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Well, it's statistically very, very rare to get in with an SAT score below 2000.
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you're saying the same thing, and I explained why you are incorrect. The average student has a 1/10 chance, if you have like a 1800-2000, you would have like a 1/15 or 1/20 chance. That's not statistically very rare compared to the original 1/10 chance. Empirically they might even admit 1/12 or 1/13 kids with 1800-2000, because most kids with low scores applying to columbia usually have strong other parts to their application.
You need to realize that just because almost everyone you knew who got into Columbia had high scores, and many low scores were rejected, it isn't grounds for a theorem that you're fcked with lower scores. I, on the other hand, have seen that people get in with 1800-2000, without a crazy hook, just because they have otherwise an all round strong application. This disproves your hypothesis.