Thank you for pointing that high IQ problem is probably a statistical effect rather than "too much of a good thing" effect. That was very interesting.
Let me attempt the problem from a simple mathematical point of view.
Let basketball playing ability, Z, is just a sum of height, X, and agility, Y. Both X and Y are Gaussian distributed with mean 0 and variance 1. Assume X and Y are independent.
So, if we know that Z>4, what is the most probable combination of X and Y?
The probability of X>2 and Y>2 is:
P(X>2)P(Y>2)=5.2e-4
The probability of X>3 and Y>1 is:
P(X>3)P(Y>1)=2.1e-4
So it is more than two times more likely for both abilities to be +2Std than one them is +3Std and the other is +1Std.
I think it can be shown rigorously that the most probable combination is Z/N for each component if there are N independent identically distributed components of an ability.
Thank you for pointing that high IQ problem is probably a statistical effect rather than "too much of a good thing" effect. That was very interesting.
Let me attempt the problem from a simple mathematical point of view.
Let basketball playing ability, Z, is just a sum of height, X, and agility, Y. Both X and Y are Gaussian distributed with mean 0 and variance 1. Assume X and Y are independent.
So, if we know that Z>4, what is the most probable combination of X and Y?
The probability of X>2 and Y>2 is: P(X>2)P(Y>2)=5.2e-4
The probability of X>3 and Y>1 is: P(X>3)P(Y>1)=2.1e-4
So it is more than two times more likely for both abilities to be +2Std than one them is +3Std and the other is +1Std.
I think it can be shown rigorously that the most probable combination is Z/N for each component if there are N independent identically distributed components of an ability.