^The preference for full pay versus financial aid must be an order of magnitude at least (more than 10 times) in order to arrive at ratios like 70/30 to 60/40. I’d guess that no family earning less than $250,000 can afford to be full pay, at least on average. That’s maybe 5% at most of American families. Wealthy people are smarter on average (but many of the kids of the wealthy will be significantly less intelligent than their parents, due to regression to the mean), but not by too much, based on all the research between SES and intelligence. Holding all the other attributes desirable to schools constant, such as leadership skills, “emotional intelligence,” “grit,” etc - and importantly there is no reason to assume that these other attributes are negatively correlated with intelligence - you’d get ratios more like 80/20 to 90/10 the other way (FA to FP) without extreme FP preferences. This analysis is roughly consistent with what another poster wrote somewhere on here about what was told to the poster at NMH - that the school had “six times” as many FA kids that they wanted to admit than they could.
I always scratch my head when I read on here that the top schools can “fill their classes with FP 99% kids” - or its corollary, “each of the Ivies can fill its class with 4.0 GPA 1600 SAT kids.” LOL. There are probably around 600 kids in the entire US who score a 99% SSAT (I guess you can add a few for the 99% ISEE, although there will be some overlap of course and double counting of the same kids). How many of those want to go to boarding school? Maybe 200? Maybe 300? The true number is probably a LOT lower, given how low the correlated boarding school stats (SAT scores, NMSF, achievement test scores) are as compared with day schools. Now, given the distribution of intelligence, how many of those 200-300 (at the very most) could possibly be from $250K+ families? Maybe 50-100, and that’s being extremely generous. Not even a small school like Groton could fill its class with them.
You could do the same exercise with colleges and the SAT. The SAT is a relatively easy test compared to what it was, say, 30 years ago, but nevertheless I think there are only approximately 650 perfect scores each year. How many of those scores also have 4.0 GPAs? Not even Dartmouth could fill its class if it wanted to.
I actually think what @someone3301 wrote makes a lot of sense, although he or she misses a few important things. (For instance, boarding schools do need to extend financial aid to a select and small group of very intelligent kids as part of their attempt to maintain the facade that they are “elite” centers of learning - there are just not enough “elite” smart, wealthy kids. The rest of the aid can be allocated to fill other needs, like athletics, virtue signalling by means of preferred and visible groups, etc.)
I admit I am an “intelligence snob” about a lot of this, and understand that many people will argue that there is not that much difference between a 84% standardized score and a 97% (or 99.8%). Well, my experience has been very different (from working with “edge smart” kids through certain gifted programs). There is a world of difference between kids separated by two standard deviations in ability (roughly the difference between 99.8 and 84), and even for those separated by only one standard deviation (roughly the difference between 97 and 99+). It’s very noticeable.
Bottom line, if you want to get those “edge smart” kids, you need a financial aid budget. You’re just not going to find many in the FP group (because the FP group is small, and smarts are distributed a little more “fairly” than income and wealth - nature’s gift to us as a society!). The dirty secret is that not many of the boarding schools are particularly interested in “edge smart” kids, probably because it would be bad business in light of their lesser ability FP cohorts. There are a very few exceptions, and I think everyone can guess which schools “scoop up” the truly extraordinary kids out there.