We live in an era of "men deserts", says the Centre for Social Justice. One day my children will look on me with worshipful eyes and say: "Mother, how did we survive the man drought of the early 21st century?" as if I'm some Mad Max of the spunkless years. This Sunday is Father's Day, the traditional time to pay tribute to any man you haven't driven screaming from your fanny ghetto after mating. But what to buy?

Luckily, everyone's got something to sell you for father's day, and some organisations aren't just hawking cufflinks and crap aftershave – they're selling a whole ethos. For example, the Centre for Social Justice is flogging the idea that we have become a nation unmanned. In a press release at the beginning of this week, the CSJ told us: "Lone parents tally heads for two million […] Around one million children grow up with no contact with their father […]some of the poorest parts of the country have become 'men deserts' because so few primary schools have male teachers".

Scary stuff. And fear must be a popular Father's Day gift, because people have been buying it and buying it and buying it. The Today programme bought it. The Telegraph bought it. The Times bought it. Newsnight bought it, in a Monday night report that opened with Paxman accepting the validity of all the CSJ's claims before asking: "Does it matter anyway?" But whoa there, Paxo! Before we assume it's a problem – let's find out if it's even true.

Checking the CSJ's claims was quite tricky at the beginning of the week, because the thinktank didn't release its report until Thursday. But it's here now, so let's do what Paxman didn't and see how those claims stack up. "Lone parents tally heads for 2 million" isn't bad: the Office for National Statistics says there were 1.7 million lone-parent households with dependent children in England and Wales in 2011, an increase from 1.4 million in 2001.

But what about the sad 1 million children who have no contact with their father? That comes from the Fatherhood Institute, which reckons the number is "between 1 and 2 million". The Fatherhood Institute may be a very fine and well-intentioned institute, but since it doesn't show its working, it may also be completely wrong. You can get a rough figure of 1 million using data from the ONS – but only if you take the highest possible figures and discount indirect contact like phone calls and email. It look as if the CSJ has just hoiked out the biggest plausible number and moved on.

Then there's the issue of male primary school teachers. It's true that men are a minority in primary teaching, and 27% of primary schools indeed have no male teachers. You may however notice that therefore 73% of primary schools do have at least one male teacher – and that proportion is increasing as more men join the profession. You might wish that there were more male primary teachers overall, but you can't really say that "so few" schools have them when actually the vast majority do.

And then we come to those vexatious "deserts", which the CSJ seems to have summoned into being by taking the smallest areas in the census (called "Lower Super Output Areas", and no I didn't have a clue what an LSOA was before this week) and conflating the LSOA with the electoral ward it's in. So the LSOA Sheffield 075G (population: 2,373) is presented as representative of the Manor Castle ward (population 21,768).

Then the CSJ excludes households without dependent children (which of course include many men living adjacent to the single mother households), then it gives the percentage of single parents as a proportion of the remainder, and then the reader is supposed to be shocked into reintroducing the married couple's tax allowance or something. Manor Castle has a bunch of issues, but it's not unusually short of men: in fact, 52% of the population is male, putting it slightly up on the Sheffield average and making it a well-irrigated man delta.

And if you find the image of a well-irrigated man delta unpleasant, consider how crude and nasty the CSJ's work is in the press release and report. Of the three headline contentions, only one is true, and the report's underlying claims are more dubious still. The CSJ assumes a direct causal link between single mothers and numerous undesirable social effects, and its proposed fix is that people should marry before they have children. Its solution is as crass as its concept of causation.

Children need more than that one caregiver – hell, they need a whole network, which is one reason why CSJ founder Iain Duncan Smith's suggestion that people just travel around chasing jobs is such a stupid answer to poverty. But those caregivers don't necessarily have to be in a sexual relationship, and fixating on marriage seems to have left the CSJ with some genuinely frightening spots of ignorance.

The report criticises agencies that intervene in cases of domestic abuse for failing to recognise fathers as caregivers, on the grounds that "many perpetrators desire a more positive relationship with their children, and this can be a powerful motivator for change." Here, the CSJ is subordinating the safety of children to the potential self-improvement of abusers. Because this report, with its wobbly stats and its exaggerated claims, isn't actually about what's best for children: it's about the fear that some women and children might be perfectly OK without a masculine hand hanging over the household.