This graph (click for larger version) shows the upper division in-major GPAs of U Oregon physics and math graduates from a 5 year period. The math GPAs were computed using a specific set of rigorous courses taken by graduate school bound majors. One might reasonably associate "mastery" of a subject with GPA > 3.5 -- roughly, the minimum threshold to be admitted to graduate school (i.e., students who earn equal numbers of A's and B's in their upper division core courses are borderline for most graduate programs). We find that no student with SAT-M less than roughly 600 was able to attain this level of mastery.
This data provides at least modest evidence for a cognitive threshold required for physics and mathematics. That is, the a priori probability that a student with SAT-M < 600 will perform well enough to be admitted to graduate school in these subjects is extremely low. Note many poorly performing students switch majors, and hence do not populate the lower left corner of our graph.
A simple but plausible model for college performance might include two factors: 1. ability (measured by SAT) and 2. conscientiousness or effort (for simplicity, an uncorrelated random variable, probably normally distributed). The predicted GPA might depend linearly on the sum of the two factors, each measured in SD units. (See bottom figure here for a similar predictor of GPA.) Applying such a model to our data, we would conclude that even a student in the, e.g., 90th percentile of work ethic has a low probability of attaining mastery if their SAT-M score is below 600.
To reiterate, SAT-M 600 seems to be the lowest score at which even a very motivated student has a chance for mastery. From the data one might guess that only for SAT-M well above 700 do students have more than a 50 percent chance of obtaining GPA > 3.5. That is, a student with average motivation or conscientiousness probably needs SAT-M well above 700 to have a high probability of obtaining mastery.
We were unable to find any similar threshold (either in SAT-V or SAT-M) in other majors, including economics, sociology, history, philosophy, biology, chemistry, etc. (More about this data set, including data on other majors, here.)
The percentile equivalents of SAT-M scores for various groups can be found here. For the total SAT-taking population, 600 is about 75th percentile. For the overall population, it might be 85th percentile. SAT-M of 750 is roughly 98th percentile for the total SAT-taking population.
For comparison, here is the analogous plot for English (black) and History (blue) majors versus SAT-R (verbal). If a threshold exists it is probably at SAT-R of 450 or so.
