Among all the accusations of scaremongering, one group of people has good reason to be fearful as EU referendum day draws near: pollsters.

Last year’s general election was a pretty traumatic experience for the market research industry. Every major polling agency failed to predict the scale of the Conservative party’s lead over Labour.

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So can we trust the numbers ahead of this month’s vote? The British Polling Council organised an academic inquiry into the general election polls, which concluded that the main problem was that the people taking part in surveys were not politically representative of the British electorate.

Pollsters have since been trying to improve the quality of their samples (the one or two thousand people they speak to each time). This is still a work in progress.

Some agencies – ComRes and Ipsos Mori being the most notable – continue to favour telephone polling for national votes. Others, such as YouGov, prefer to use their own online panels.

The different contact modes have taken on greater significance with the EU referendum, because they seem to contradict each other. Telephone polls suggest a convincing lead for remain, while online polls imply a neck-and-neck race to the finish.

Neither approach is perfect. Cold-call scammers have ruined telephone response rates for legitimate market research calls, while online panels tend to attract people with strongly held views who seek out surveys as a means of voicing them.

Survey research is, and always has been, a blunt instrument. It tries to distil complex thought processes into simple, multiple-choice questions, and sacrifices some nuance in the process.

More subtlety is lost when a poll is turned into a newspaper headline. We don’t know exactly what impact the reporting of polls has on voter behaviour, but it can certainly colour a campaign. The powerful “vote Labour, get SNP” attack at the last election depended on the notion, based on the polls, that the SNP would be kingmakers in a hung parliament.

Instead of getting too caught up in the horse race, it is important to look at the context behind the data.

The remain campaigners enjoy the natural benefit of advocating the status quo (“better the devil you know”). Their coalition of Conservative Europhiles, most of the Labour party, the Lib Dems, the Greens, the SNP, Plaid Cymru, international leaders, economists and businesses is much broader than Vote Leave’s.

Polls tend to understate support for incumbents and status quo options. We saw this at the last two general elections, at the Scottish independence referendum, and at the alternative vote referendum.

The requirement for balance in the broadcast media can give the misleading impression that 50% of voters are in favour of EU membership and 50% against. Even the press, which can be as biased as they like, revel in the sport of a close contest.

Many voters say they are undecided. The pattern in similar sovereignty referendums, in places such as Scotland and Quebec, has been for the side calling for change to struggle to win over any undecided voters in the final weeks of the campaign.

The implication is that when faced with a “stick or twist?” situation, voters’ default position is to stick – and this is supported by findings in psychology and behavioural economics.

When looked at from this perspective, there is a crucial similarity between the online and telephone polls. In both cases, leave averages about 40-45%, and there is little evidence as yet that they can cross the magic 50% line.

Some have argued that leave’s higher levels of passion mean they can win as remain voters stay at home. In fact, a large chunk of leave’s support base is those groups who are historically least likely to vote: the less affluent, the less educated, and the less politically engaged.

Turnout at the Scottish independence referendum was an enormous 85%. Predictions of paltry turnout perhaps underestimate the importance most British adults attach to this vote.

Comparing the polls with previous referendums and setting them against the backdrop of David Cameron’s success at 2015’s general election, history says remain should win comfortably.

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Brussels, immigrants, sovereignty has not been a winning trifecta at any previous British election.

However, the landscape can change. Cameron’s popularity ratings have begun to fall, and his relationship with middle England is going through some of the same troubles that Tony Blair experienced at the same point in his premiership. We know that remain cannot win without attracting Conservative voters, and the prime minister was thought to be the campaign’s best chance of ensuring that happened.

Among the pollsters themselves, the hyper-confidence that preceded last year’s election has mostly given way to cautious humility. They know they could still get it wrong.