I meant both tests but definitely the subject tests.
Usually the CEEB, SAT, and GRE tests are designed to have Gaussian distribution with mean 500 and standard deviation 100. So, can get a table of the cumulative Gaussian distribution and see what percentile 4 standard deviations above the mean is; I would guess less than 1%.
I got 800 on the GRE subject math test, the only 800 I got on any of those tests, and that 800 always intimidated my wife, MUCH smarter than I am, PBK, etc.
Why especially the subject tests? Because the question was how to skip a Bachelor's degree and do not pass GO, do not collect $100, and go directly to grad school. There the grad school may still want a Bachelor's but for anything less really good scores on the relevant GRE subject tests may be the difference. Show up with 750 or better on GRE subject math, physics, and computer science, and offer to take the Ph.D. quals right away, and may, just MAY, be permitted to 'enroll'. Blow away the quals, publish a paper or two, even in a conference, maybe knock out some code just to prove are not all theory, and may be regarded as a good student. Then will be closer to the front of the line for various kinds of financial aid.
How to skip a Bachelor's is chancy. For the importance and potential of good independent work, at the best research universities that's rock solid. Read the story of the guy who gave the name a 'good' algorithm, Jack Edmonds. Read what Feynman did at Princeton. Read what Gleason did at Harvard. Independent work was just crucial in how I got my Ph.D.: It helped that I did the research for my dissertation independently in my first summer and worte a 50 page manuscript. Then it helped that took a 'reading course', selected a long outstanding problem, and found a solution which also solved a problem in a famous paper by Arrow, Hurwicz, and Uzawa (poor Uzawa was left out of the prize). It helped that the department chair taught a flunk out course, an advanced, second, course in linear algebra and I took it as my first course in linear algebra and blew everyone else away. How? I'd done a LOT in linear algebra independently and in my career in 'scientific programming' before grad school. It even helped that I was the only student that year who showed that there are no countably infinite sigma algebras. That's the kind of stuff that can help one get through grad school.
Skipping the Bachelor entirely might be one use case for the GRE subject test, but I think a more common use case is where the applicant to grad school has a Bachelor is a different field (Math grad going to CS school). I imagine one wouldn't need blow out grades in that case.
Also, I don't think what you're saying about the Gaussian distribution is true for GRE Subject tests as they have significantly varying distributions. Check out this table:
I had a 'qualification': "Usually the CEEB, SAT, and GRE", and with that we both can be right!
For those tests, at the level of detail of the distribution of the scores, it can be tough to get solid data.
But it is easy and common in educational testing to scale 'raw' scores so that the 'scaled scores' are Gaussian.
Also in educational testing, it is easy and common to have enough data on individual questions so that the distribution will be known fairly accurately for a test made of such questions -- my father did that for years as the main 'educational architect' at one of the world's largest and most important technical schools, with 40,000 students there at any one time.
Having those tests be accurately Gaussian more than
2.5 standard deviations away from the mean is likely challenging. I've often suspected that on some of the SAT tests they didn't give any 800 scores.
There have been suggestions that those testing companies have not always been very open about just what they were doing in their details!
Usually the CEEB, SAT, and GRE tests are designed to have Gaussian distribution with mean 500 and standard deviation 100. So, can get a table of the cumulative Gaussian distribution and see what percentile 4 standard deviations above the mean is; I would guess less than 1%.
I got 800 on the GRE subject math test, the only 800 I got on any of those tests, and that 800 always intimidated my wife, MUCH smarter than I am, PBK, etc.
Why especially the subject tests? Because the question was how to skip a Bachelor's degree and do not pass GO, do not collect $100, and go directly to grad school. There the grad school may still want a Bachelor's but for anything less really good scores on the relevant GRE subject tests may be the difference. Show up with 750 or better on GRE subject math, physics, and computer science, and offer to take the Ph.D. quals right away, and may, just MAY, be permitted to 'enroll'. Blow away the quals, publish a paper or two, even in a conference, maybe knock out some code just to prove are not all theory, and may be regarded as a good student. Then will be closer to the front of the line for various kinds of financial aid.
How to skip a Bachelor's is chancy. For the importance and potential of good independent work, at the best research universities that's rock solid. Read the story of the guy who gave the name a 'good' algorithm, Jack Edmonds. Read what Feynman did at Princeton. Read what Gleason did at Harvard. Independent work was just crucial in how I got my Ph.D.: It helped that I did the research for my dissertation independently in my first summer and worte a 50 page manuscript. Then it helped that took a 'reading course', selected a long outstanding problem, and found a solution which also solved a problem in a famous paper by Arrow, Hurwicz, and Uzawa (poor Uzawa was left out of the prize). It helped that the department chair taught a flunk out course, an advanced, second, course in linear algebra and I took it as my first course in linear algebra and blew everyone else away. How? I'd done a LOT in linear algebra independently and in my career in 'scientific programming' before grad school. It even helped that I was the only student that year who showed that there are no countably infinite sigma algebras. That's the kind of stuff that can help one get through grad school.