That’s a really interesting point. I’m not sure how to do the math myself, so I wonder if anyone can help verify this. Also, is this assuming a population-average upbringing/education? What if we give the clones the best upbringing and education that money can buy? (I assumed a budget of $10 million per clone in the OP.)
For a normally distributed property 1/billion is +6 sigma, while +3 sigma is 1⁄750. If a property is normally distributed, the clones share 50% of the variation, and von neumann is 1 in a billion, then I think it’s right that our median guess for the median clone should be the 1 in 750 level.
(But of your 100,000 clones several of them will be at the one in a trillion level, a hundred will be more extreme than von neumann, and >20,000 of them will be one in 20,000. I’m generally not sure what you are supposed to infer from the “one in X” metric. [Edited to add: all of those are the fractions in expectation, and they are significant underestimates because they ignore the uncertainty in the genetic component.])
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.)
500 seems too small. If someone is 1 in 30,000 on A and 1 in 30,000 on B, then about 1 in a billion will be at least as extreme as them on both A and B. That’s not exactly the number that we’re looking for but it seems like it should give the right order of magnitude (30,000 rather than 500).
And it seems like the answer we’re looking for should be larger than 30,000, since people who are more extreme than them on A+B includes everyone who is more extreme than them on both A and B, plus some people who are more extreme on only either A or B. That would make extreme scores on A+B more common, so we need a larger number than 30,000 to keep it as rare as 1 in a billion.
I might be totally mistaken here, but the calculation done by Donald Hobson and Paul seems to assume von Neumann’s genes are sampled randomly from a population with mean IQ 100. But given that von Neumann is Jewish (and possibly came from a family of particularly smart Hungarian Jews; I haven’t looked into this), we should be assuming that the genetic component is sampled from a distribution with higher mean IQ. Using breeder’s equation with a higher family mean IQ gives a more optimistic estimate for the clones’ IQ.
That’s a really interesting point. I’m not sure how to do the math myself, so I wonder if anyone can help verify this. Also, is this assuming a population-average upbringing/education? What if we give the clones the best upbringing and education that money can buy? (I assumed a budget of $10 million per clone in the OP.)
For a normally distributed property 1/billion is +6 sigma, while +3 sigma is 1⁄750. If a property is normally distributed, the clones share 50% of the variation, and von neumann is 1 in a billion, then I think it’s right that our median guess for the median clone should be the 1 in 750 level.
(But of your 100,000 clones several of them will be at the one in a trillion level, a hundred will be more extreme than von neumann, and >20,000 of them will be one in 20,000. I’m generally not sure what you are supposed to infer from the “one in X” metric. [Edited to add: all of those are the fractions in expectation, and they are significant underestimates because they ignore the uncertainty in the genetic component.])
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.)
500 seems too small. If someone is 1 in 30,000 on A and 1 in 30,000 on B, then about 1 in a billion will be at least as extreme as them on both A and B. That’s not exactly the number that we’re looking for but it seems like it should give the right order of magnitude (30,000 rather than 500).
And it seems like the answer we’re looking for should be larger than 30,000, since people who are more extreme than them on A+B includes everyone who is more extreme than them on both A and B, plus some people who are more extreme on only either A or B. That would make extreme scores on A+B more common, so we need a larger number than 30,000 to keep it as rare as 1 in a billion.
I might be totally mistaken here, but the calculation done by Donald Hobson and Paul seems to assume von Neumann’s genes are sampled randomly from a population with mean IQ 100. But given that von Neumann is Jewish (and possibly came from a family of particularly smart Hungarian Jews; I haven’t looked into this), we should be assuming that the genetic component is sampled from a distribution with higher mean IQ. Using breeder’s equation with a higher family mean IQ gives a more optimistic estimate for the clones’ IQ.