# Average measurements

## Malcolm Gillies discusses the fees revolution’s mean, median and mode

By the law of averages, Vince Cable, the business secretary, and David Willetts, the universities and science minister, might have had reason to expect that, even after the University of Cambridge’s opening salvo of £9,000 in the Fees War, English universities would still behave as planned: filling out the full range of charges down to £6,000, and so achieving a sectoral average of £7,500.

But as the war hotted up, it became clear that lowest and highest were not going to be so far apart. On average, aspiration to excellence has triumphed over cost containment or student affordability.

Averages are clearly not the forte of every journalist, or faculty administrator, or vice-chancellor. I’m sure that many a fees list has been shredded as universities went through this ritual of half-blind calculation in preparation for the submission of their access agreements on 19 April.

In March, when Times Higher Education unveiled its countdown listing for 2012-13 tuition fees for home students, its first online readers’ comment was posted by Peter Coles, professor of theoretical astrophysics at Cardiff University. He accused THE of “dodgy” arithmetic: “What you have done is just tot up the fees above, counting each institution as equal. The correct average per student should be weighted by the number of students at each institution.”

Now, THE had, in fact, employed a legitimate average - an institutional one - but what everyone (especially the Treasury) really wanted to know was the weighted student average.

So the magazine gracefully withdrew into the formulation of a “rough and ready average” for the “declared headline” fee. By 8.48pm on 19 April this stood at £8,713, but by 5pm on 21 April it was £8,652. Last-minute declarers preferred somewhat lower prices, it seems. And quite a few did not make public declarations at all.

Most people think of the average as the arithmetic mean: you know, you add up the figures in a list and divide by the number of items. Of course, there are those testy other means - geometric and harmonic - that are a bit more complex. (I’ve always been suspicious of the harmonic variety. You will remember the example of average speed: you go to work at 60 miles per hour, you travel home at 40mph. The average speed turns out to be…48mph.)

Coles and THE came into conflict over how the arithmetic mean was being applied: whether the meaningful unit was the institution or the student. As Coles observed, many larger universities were declaring at £9,000, and so his average, weighted by student size, would be above the one-vote-one-value institutional average.

But then, as the numbers piled in on Access Day, the press started to announce that more than two-thirds of England’s universities were hoping to charge way above the average fee - and just about everyone was pitching above the Treasury’s estimated average. Now, this probably confused many who had recently studied GCSE maths. For as the BBC Schools website states, it teaches that “an average indicates the typical value of a set of data and the main types are mean, median and mode”.

So, while the (arithmetic) mean looks like it will settle at about £8,600 a year, the median may be close to, or at, £9,000. That is, if you lined up every UK/European Union undergraduate in the land and went to the middle one in the line, on current information, he or she will probably pay £9,000, or very close to it. And the mode, the figure around which most fees cluster, would definitely be £9,000, not a penny more (that’s not allowed), but also not a penny less. All of these values our students know as different kinds of average, different kinds of typical value.

So much for the maths: everyday speech poses another problem - average also means “mediocre”. The spin doctors in our marketing departments know to avoid the word like the plague. You just can’t write, “The University of Such-and-Such is an average institution in the middle of England”, and certainly not, “For the past 30 years, the University of Such-and-Such has been a below-average university”.

Both statements may be mathematically true, of course, and, however hard we try - whatever level of excellence we may attain - the proportion of below-average universities is likely to be about the same in 10 years’ time, along with the number of below-average students.

But how many universities are “below average”? Here we return to Coles’ point: the universities at the top of most league tables are, on “average”, bigger than those further down. As a result, a university about one-third of the way down might actually contain the median student for the sector.

So, if you lined up all the students from the league table leader (say, the University of Oxford), then those from number two and so on, and then went to the middle person in the row and asked which university they came from, they could well be studying at the institution about one-third of the way down the table. But does that mean, then, that about two-thirds of our universities are below average?

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