For another data point, Andover (which many consider a very generous school…“youth from every quarter” and all that) lists an average grant of $38,100 for boarding students (SAS is 100% boarding) on their site.
My point in coming back to post that fact is that I think people sometimes focus on the “big name” schools because (in addition to prestige and perceived quality of education) they think only the big name schools have money. Not always true. And to build on something that soxmom wrote, at the most selective schools, I’m guessing a mid-80s SSAT is on the low end of things and at best, just average in the applicant pool.