In light of the fact that the most prolific and horrific commenter on this blog (Mugabe, aka chocolate babies)claims to have scored 800 on all three sections of the old GRE, I wanted to explore this test further.  Some people believe him; others are very skeptical.  Some ask why I tolerate him at all.  The reason is I’m a horror fan, and thus have high tolerance for freakish behavior.  The other reason is I LOVE HIS AVATAR!

mugabe

It’s hilariously ironic to have someone with the avatar of a black man expressing such aggressively alt-right opinions, and even though there’s the stereotype of blacks being less smart, the particular black avatar he chose looks fiercely intelligent, and I’ve finally figured out why.  It’s because it looks like a black Ben Stein, who is rumoured to have a freakishly high IQ.

ben

Enough about Mugabe.  Let’s talk about the GRE.

The GRE and the SSS (the super self-selected)

During the late 20th century, Americans who took the SAT, were self-selected to have an above average IQ (108; U.S. norms),  but those who were academically ambitious and confident enough to take both the SAT, and then several years later, the GRE, appear to have been far more self-selected for IQ than I could have ever imagined.  Indeed based on their verbal SAT distribution (see image below), they had a mean IQ of 120 and an SD of 14, compared to the general U.S. distribution with a mean of 100 and an SD of 15. I will refer to this group as the super self-selected (SSS).

gredata

Perfect verbal score on old GRE = IQ 158 (U.S. norms); IQ 157 (U.S. white norms)

Now among the SSS, the GRE verbal (circa 1990) had a mean and SD of 510.1 and 107.7 respectively (see image above), so assuming a normal curve, only one in 261 of these highly self-selected people would score a perfect 800 (a Z score of 2.69 with respect to the SSS).  But because the SSS has a mean IQ of 120 and an SD of 14, a Z score of 2.69 equates to an IQ of 2.69(14) + 120 = 158 (U.S. norms)

Perfect verbal score AND perfect quantitative score on old GRE = IQ 162 (U.S. norms); IQ 161 (U.S. white norms)

Now of those one in 261 SSS who scored 800 on the GRE Verbal (Z = 2.69), how many would score 800 again  on the GRE Quantitative?  Well, given a 0.56 correlation between GRE V and GRE Q in this population, the expected Q score of someone with a Z score of 2.69 on V, would of course be 0.56(2.69) = 1.51 (standard error = 0.83).

Given that the GRE Q has a mean and SD of 573.4 and 125.6 respectively in this  population, a score of 800 has a Z score of 1.8, which is 0.35 standard errors higher than the expected Q score of an 800 V, which means that only one in 2.71 of them should repeat their 800 on the Q section.  Since scoring 800 on V is already a one in 261 performance (in the SSS), getting another 800 on Q becomes a one in 261(2.71) = 707 performance.

Assuming a normal curve, one in 707 is an incredible three standard deviations above average, but because the SSS has a mean IQ of 120 and an SD of 14, it equates to an IQ of 3(14) + 120 = 162 (U.S. norms)

Perfect score on ALL THREE SECTIONS of the old GRE = IQ 164 (U.S. norms); IQ 165 (U.S. white norms)

Now of the one in 707 SSS who scored 800 on both GRE V and GRE Q, how many did it yet again on the analytical section?  Given that the correlation, in the SSS, between GRE V and GRE A is 0.65, and the correlation between GRE Q and GRE A is 0.73, and the correlation between GRE V and GRE Q is 0.56, the following multiple regression equation can be derived:

Expected Z on GRE A = 0.53(Z on GRE Q) + 0.35(Z on GRE V)

Applying this formula to someone with a perfect V (Z = 2.69) and a perfect Q (Z = 1.8), gives:

Expected Z on GRE A = 0.53(1.8) + 0.35(2.69)

Expected Z on GRE A = 0.954 + 0.942

Expected Z on GRE A = 1.9 (standard error = 0.62)

Now in the SSS, the GRE A has a mean of 579.7 and an SD of 117.6, so a score of 800 has a Z score of 1.87, which is 0.05 Standard errors below the expected score of someone who scored 800 on V and Q.  What this means is that of the one in 707 SSS who scored perfect on BOTH V and Q, one in 1.9 would also score perfect on A.

Thus, those who achieve the perfect trifecta have a rarity of one in 707(1.9) = 1,343, within an already highly filtered group.  In a normal curve, one in 1,343 is 3.17 standard deviations above the mean, but because the SSS has a mean IQ of 120 and SD of 14, it equates to:

IQ = 3.17(14) + 120

IQ = 44 + 120

IQ = 164 (U.S. norms)

In theory, if all American young adults in the late 20th century had taken the GRE, only about one in 100,000 should have scored perfect on all three sections, and only about one in 136,000 white American young adults should have done so.

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