The significance of these figures can be gauged from the fact that across the UK as a whole only 7 per cent of pupils attend independent schools, while in Scotland the proportion is just 4 per cent. The conclusion to be drawn is that St Andrews and Edinburgh need to do better in terms of recruitment from the state sector if aspirations for social inclusiveness are to be fulfilled.
Both are excellent universities and undoubtedly attractive to applicants.
There are, however, some important issues that need to be considered.
First, are Scottish school-leavers disadvantaged? Over the years, St Andrews and Edinburgh have been subject to the criticism that their identity as distinctively Scottish institutions has been diluted by the large numbers of English students who are admitted.
Many academic staff also have a background in English education. One does not have to subscribe fully to George Davie's "Anglicisation" theory of Scottish higher education to feel that the criticism seems to have some face validity. This point is not unconnected to the fact Edinburgh is certainly the most "Anglified" of Scottish cities and Scottish voices often seem outnumbered in the streets of St Andrews.
These observations should not be dismissed as mere nationalism. At a time when the identity of post-devolution Scotland is still being forged, serious thought needs to be given to the opportunities available to young Scots. It is a legitimate question to ask whether the universities, as major public institutions, are playing their part fairly and effectively.
One interpretation of the HESA statistics is that they demonstrate the power of "social capital" in maintaining privilege. The parents of young people who attend independent schools are not only likely to be financially secure, they are also likely to be able to assist their offspring in negotiating the system of entry to higher education, often drawing on the professional and social networks to which they belong.
"Capital" comes in a variety of forms - financial, cultural, psychological, intellectual - and serves to give social advantage to those who possess it.
It is perfectly natural for parents to want to help their children, but the effect may be to widen the gap between different social classes.
It is revealing that in the political lexicon of new Labour, the term "class" is rarely heard. Perhaps we need to reinstate it in public discourse. The effects of all this extend beyond education. In St Andrews, for example, well-heeled parents buy former council houses for students at inflated prices, thereby making it harder for young locals to get on the housing ladder. Universities need to take account of the wider consequences of their recruitment policies.
I should declare a personal interest. I am a first-generation graduate who was fortunate enough to attend university in the expansionist period of the 1960s. I was not any cleverer than my brothers and sisters, just luckier, being the youngest member of the family. I want as many people from my sort of background to have the same opportunity to experience the benefits of higher education.
The best universities have a public responsibility to give a lead in promoting greater inclusiveness. Otherwise they will simply be seen as agencies which serve to consolidate and maintain social privilege.
Walter Humes is professor of education at Aberdeen University.