In my view, St Andrews is a bit of an oddity among top UK universities in that it is more undergrad focused than most. Like, St Andrews last reported 10235 total students, of which 8388 (about 82%) were undergraduates, whereas LSE reported 13728 students, of which 6175 (about 45%) were undergraduates. Oxford reported being about 47% undergraduate, Cambridge 52%, and so on.
I think in US terms, St Andrews is sort of like the Dartmouth of top UK colleges in that sense–although even Dartmouth is still only around 66%. But that is a lot more than, say, Harvard, about 33%, or Chicago, about 39%, and so on.
Or alternatively, you could see it as like a William & Mary (its US partner for its joint degree program). W&M is about 72% undergrad.
For many undergrad courses, St Andrews is then quite selective. The Guardian league tables include an average entry tariff column (this is the “typical UCAS scores of young entrants (under 21) to the department”), and while that doesn’t fully capture all elements of selectivity, it is a reasonable proxy in many cases.
So for their undergrad course in Economics, for example, it goes Cambridge 224, St Andrews 221, Oxford 211, LSE 204. I don’t think St Andrews is actually more selective than Oxford, once you factor in the interview stage and such. But it is definitely quite selective in the sense it is popular enough to be able to turn away UK applicants who do not have very high UCAS scores.
For the Business course, it goes Oxford 220 (Cambridge doesn’t have this course), Strathclyde 210, St Andrews and Glasgow tied at 202 . . . LSE is actually down at a tie for #9 with Bath and City at 183.
Of course this is necessarily for UK students–US students at these colleges do not have meaningful UCAS scores. So it is plausible St Andrew’s courses are often more selective for UK applicants than US applicants (this is certainly the reputation, the suggestion being St Andrews wants the extra cash from US students).
I don’t know enough about Bocconi or IE to really comment on them, but they are such different types of institutions, and obviously in a different country, so I think it might be hard to really compare.
But I would say that for undergrad purposes, St Andrews, more so than LSE, or very much more so than Bocconi or IE, likely appeals more to the sort of US students who are otherwise looking at general interest private universities in the US, or even LACs (or William & Mary, which is on the same model despite being a public). And then Oxbridge in the UK.
Indeed, in my (admittedly limited) experience, a US kid applying to St Andrews for undergrad may ONLY apply to one of Oxbridge as well. The next-best choice for what they are looking for might actually be Durham, but they might not see it worth applying to Durham.
LSE in my circles is then much more known as a university you might consider for a grad degree (and conversely, St Andrews less so). But I know some US undergrad do consider LSE. I just don’t know whether they also consider Bocconi or IE.