Could the undecided voter swing Thursday’s referendum?

There has been plenty of shouting on both sides of the referendum on the U.K.'s membership of the European Union, but with Remain and Leave still extremely close in the polls, much could rest on those still making up their minds.

Most recent polls suggest 10-11 percent of those eligible to vote are still making up their minds, and the outcome of the historic referendum may hang on whether they get to the polling booth to decide.

"Who bothers to make it to the polling station is extremely important," John Curtice, professor of politics at Strathclyde University and the polling expert who was first to call the result of the 2015 general election as votes closed, told CNBC.

Dilemma confused
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"This is a referendum which most people made up their minds on months ago. For those saying 'Don't Know' in the polls, that is often a surrogate for 'Don't Care'. The proportion of people who will make it and have no idea what they are going to do is minimal, but I suspect that many don't knows have an inclination one way or the other but are still pondering is greater."

There has been a series of interventions in the debate as polling day approaches. The Remain campaign has had a succession of serious economists issuing doom-laden warnings about what could happen in the event of a U.K. exit from the EU, plus celebrities including David Beckham and Daniel Craig backing them.

However, Leave have had a clearer message to sell, of taking back control from a perceived weighty and bureaucratic EU.

"There's many a voter out there who agrees with Remain on the economy and Leave on immigration, and the implications of risk are not yet kicking in for Remain," Curtice said.

Stock markets appeared to have priced in a late swing to Remain, with a relatively calm trading day on Wednesday after turbulence earlier in the week.

Curtice, who said that the vote is too close to call, added: "Higher turnout is not necessarily good for Remain," which counteracts received wisdom that higher turnout should benefit the Remain campaign.

Peter Kellner, the former President of YouGov, told CNBC: "My hunch is that Remain is probably slightly ahead and I would expect that lead to grow in the hours that are left, but it's a judgment, it's definitely not a certainty."

"The evidence is that the undecided who might still vote will vote remain. As we got close to polling day, the people who are making up their minds late will think a lot about the economy, their own prosperity, that of their families, and veer away from the risk of Brexit."

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