In the past three weeks, readers of mainstream UK newspapers have learned a number of things about the UK social security system and those who rely on it. They have learned that 878,000 claimants have left employment and support allowance (ESA) to avoid a tough new medical assessment; that thousands have rushed to make claims for disability living allowance (DLA) before a new, more rigorous, assessment is put in place; and that one in four of those set to be affected by the government's benefit cap have moved into work in response to the policy. These stories have a number of things in common. Each is based on an official statistic. Each tells us about how claimants have responded to welfare policy changes. Each includes a statement from a member of the government. And each is demonstrably inaccurate.

When we say inaccurate, we are choosing our words carefully. Politicians are inevitably selective in the data they choose to publicise, picking the figures that best suit whatever story they want to tell. This can mean that stories that are technically accurate can nonetheless be potentially misleading. Within reasonable limits that is in itself neither improper nor unethical: indeed, it is virtually unavoidable. But here are some examples that are not just misleading: they assert that official government statistics say things they do not.

First, the claim that "more than a third [878,000] of people who were on incapacity benefit [who] dropped their claims rather than complete a medical assessment, according to government figures. A massive 878,300 chose not to be checked for their fitness to work [our italics]." For the Conservative party chairman, Grant Shapps, the figures "demonstrate how the welfare system was broken under Labour and why our reforms are so important".

In fact, every month, of the roughly 43,000 people who leave ESA, about 20,000 have not yet undergone a work capability assessment (WCA); a number that over four years or so adds up to the headline 878,000. There is no mystery about this: there is an inevitable gap between applying for the benefit and undertaking the WCA. During that time, many people will see an improvement in their condition and/or will return to work (whether or not their condition improves). DWP research has shown that overwhelmingly these factors explain why people drop their claims before the WCA; it also showed that it was extremely rare for claimants not to attend a WCA. In stating, in effect, that official figures showed the opposite of this, the story was simply wrong.

Iain Duncan Smith's assertion about a surge in DLA claims turns on the fact that DLA is being abolished for new claims and replaced with a new benefit, personal independence payment (PIP), for which most claimants will require a face-to-face assessment (for DLA, other forms of medical evidence could be used to support claims). He said: "We've seen a rise [in claims] in the run-up to PIP. And you know why? They know PIP has a health check. They want to get in early, get ahead of it. It's a case of 'get your claim in early'.''

Some very specific figures were cited: "In the north-east of England, where reforms to disability benefits are being introduced, there was an increase of 2,600 in claims over the last year, up from 1,700 the year before, the minister told the Daily Mail. In the north-west, there were 4,100 claims for the benefits over the past 12 months, more than double the 1,800 in the previous year, he said."

But these figures, to be found on DWP's website, in fact represent the change – successful new claims minus those leaving the benefit – in the total DLA caseload from August 2011 to August 2012, crucially including pensioners and children who are not affected by the change from DLA to PIP. They do not constitute even indicative evidence of a DLA "closing down sale". So what happens if we look at new claims, or indeed the total caseload, for those (between 16 and 64) who will be actually affected by the change? In fact, both fell, in both regions, between those two dates. These falls – well within the normal quarterly variation – tell us little, except to show conclusively that Duncan Smith's statements are supported by no evidence that he has offered whatsoever.

Finally, the coalition's flagship "benefit cap". On this occasion, not only did Duncan Smith misrepresent what his own department's statistics meant, but he chose to directly contradict his own statisticians, claiming: "Already we've seen 8,000 people who would have been affected by the cap move into jobs. This clearly demonstrates that the cap is having the desired impact."

But the official DWP analysis, from which the 8,000 figure is drawn, not only does not say this, it says the direct opposite: "The figures for those claimants moving into work cover all of those who were identified as potentially being affected by the benefit cap who entered work. It is not intended to show the additional numbers entering work as a direct result of the contact [their emphasis]."

As DWP analysts know only too well, people move off benefits into work all the time. Unless it is shown that these flows have increased for those affected, and by more for them than for other claimants – and no such analysis has yet been published, either by DWP or anybody else – we know nothing about whether the policy has had any impact (this claim is now being reviewed by the UK statistics authority).

None of this should be taken as comment on the merits of the policies in question. But these misrepresentations of official statistics cross a line between legitimate "spin", where a government selects the data that best supports its case, and outright inaccuracy.

Public cynicism about official statistics is often misplaced – the UK, like most democracies, strictly limits the ability of governments to influence the production and dissemination of official data, often, no doubt, to the frustration of ministers. These restrictions on what government can do with official data are an unsung but essential element in modern democratic governance. When government seeks to get around these limitations by, in effect, simply making things up, this is not just an issue for geeks, wonks and pedants – it's an issue for everyone.

• This article was amended on 19 April 2013. The original said 130,000 people leave employment and support allowance every month; that is in fact how many people leave ESA each quarter.