How are you computing the average? Princeton’s and Yale’s 2019-20 CDS reports both report exactly the same 25th/75th percentiles for SAT composite of 1470 and 1560.
Answering your question more generally, a large difference in SAT scores between similarly selective (not the same as similar “prestige”) colleges can suggest that one may prioritize SAT more than the other. However, there are also many other contributing factors, so the conclusion is not as straightforward as it may appear. And 10 points is certainly not a large difference in scores.
For example, Caltech was mentioned. I suspect that Caltech has a unique applicant pool that has a higher concentration of high test score students than most colleges with a similar admit rate. Caltech’s higher test scores may have more to do with he applicant pool than it does Caltech prioritizing scores.
Or the higher test scores may relate to Caltech prioritizing something else that is correlated with scores, rather than the scores themselves. For example, MIT’s website writes the following. They imply that they do not focus on scores and instead have very high scores because they care about things that are correlated with scores, like their IMO medalist example. Along the same lines, I’d expect that Caltech will still have higher median scores than HYPS even though they are now test blind (if test scores were reported).
" Now, I and others are on the record as saying that we admit people, not test scores, and that in any case there is really not a difference in our process between someone who scores, say, a 740 on the SAT math, and someone who scores an 800 on the SAT math. So why, as the commentor asks, is there such a difference in the admit rate? Aha! Clearly we DO prefer higher SAT scores!
Well no, we don’t. What we prefer are things which may coincide with higher SAT scores. For example, a student who receives a gold medal at the IMO is probably more likely to score an 800 on the math SAT than a 740. But if we take an IMO medalist (with an 800) over random applicant X (with a 740), does that mean we preferred an 800 to a 740? No. It means we preferred the IMO medalist, who also happened to get an 800!"
Some clues that a college may prioritize scores can sometimes be found in Naviance type scattergrams, if you have enough samples. For example, scattergrams for Vanderbuilt used to (I haven’t checked recently) show a wall of acceptances at their 75th SAT score. The acceptances rate was extremely high for applicants above 75th percentile, but quite a bit lower below 75th percentile.
Another clue is a larger number of NMSs that one would expect based on selectivity, including sponsored ones. For example, the number of NMS by school in 2018-19 report (didn’t find more recent one online) is below. Vanderbilt had more NMSs than HYPS, which is another clue that Vanderbuilt may prioritize scores.
Vanderbuilt – 222
Harvard – 207
Yale – 140
Stanford – 124
Princeton – 114