Also, given imperfect correlation it is unclear how one should convert the scores. If I pick someone with SAT in top 1% I shouldn’t expect IQ in the top 1% because of regression towards the mean.
The correlation is the slope of the regression line in coordinates normalised to unit standard deviations. Assuming (for mere convenience) a bivariate normal distribution, let F be the cumulative distribution function of the unit normal distribution, with inverse invF. If someone is at the 1-p level of the SAT distribution (in the example p=0.01) then the level to guess they are at in the IQ distribution (or anything else correlated with SAT) is q = F(c invF(p)). For p=0.01, here are a few illustrative values:
The standard deviation of the IQ value, conditional on the SAT value, is the unconditional standard deviation multiplied by c’ = sqrt(1-c^2). The q values for 1 standard deviation above and below are therefore given by qlo = F(-c’ + c invF(p)) and qhi = F(c’ + c invF(p)).
There are subtleties though. E.g. if we take some programming contest finalists / winners, and take their IQ scores, those are regressed towards the mean from their programming contest performance. Their other abilities will be regressed towards the mean from the same height, not from IQ. This might explain the dramatic cognitive skill disparity between, say, Mensa and some professional group of same IQs.
The correlation is the slope of the regression line in coordinates normalised to unit standard deviations. Assuming (for mere convenience) a bivariate normal distribution, let F be the cumulative distribution function of the unit normal distribution, with inverse invF. If someone is at the 1-p level of the SAT distribution (in the example p=0.01) then the level to guess they are at in the IQ distribution (or anything else correlated with SAT) is q = F(c invF(p)). For p=0.01, here are a few illustrative values:
The standard deviation of the IQ value, conditional on the SAT value, is the unconditional standard deviation multiplied by c’ = sqrt(1-c^2). The q values for 1 standard deviation above and below are therefore given by qlo = F(-c’ + c invF(p)) and qhi = F(c’ + c invF(p)).
There are subtleties though. E.g. if we take some programming contest finalists / winners, and take their IQ scores, those are regressed towards the mean from their programming contest performance. Their other abilities will be regressed towards the mean from the same height, not from IQ. This might explain the dramatic cognitive skill disparity between, say, Mensa and some professional group of same IQs.