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.
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.