Behind the headline polling data, you sometimes find a real story. YouGov produces a daily poll on behalf of The Sun and Sunday Times newspapers. In recent times the poll has been remarkably steady – boring even – with Labour’s support remarkably solid, at around 40%, and the Conservatives just a few points behind it. But YouGov polls for more than just voting intention. And, occasionally, it asks a question that reveals a deeper truth about the electorate. That happened this week when it tested attitudes to gay marriage.

The surprise was not that a plurality (43%) support the proposal; nor that a strident minority (15%) oppose both it and the existing civil partnerships. The shock was that though David Cameron has openly said he thinks it right that gay marriage be allowed – a large majority of the public do not believe him. A whopping 63% of those polled agreed with this statement: “he does not believe it is right, but is doing it for political reasons.” I found this extraordinary – and depressing. And I wasn’t the only one.

Peter Kellner, the president of YouGov, was so upset by the findings he felt moved to blog on the subject. Kellner, a dyed-in-the-wool Labour supporter who is married to the Labour peer Baroness Ashton, found himself defending the Prime Minister. “For what it’s worth,” wrote Kellner, “my personal view – as someone who has never voted Conservative and can’t imagine ever doing so – is that the Prime Minister is absolutely sincere in supporting reform, and that it is not only wrong but ridiculous to suggest that he is motivated by the (probably non-existent) electoral advantages of speaking as he has. Yet despite the absence of any hard evidence that he is acting cynically on this occasion, most people, including most Tories (58%), appear simply to assume that political calculation trumps all. This is bad news, not only for Mr Cameron, and indeed not only for the Conservatives, but for the reputation of our political system.”

As many managers and leaders will know, coming across as sincere is crucial. When you are being sincere, yet nobody believes you, you are in a hugely frustrating position.

So how to overcome it? Normally I’d invoke the old journalistic adage: “tell them, tell them again and, if in doubt, tell them a third time”. But, in this case, the unfortunate Prime Minister may have to tell them many more times than three. Even repeating it via loudhailer atop a Gay Pride float may not be enough to convince what is a depressingly cynical electorate.