Is this true?

You are to be commended for asking whether the statement is true. I'd like to check that author's references, but I have reason to believe that he is on the right track.

I'd wager that the average IQ of top-rank mathematicians (say, the top 100 as judged by their peers) is at least three standard deviations above the mean, and more likely closer to 4.

You have indicated your view of the sources you have read by proposing a wager. I'll gladly stake a large amount of money on that wager, because you would surely lose. There have already been studies of the issue, after all. Definitional problems here for settling the wager among you, me, and our seconds would include defining just who the top 100 mathematicians are (I'd expect a lot of debate on that point), and which brand of IQ test should be taken to be the most definitive, as each IQ test battery disagrees with each other IQ test battery. But I have no doubt I would win the bet.

For one thing, four standard deviations above the population median (IQ score of 160 by current standard scoring conventions) is the very peak of reliable scoring on any currently normed brand of IQ test. Commenting on the higher numerical scores found in the scoring tables of the obsolete Stanford-Binet Form L-M IQ test, Christoph Perleth, Tanja Schatz, and Franz J. Mönks (2000) comment that "norm tables that provide you with such extreme values are constructed on the basis of random extrapolation and smoothing but not on the basis of empirical data of representative samples." "Early Identification of High Ability". In Heller, Kurt A.; Mönks, Franz J.; Sternberg, Robert J. et al.. International Handbook of Giftedness and Talent (2nd ed.). Amsterdam: Pergamon. p. 301. ISBN 978-0-08-043796-5. Lewis Terman recognized the core problem with IQ scores at the high end a long time ago:

"The reader should not lose sight of the fact that a test with even a high reliability yields scores which have an appreciable probable error. The probable error in terms of mental age is of course larger with older than with young children because of the increasing spread of mental age as we go from younger to older groups. For this reason it has been customary to express the P.E. [probable error] of a Binet score in terms of I.Q., since the spread of Binet I.Q.'s is fairly constant from age to age. However, when our correlation arrays [between Form L and Form M] were plotted for separate age groups they were all discovered to be distinctly fan-shaped. Figure 3 is typical of the arrays at every age level.

"From Figure 3 it becomes clear that the probable error of an I.Q. score is not a constant amount, but a variable which increases as I.Q. increases. It has frequently been noted in the literature that gifted subjects show greater I.Q. fluctuation than do clinical cases with low I.Q.'s . . . . we now see that this trend is inherent in the I.Q. technique itself, and might have been predicted on logical grounds." (Terman & Merrill, 1937, p. 44)

Alan S. Kaufman has a great discussion of error of estimation in IQ testing and variance in scores between one IQ test and another in his recent book IQ Testing 101. For much, much more on this issue, see

http://en.wikipedia.org/wiki/User:WeijiBaikeBianji/Intellige...

for further reading suggestions.

A quotation, from an interview with Stephen Hawking in the New York Times:

"Q: What is your I.Q.?

"A: I have no idea. People who boast about their I.Q. are losers."

http://www.nytimes.com/2004/12/12/magazine/12QUESTIONS.html