In general, if we have two vectors X and Y which are jointly normally distributed, we can write the joint mean μ and the joint covariance matrix Σ as
μ=[μXμY],Σ=[KXXKXYKYXKYY]
The conditional distribution for Y given X is given by Y|X∼N(μY|X,KY|X), defined by conditional mean
μY|X=μY+KYXKXX−1(X−μX)
and conditional variance
KY|X=KYYKXX−1KXY
Our conditional distribution for the IQ of the median rationalist, given their SAT score is N(0+(0.8⋅1⋅(2.42−0)),1−(0.8∗1∗0.8))=N(1.94,0.36) (That’s a mean of 129 and a standard deviation of 9 IQ points.)
Our conditional distribution for the IQ of the median rationalist, given their height is N(0+(0.2∗1∗(1.85−0)),1−(0.2∗1∗0.2))=N(0.37,0.96) (That’s a mean of 106 and a standard deviation of 14.7 IQ points.)
Our conditional distribution for the IQ of the median rationalist, given their SAT score and height is N(0+([0.80.2][10.160.161]−1[2.421.85]),1−([0.80.2][10.160.161]−1[0.80.2]))=N(2.04,0.35)(That’s a mean of 131 and a standard deviation of 8.9 IQ points)
Unfortunately, since men are taller than women, and rationalists are mostly male, we can’t use the height as-is when estimating the IQ of the median rationalist (maybe normalizing height within each sex would work?).
Eric Neyman is right. They are both valid!
In general, if we have two vectors X and Y which are jointly normally distributed, we can write the joint mean μ and the joint covariance matrix Σ as
μ=[μXμY],Σ=[KXXKXYKYXKYY]The conditional distribution for Y given X is given by Y|X∼N(μY|X,KY|X),
defined by conditional mean
μY|X=μY+KYXKXX−1(X−μX)
and conditional variance
KY|X=KYYKXX−1KXY
Our conditional distribution for the IQ of the median rationalist, given their SAT score is N(0+(0.8⋅1⋅(2.42−0)),1−(0.8∗1∗0.8))=N(1.94,0.36)
(That’s a mean of 129 and a standard deviation of 9 IQ points.)
Our conditional distribution for the IQ of the median rationalist, given their height is N(0+(0.2∗1∗(1.85−0)),1−(0.2∗1∗0.2))=N(0.37,0.96)
(That’s a mean of 106 and a standard deviation of 14.7 IQ points.)
Our conditional distribution for the IQ of the median rationalist, given their SAT score and height is N(0+([0.80.2][10.160.161]−1[2.421.85]),1−([0.80.2][10.160.161]−1[0.80.2]))=N(2.04,0.35)(That’s a mean of 131 and a standard deviation of 8.9 IQ points)
Unfortunately, since men are taller than women, and rationalists are mostly male, we can’t use the height as-is when estimating the IQ of the median rationalist (maybe normalizing height within each sex would work?).