Britain’s vote to leave the European Union has prompted a great deal of speculation about whether the outcome really does represent the will of the entire electorate. Citizens and commentators have asserted that the result may well have been different had various groups of potential voters gone to the polls in greater numbers. If only more … (fill in your choice of young people, ethnic minorities, Londoners, Scots, university graduates, etc.) had voted, then Remain would have won. At least that’s the argument.

But why restrict this discussion about increased turnout to specific groups? Let’s dare to be democratic. What would have happened if all eligible voters had exercised their franchise by casting a vote in the referendum? Would Britain have voted for Brexit or would the country instead have opted to remain in the EU?

To answer this question, we need to put our nerd hats on for a moment. We need to use data gathered in our Essex Continuous Monitoring EU referendum survey.

As we already know, Leave won 51.89% of the referendum vote and Remain won 48.11%. The Electoral Commission reports that the overall turnout was 72.21%. These figures imply that 34.73% of the entire electorate voted to Remain. But what about the people who did not take part in the vote?

A question in our post-referendum survey asked people who did not vote how they would have voted had they gone to the polls. It turns out that 39.1% would have voted remain. Given that the Electoral Commission’s records indicate that 27.79% of eligible voters didn’t turn out, this would have given an additional 10.87% points to Remain (27.79 x .391).

But the story does not end there. Another 32.2% of the respondents in our survey who did not vote said, after the referendum, they didn’t know how they would have voted. This amounts to 8.95% of the entire electorate (27.79 x .322).

To determine how these people would have voted, we use a question in the pre-referendum survey (conducted on June 19 and 20, just a few days before the event) which asked them how they were going to vote. Of those who didn’t know, 53.1% reported after the referendum survey that they opted for Remain. Using this number to estimate how many of the 8.95% of the electorate would have voted Remain suggests 4.75% (8.95 x .531) would have done so.

Now, if we combine these calculations (34.73 + 10.87 + 4.75) then we are left with the finding that if everybody had voted at the referendum then 50.35% would have voted Remain. That’s only a narrow win, but still a different result from that which emerged in reality.

## A million referendums

Remainers, however, should not get too excited. This figure is still not conclusive evidence that Remain has majority support across the electorate as a whole. Rather, the 50.35% result is only an estimate of Remain’s strength and one that fails to account for the uncertainty in the survey data which are drawn from a sample of eligible voters.

As always, it is important to respect sampling uncertainty in survey data. To do so, we compute a standard 95% confidence interval or an “uncertainty boundary” which tells us how varied the results would have to be in order to be 95% sure that the actual outcome would be inside the boundary. Our calculations suggest that Remain’s strength in the electorate would have varied from 48.65% to 52.05%. So, even if everyone had gone to the polls, Remain could still have lost.

How likely would a Remain loss have been? Although we cannot be certain what would have happened if everyone had voted, we can gain additional insights into the likelihood of a Remain victory.

Keep your nerd hat on and imagine conducting many (say, a million) referendums with a random component distributed about a mean of 50.35% with a standard deviation of 0.85% (a measure of how variable our survey estimates were of Remain’s strength).

Assuming a normal distribution for these contests, Remain’s total is greater than Leave’s in 66.03% of the million referendums. Leave wins 33.97% of them. So, had everyone voted then the odds of a Remain victory would have been substantial but not overwhelming (1.94 to one).

Of course, UK voters did not have a million chances to vote to stay in the EU. They had one, and a majority of those who cast a ballot opted to leave. Brexit may not reflect the sentiment of the entire electorate but the result of the referendum reflects how democracy works. This is a longstanding constitutional principle and it was honoured on June 23. If you don’t participate, your voice is not heard.

*This article also appears on the UK in A Changing Europe blog*