The component should have a smaller standard deviation, though. If A and B each have stdev=1 & are independent then A+B has stdev=sqrt(2).
I think that means that we’d expect someone who is +6 sigma on A+B to be about +3*sqrt(2) sigma on A in the median case. That’s +4.24 sigma, or 1 in 90,000.
They are +4.2SD on the genetic component of the property (= 1 in 90,000), but the median person with those genetics is still only +3SD on the overall property (= 1 in 750), right?
(That is, the expected boost from the abnormally extreme genetics should be the same as the expected boost from the abnormally extreme environment, if the two are equally important. So each of them should be half of the total effect, i.e. 3SD on the overall trait.)
With A & B iid normal variables, if you take someone who is 1 in a billion at A+B, then in the median case they will be 1 in 90,000 at A. Then if you take someone who is 1 in 90,000 at A and give them the median level of B, they will be 1 in 750 at A+B.
(You can get to rarer levels by reintroducing some of the variation rather than taking the median case twice.)
The component should have a smaller standard deviation, though. If A and B each have stdev=1 & are independent then A+B has stdev=sqrt(2).
I think that means that we’d expect someone who is +6 sigma on A+B to be about +3*sqrt(2) sigma on A in the median case. That’s +4.24 sigma, or 1 in 90,000.
They are +4.2SD on the genetic component of the property (= 1 in 90,000), but the median person with those genetics is still only +3SD on the overall property (= 1 in 750), right?
(That is, the expected boost from the abnormally extreme genetics should be the same as the expected boost from the abnormally extreme environment, if the two are equally important. So each of them should be half of the total effect, i.e. 3SD on the overall trait.)
Oh, you’re right.
With A & B iid normal variables, if you take someone who is 1 in a billion at A+B, then in the median case they will be 1 in 90,000 at A. Then if you take someone who is 1 in 90,000 at A and give them the median level of B, they will be 1 in 750 at A+B.
(You can get to rarer levels by reintroducing some of the variation rather than taking the median case twice.)