I heard somewhere that IQ scores are normally distributed by definition, because they are calculated by projecting the measured rank onto the normal distribution with mean 100 and stddev 15. Can’t seem to find a reference on Wikipedia though, so maybe that’s not true.
IQ distributions are calibrated based on a reference sample, such that the reference sample has mean 100 and std 15 and follows a normal distribution. I believe the reference sample is generally British nationals or European Americans, so that interracial comparisons are sensible.
That doesn’t mean that the distribution of all test-takers follows a normal distribution with mean 100 and std 15.
Precisely. If you are looking at some third world nation, well, there’s all those kids who have various nutritional deficiencies, their IQs are impaired. The mean is lowered considerably, but that’s through introduction of extra variables into the (approximate) sum.
If you don’t take that into account and assume that only the mean in the distribution has changed, you get entirely invalid results at the high range due to how rapidly the normal distribution falls off far from the mean (as exponent of a square). For example if you were to calculate number of some rare geniuses out of the reference population (say, 300 millions with mean of 100 and standard deviation of 15), and from the world population assuming some lower mean and same standard deviation, for sufficiently rare “genius” you’ll get a smaller number of geniuses in the whole world than in that one reference population (which is ridiculous).
edit: which you can see by noting that this with c smaller than b grows as x grows (i.e. ratio of prevalences between two populations grows with distance from the mean).
The example I’d give here is India, where you have lots of mostly distinct ethnic groups, and so it’s reasonable to expect that the true distribution is a mixture of Gaussians. Knowing the Indian average national IQ would totally mislead you on the number of Parsis with IQs of 120 or above, if all you knew about Parsis was that they lived in India.
(It’s not clear to me that malnourishment leads to multiple modes, rather than just decreasing the mean while probably increasing the variance, because I think damage due to malnourishment is linear, and it’s probably the case that many different levels of severity of malnourishment are roughly equally well represented.)
In the limit, the mixture of Gaussians is a Gaussian.
It’s not clear to me that malnourishment leads to multiple modes, rather than just decreasing the mean while probably increasing the variance
Theoretically, malnourishment (given that only a part of the population suffers from it) should lead to a negatively skewed distribution. And yes, with a lower mean and higher variance.
In the limit, the mixture of Gaussians is a Gaussian.
Nope. The sum of Gaussian random variables is a Gaussian random variable, but a mixture Gaussian model is a very different thing. (In particular, mixture Gaussians are useful for modeling because their components are easy to deal with, but if you have infinite mixtures you can faithfully represent an arbitrary distribution.)
Theoretically, malnourishment (given that only a part of the population suffers from it) should lead to a negatively skewed distribution.
(It’s not clear to me that malnourishment leads to multiple modes, rather than just decreasing the mean while probably increasing the variance, because I think damage due to malnourishment is linear, and it’s probably the case that many different levels of severity of malnourishment are roughly equally well represented.)
Not everyone’s malnourished, though—a significant number of people are into diminishing returns, nutrition wise. It’s very nonlinear in the sense that as long as there’s adequate nutrition, it plate-outs—access to more nutrition does not improve anything.
I heard somewhere that IQ scores are normally distributed by definition, because they are calculated by projecting the measured rank onto the normal distribution with mean 100 and stddev 15. Can’t seem to find a reference on Wikipedia though, so maybe that’s not true.
IQ distributions are calibrated based on a reference sample, such that the reference sample has mean 100 and std 15 and follows a normal distribution. I believe the reference sample is generally British nationals or European Americans, so that interracial comparisons are sensible.
That doesn’t mean that the distribution of all test-takers follows a normal distribution with mean 100 and std 15.
Precisely. If you are looking at some third world nation, well, there’s all those kids who have various nutritional deficiencies, their IQs are impaired. The mean is lowered considerably, but that’s through introduction of extra variables into the (approximate) sum.
If you don’t take that into account and assume that only the mean in the distribution has changed, you get entirely invalid results at the high range due to how rapidly the normal distribution falls off far from the mean (as exponent of a square). For example if you were to calculate number of some rare geniuses out of the reference population (say, 300 millions with mean of 100 and standard deviation of 15), and from the world population assuming some lower mean and same standard deviation, for sufficiently rare “genius” you’ll get a smaller number of geniuses in the whole world than in that one reference population (which is ridiculous).
edit: which you can see by noting that this with c smaller than b grows as x grows (i.e. ratio of prevalences between two populations grows with distance from the mean).
The example I’d give here is India, where you have lots of mostly distinct ethnic groups, and so it’s reasonable to expect that the true distribution is a mixture of Gaussians. Knowing the Indian average national IQ would totally mislead you on the number of Parsis with IQs of 120 or above, if all you knew about Parsis was that they lived in India.
(It’s not clear to me that malnourishment leads to multiple modes, rather than just decreasing the mean while probably increasing the variance, because I think damage due to malnourishment is linear, and it’s probably the case that many different levels of severity of malnourishment are roughly equally well represented.)
In the limit, the mixture of Gaussians is a Gaussian.
Theoretically, malnourishment (given that only a part of the population suffers from it) should lead to a negatively skewed distribution. And yes, with a lower mean and higher variance.
Nope. The sum of Gaussian random variables is a Gaussian random variable, but a mixture Gaussian model is a very different thing. (In particular, mixture Gaussians are useful for modeling because their components are easy to deal with, but if you have infinite mixtures you can faithfully represent an arbitrary distribution.)
Yep, I should have mentioned that also.
Yes, you are correct, I got confused between a sum and a mixture.
Not everyone’s malnourished, though—a significant number of people are into diminishing returns, nutrition wise. It’s very nonlinear in the sense that as long as there’s adequate nutrition, it plate-outs—access to more nutrition does not improve anything.