The Guardian ran an innumerate little piece today headed ‘David Cameron’s Oxford college admits fewest state school applicants.’ Before we even look at the substance (hah!) of the article, we might briefly pause to think about what we expect the implications to be of this story. Let’s just let the emotion build as we read the headline. ‘David Cameron’s’ – what, wait, he’s the Prime Minister isn’t he?; ‘Oxford’ uh-oh, privilege and all that, there’s that picture of that drinking club he was a member of where they beat pigs to death with champagne corks and then have sex with their corpses; ‘college’ yeah, well, they have those at Oxford don’t they?; ‘admits fewest’ oh dear, it’s exclusive! It’s all about exclusivity. They’re keeping people out. They’re stopping people from going there!; ‘state school applicants’ Aagh! It’s class war! The vile and perverted establishment toffs of Oxford are slapping down the poor downtrodden people of the state schools!

Now let’s focus on the little statistic that the Education editor forced the writer to try to make a story out of. ‘Go on Sally, you must be able to make something out of this!’

Brasenose College, where Cameron went, apparently admits 11% of state school applicants, the lowest proportion of any Oxford college. Well, looks like we can’t fault the headline yet (Ooh, except that ‘lowest proportion’ doesn’t necessarily mean fewest. If one college receives one application, and accepts it, that’s a 100% acceptance rate, while if another college receives six applicants and accepts two of them that’s only 33%, but twice as many students. But let’s press on for the moment. I’ve a feeling it’s going to get worse.) (But I wonder how many readers skimmed this and thought ‘Tsk! Brasenose only has 11% of its undergraduates from state schools’, which is not the same at all. Indeed, the report where the statistics are published itself laments the fact that teachers vastly under-rate the proportion of Oxbridge students from state school background, which is in fact roughly 60% – yes, that’s right, the majority.)

So what was happening at the other end of this league table? Which mighty Oxford college was a hero of egalitarianism, throwing its doors open to the poor, oppressed working classes? St Peter’s, who let in 30% of their state school applicants. So wait, you’re thinking, even St Peter’s isn’t doing that great. They turn away 70% of their state school applicants? Heartless bastards.

Reality check, readers. More people apply to study at Oxford than can actually do so. This really shouldn’t need too much unpacking. Lots of people apply. Only a fairly small proportion of them get in. Add to that the fact that some people apply and then end up not going even if they are accepted (because they go somewhere else) and that Brasenose will accept people who don’t even apply to them – it’s not clear whether they are included in the statistic – through the ‘pool’ system, and suddenly an acceptance rate of 11% doesn’t look that bad.

But wait, you’re saying, don’t try to sell us that one. Let’s hear how the over-privileged coddled sons and daughters of the rich do, the ones who have been swanning around their cloisters in their boaters and being spoonfed extra lessons by teachers of brilliance. I bet Brasenose takes loads of them. Well, 13.5% is the answer. If you’re a state school applicant to Brasenose waiting to be accepted, you know that you’ve got an 11% of success. The girl from the posh end of town is a tiny bit more likely than you to succeed. If all 100 people in your sixth form applied to Brasenose, 11 people would get in. If all 100 people in hers did, 2 more would. (But hey, one of them could have been you. No wait, you didn’t go to her school. That’s the point. Damn.) Well, that’s nearly 20% more people. Quite a large advantage. You’d think.

However, these are statistics after the fact, and using them to point to some kind of empirical value is notoriously dangerous. If I flip a coin once, and it comes up Heads, I am not entitled to conclude that it is biased towards Heads. Even if I flip it three times and it comes up Heads every time, that will still happen 12.5% of the time with a perfectly fair coin. I have to get Heads another four times to get that probability down below 1%. Even then, concluding that the coin is biased on this basis will be wrong on average every 128 times I do it. In other words, if I threw away all the coins that seemed biased using this test, I’d be throwing away a good coin nearly one hundredth of the time.

In addition, as the report acknowledges, some subjects are much over-subscribed than others: your chances of getting into Cambridge to read Economics (13%) is much lower than your chance of getting in to read Classics (48%). This much variance causes so much noise to the headline statistic as to make it meaningless. What if state schools pupils are much more likely to be applying for Economics than for Classics (which seems perfectly likely)? What if Brasenose takes students for Economics (it does) but not for Classics (it does, but that’s not the point). In any case, the overall acceptance rate across all subjects for Brasenose, a small, over-subscribed college, is 12%.

So given that all these statistics are only averaged across the three years 2012 – 2014, the fact that the 12% splits between 13.5% for private schools, and 11% for the maintained sector really really shouldn’t be headline news. It would only mean a few private students getting a bit lucky (though precisely what that luck might constitute might be pause for thought) for the whole thing to be evidence of nothing at all.

The statistic underlying the whole debate is the famous one that only 7% of the school population is at an independent school, and yet they make up 43.7% of Oxford students. It is worth wondering what we would like these numbers to be such that we felt that the world was as fair as it could be – if indeed these numbers represent genuine unfairness. Would we really like the two numbers to be the same? That only 7% of Oxford was ex-private school, so that private school was no measure of enhanced success or chance in life. A few moments reflection would show that this is not a stable proposition. While some parents send their children to private school for the traditions, the uniforms, the quaint buildings and the eccentric teachers, there are very few who would be content to dispense with the academic standards. If a school did no better than the local comprehensive, it would die in short order, leading to the 7% becoming 0%. Paying parents expect results, and a high proportion of Oxbridge entrants is one of their measures of success. In addition, most private schools are selective. We expect their pupils to be better than those from state schools, not least because they represent talent that has been siphoned out of the local state sector (one of the most powerful arguments, incidentally, against private education that there is). If only 7% of Oxbridge were made up of those 7% of pupils in independent schools, the schools would actually have to have damaged them in comparison to how they would have done in the state sector.

How then, is this gap of opportunity to be closed? There is, of course, an artificial means to engineer this, which has already begun to some degree – positive discrimination in favour the state school pupil: lower offers to encourage applications from the disadvantaged. It is not clear whether this translates into lower expectations once such pupils arrive, and a lower grade boundary for a top degree, but it seems unlikely that if so, it would be followed through with tolerance for poorer performance in the professional and international workplace.

What is needed then, is not for universities (and anxious broadsheets) to obsess about dubious statistics, but for the state education sector to be given the resources it needs to put the private schools out of business by sheer competition.

I’m not holding my breath.