When a test mean is 100 and the standard deviation is 15, then a score of 160 should appear in approximately 1 out of 31,560 people. Correct?
That's the idea.
1. How is it possible to assess the rarity of scores above, say 145, when the normative samples for these tests are only a few thousand?
I don't know, but I tend to think that the higher the results are, the less accurate they may be, partly because of difficulties in crafting test questions for more and more rarefied and refined sorts of intelligence (I'm a layperson so go easy on me), but also because of lack of testing samples. I also wonder whether having a bad or good test day might not influence results much more in the upper ranges-- I'd be curious to know how much a single extra right answer would influence the final score on various tests. A person who can score an average of 180 on a modern IQ test might fluctuate more wildly than someone who averages lower.
2. Why is it that achievement scores of 160 are anecdotally way more common than IQ scores of 160?
I've wondered that myself. I think it's a combination of factors. IQ tests may be harder to construct so that they accurately test what they're designed to test, especially in the upper ranges, while intelligence that doesn't show well on a particular IQ test might easily still result in high achievement; hothousing effects may be much more exaggerated in younger testees; and there may be high achievers who simply aren't as natively intelligent, but "play up" due to determination and hard work.
3. On achievement tests like the WIAT and WJ, is it possible to interpret a raw score that exceeds the minimum required for a 160? (some kind of 'extended norms'?)
I don't know about the WIAT, but the WJ-III (Ach) is already able to generate scores above 160 without the need for extended norms or the like.
