Joanne Gibbons was sentenced to community service for claiming income support while holding down two paid jobs. Through accumulated payments of £66-a-week, the court heard, she collected £3,140 to which she wasn't entitled.

Predictably, the Daily Mail is outraged. But here's the strange twist: had Gibbons claimed the benefits to which she was actually entitled, she could have collected £130 a week through family tax credits and child benefit. In total, Gibbons' fraudulent claims cost the taxpayer around £3,100 less than claiming what she was actually entitled to.

It's the reaction to Gibbons' claims which are particularly noteworthy. Matthew Sinclair, chief executive of the Taxpayers' Alliance – an organisation rarely troubled by wealthy people's tax avoidance – tells the Mail:

"It beggars belief that somebody going to the lengths of making fraudulent claims would have actually received more in benefits had they been honest. "It just goes to show that the current system is broken and doesn't provide the right incentives for claimants to go back to work."

This quote suggests Sinclair is perhaps even less numerate than the "benefits cheat" he's deriding. Gibbons was entitled to £130 a week in legitimate benefits, while working on two low-income jobs. This total was higher than the £66 a week out-of-work benefit she was improperly claiming (though some of the £130 a week could be claimed in or out of work).

In what sense is a system which tops up low wages a disincentive to work? Sinclair appears lost in lazy rhetoric – an all-too-common failing when it comes to chastising the millions of families, most of whom with at least one adult in work, who rely on the benefit system.

The British public believe benefit fraud is a big problem. A recent poll by the TUC showed people believe 27% of the welfare budget is fraudulently claimed.

The reality is very different. Last year, 0.7% of total benefit expenditure was overpaid due to fraud, according to the DWP's official estimates. This totalled £1.2bn over the year. Nor is fraud getting worse – even against a background of benefit cuts and long-term unemployment fraud made up a smaller share of the welfare bill last year than it did in 2010/11 or 2009/10.

Indeed, welfare fraud is smaller than accidental overpayments due to error, which totalled £2.2bn (£1.4bn of which due to official error). It's also smaller than the amount of money underpaid to those entitled to it: £1.3bn.

In other words, if we wiped out benefit fraud tomorrow – but also eliminated the errors that deprive people of money to which they are entitled – the welfare bill would grow, not shrink.

In the context of the UK's £700bn public spending, and £150bn+ welfare bill (of which pensions and in-work benefits make up the substantial majority), benefit fraud is a relatively small revenue loss. But how does it compare to another textbook villain: tax avoidance?

Put simply, it is comparatively tiny. HMRC consistently estimates the UK's tax gap – the gap between what HMRC thinks it should receive versus what it actually gets – at more than £30bn per year. Others estimate this is far, far higher.

Of this, even conservative estimates suggest around a sixth – £5bn a year – is lost through tax avoidance, tricks to reduce tax bills which fall within the letter (if not spirit) of the law, but often fall outside what's regarded as acceptable by the public. A further sixth, at least, is estimated to be due to wholesale tax evasion: simply illegally not paying the tax that's owed.

These conservative estimates alone outweigh benefit fraud by a factor of eight, but this time not done in tens (or at most hundreds) of pounds per week by people struggling to get by; but rather by people who could afford to pay more, but prefer not to.

Benefit underpayments save us more money than benefit fraud costs us. By the most conservative estimates, tax avoidance and tax evasion outweighs benefit fraud eightfold. But the constant target of argument – "scroungers", "benefit cheats", and more, isn't the well-heeled middle classes who knock a little off their tax return, or the high-rollers with elaborate offshore schemes.

Instead, it's those at the bottom of society – for the government, perhaps, it makes it easier to sell the public swingeing cuts to the safety net that millions of families, both in and out of work, rely on to get by. For the Mail, it's easier to sell papers by buying into the easy preconceptions of their readers than bothering to challenge them.

Unfortunately, all too often, that's a view the Labour party – and others on the left – seem all too happy to go along with. If we must have national villains, surely we can do better than these?