In 1997, 65 percent of students with three As at A'level were from state schools, but only 50 percent of successful Cambridge applicants. In another recent year, if you had three As at A'level, then your chances of getting into Oxbridge were 50 percent if you were from an independent school, but only 35 percent if you were from a state school. On the face of it, these figures appear to be scandalous, and the clearest possible evidence of bias by universities such as Oxford and Cambridge against applicants from state schools.

Since to write about this issue is to enter a minefield, let me say first that this brief note has nothing to do with politics. There are difficult and important issues about how the top universities should respond to the fact that some applicants have been given an enormous educational advantage at school level, and I shall have nothing to say about them. All I want to do is point out that the figures I have just mentioned are not in themselves a proof of bias. And that is a very simple mathematical observation - though, alas, not simple enough to be understood by politicians and educational commentators.

Admissions to Cambridge are complicated by the fact that selection is made partly on the basis of promise, and it is much harder to make guesses about future performance than it is to judge current aptitude. So let us abstract away a little from the actual admissions procedure and imagine a simpler, less realistic situation in which there was a clear notion of current academic ability, and the aim of the admissions procedure was simply to assess this ability and select the most able candidates. Suppose that, of pupils with top grades at A'level, a significantly higher proportion of pupils from independent schools than from state schools were chosen. Would this be proof of bias?

There is one important fact that is relevant to the discussion, which is that a much higher proportion of pupils from independent schools obtain top grades in the first place. It is because of this that such figures would not on their own be proof of bias at all. There are two ways of seeing why not, one more intuitive, the other a bit more technical.

To see it intuitively, let us imagine a less politically sensitive situation. Suppose that you have started a company and wish to employ 100 people, and for some reason the only thing that matters is that applicants should be as tall as possible. Suppose also that you are not prepared even to consider applicants under five foot seven. Finally, suppose that the applicants do not have much idea who else will be applying, so that you do not have people of five foot seven and a half not applying because they think they will have no chance.

Now suppose that when the applications come in, a third of the applicants are women, and that the hundredth tallest applicant is six foot tall. What proportion of successful applicants do you expect to be women? Will it be a third? No, because as height increases in that sort of range, the proportion of women with the given height decreases very rapidly: while it is not particularly uncommon for a woman to be five foot seven, women over six foot tall are extremely rare.

Here, I am imagining that a height of five foot seven is taken
as a sort of "A grade", but that, as with Oxbridge applications,
a small selection has to be made from a large group of people who
all have top grades. What this example shows is that you cannot
just assume that the way ability is distributed * within *
the cohort of top-grade school pupils is the same in the two sectors.
Indeed, as with height, if the average ability (as measured by A'level
grades) is higher in one sector than another, then you expect
that the average ability within the top grade to be higher in
that sector as well.

Another example shows this even more clearly. Of all school pupils who obtain an A, B or C at A'level, the proportion who get an A is significantly higher amongst pupils from independent schools than it is amongst pupils from state schools. Does this show that the A'level examination boards are biased in favour of independent schools? Of course it doesn't. Suppose that the current A grade at A'level were split into two, with the top half getting an A1 and the bottom half getting an A2. One might expect that equal numbers of independent-school pupils would get an A1 as an A2, but in fact that would be very surprising indeed. As you go up the grades, the proportions from independent schools increase. Why would this phenomenon suddenly stop when you got to the last two (new) grades?

Here is a much quicker way of stating the argument, for anybody who has done a small amount of probability. It is plausible that the spread of ability within the independent-school sector should be roughly normal, and also normal within the state-school sector. Moreover, the A'level figures show that the average of the distribution for the independent-school applicants is higher than that for state-school applicants. Once again, I stress that I am not talking about intrinsic ability - which one might define as the academic potential at birth of a pupil who is placed in a happy and highly stimulating educational environment. This sort of ability, which is probably what politicians have in mind, is unlikely to be any different on average in the two sectors.

It is not clear what one should expect of the standard deviations of these two normal-ish distributions, so let us suppose for simplicity that they are the same. (If the deviation amongst independent-school pupils were very small - contrary to popular belief - then the argument I am about to give would no longer work.) The following is then an uncontrovertible mathematical fact. Suppose you have a set X partitioned into two subsets A and B, and suppose that f is a function defined on X and that the restrictions of f to A and B are normally distributed with the same standard deviations. Finally, suppose that the average of f is significantly higher over A than it is over B. Now choose a random element x of X and consider the probability that x belongs to A given that f(x) is greater than t. This probability increases with t, and the increase is very considerable when t is a couple of standard deviations higher than the average of f over B.

In other words, the maths says that the figures that appear to show bias are of exactly the kind that one would in fact expect if there were no bias. So, by all means criticize Oxford and Cambridge for not doing more to compensate for the fact that state-school pupils may be less well prepared than independent-school pupils, but be aware that if you want the proportions admitted from those with top grades to be the same in the two sectors, then you are advocating a very strong form of positive discrimination.