin flip graph, to get the percentage points in, we’re going down from a 8% point to a 7.5% point. 0.5%? This isn’t a significant change and it only gets smaller when the numbers get bigger (towards the 34 million coin flips) relative to the size of the voting populous. It’s totally unsounds and only cataclysmic idiots could take action on a chance result.

Hence, from a statistical perspective, it was handled like an absolute shambles! The most amateur way you could have counted the vote, but it is a reflection of the lack of mathematical and statistical literacy throughout the country, including both the voters and within the civil service. I mean think about this. Imagine the EU Referendum was won by one vote. In two years time, some of the voters will have died. Given the older generation made up more of the Leave vote, this would more likely come from Leave. Hence, you could end up with 2 deaths, both from Leave (there is a confidence interval for that too) and suddenly, by 2019, the majority of people who voted, voted to remain. Yet another reason why the referendum being run the way it was, was a shambles.

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