The analogy that I’m objecting to is, if you looked at e.g. the total for a ledger or a budget, it is an index that sums together expenses in a much more straightforward way. For instance if there is a large expense, the total is large.
Meanwhile, IQ scores are more like the geometric mean of the entries on such an entity. The geometric mean tells you whether the individual items tend to be large or small, which gives you broad-hitting information that distinguishes e.g. people who live in high-income countries from people who live in low-income countries, or large organizations from individual people; but it won’t inform you if someone got hit by a giant medical bill or if they managed to hack themselves to an ultra-cheap living space. These pretty much necessarily have to be low-rank mediators (like in the g model) rather than diverse aggregates (like in the sum model).
(Well, a complication in this analogy is that a ledger can vary not just in the magnitude of the transfers but also qualitatively in the kinds of transfers that are made, whereas IQ tests fix the variables, making it more analogous to a standardized budget form (e.g. for tax or loan purposes) broken down by stuff like “living space rent”, “food”, “healthcare”, etc..)
So, the arithmetic and geometric mean agree when the inputs are equal, and, the more unequal they are, the lower the geometric mean is.
I note that the subtests have ceilings, which puts a limit on how much any one can skew the result. Like, if you have 10 subtests, and the max score is something like 150, then presumably each test has a max score of 15 points. If we imagine someone gets five 7s and five 13s (a moderately unbalanced set of abilities), then the geometric mean is 9.54, while the arithmetic mean is 10. So, even if someone were confused about whether the IQ test was using a geometric or an arithmetic mean, does it make a large difference in practice?
The people you’re arguing against, is it actually a crux for them? Do they think IQ tests are totally invalid because they’re using an arithmetic mean, but actually they should realize it’s more like a geometric mean and then they’d agree IQ tests are great?
If you consider the “true ability” to be the exponential of the subtest scores, then the extent to which the problem I mention applies depends on the base of the exponential. In the limiting case where the base goes to infinity, only the highest ability matter, whereas in the limiting case where the base goes to 1, you end up with something basically linear.
As for whether it’s a crux, approximately nobody has thought about this deeply enough that they would recognize it, but I think it’s pretty foundational for a lot of disagreements about IQ.
The analogy that I’m objecting to is, if you looked at e.g. the total for a ledger or a budget, it is an index that sums together expenses in a much more straightforward way. For instance if there is a large expense, the total is large.
Meanwhile, IQ scores are more like the geometric mean of the entries on such an entity. The geometric mean tells you whether the individual items tend to be large or small, which gives you broad-hitting information that distinguishes e.g. people who live in high-income countries from people who live in low-income countries, or large organizations from individual people; but it won’t inform you if someone got hit by a giant medical bill or if they managed to hack themselves to an ultra-cheap living space. These pretty much necessarily have to be low-rank mediators (like in the g model) rather than diverse aggregates (like in the sum model).
(Well, a complication in this analogy is that a ledger can vary not just in the magnitude of the transfers but also qualitatively in the kinds of transfers that are made, whereas IQ tests fix the variables, making it more analogous to a standardized budget form (e.g. for tax or loan purposes) broken down by stuff like “living space rent”, “food”, “healthcare”, etc..)
So, the arithmetic and geometric mean agree when the inputs are equal, and, the more unequal they are, the lower the geometric mean is.
I note that the subtests have ceilings, which puts a limit on how much any one can skew the result. Like, if you have 10 subtests, and the max score is something like 150, then presumably each test has a max score of 15 points. If we imagine someone gets five 7s and five 13s (a moderately unbalanced set of abilities), then the geometric mean is 9.54, while the arithmetic mean is 10. So, even if someone were confused about whether the IQ test was using a geometric or an arithmetic mean, does it make a large difference in practice?
The people you’re arguing against, is it actually a crux for them? Do they think IQ tests are totally invalid because they’re using an arithmetic mean, but actually they should realize it’s more like a geometric mean and then they’d agree IQ tests are great?
If you consider the “true ability” to be the exponential of the subtest scores, then the extent to which the problem I mention applies depends on the base of the exponential. In the limiting case where the base goes to infinity, only the highest ability matter, whereas in the limiting case where the base goes to 1, you end up with something basically linear.
As for whether it’s a crux, approximately nobody has thought about this deeply enough that they would recognize it, but I think it’s pretty foundational for a lot of disagreements about IQ.