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From Wikipedia's page on Mental age, we learn that:

Originally, the differences between mental age and chronological age were used to compute the intelligence quotient, or IQ. This was computed using the ratio method, with the following formula: mental age/chronological age * 100 = IQ. No matter what the child's chronological age, if the mental age is the same as the chronological age, then the IQ will equal 100.[4] An IQ of 100 thus indicates a child of average intellectual development. For a gifted child, the mental age is above the chronological age, and the IQ is higher than 140; for a mentally retarded child, the mental age is below the chronological age, and the IQ is below 70.

In this way, precocious children can get exceptionally high scores (e.g. scores above 160, s.d. 15), even if they answer correctly only a fraction of the total number of questions, provided the test is sufficiently hard in comparison with their actual chronological age. However, I am not aware of any tests for adults that discriminate accurately above the threshold of IQ $\sim$ 160 - their accuracy is even less than 4 s.d., taking into account a ceiling effect.

  • How can psychometry assess an IQ of 170 or 180 in adults when the rarity of such scores is 1 in several millions?
  • Can the IQ scores of exceptionally and profoundly gifted children be extrapolated into adulthood?

  • Is it true that that a profoundly gifted child with an IQ of 170 would score better than 99.9998% of the general population?

  • It seems that two different definitions of IQs are used here: is there a correlation between them?
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Is it true that he will score better than 99.9998% of the general population? From my understanding, prior to a certain age the IQ score is compared to the child's age group (thus the 'general population' is actually the 'comparative population')- whereas among adults the score in question is compared to the entire population of adults (actual 'general population'). You may also be interested in the question cogsci.stackexchange.com/questions/22/… –  BenCole Apr 8 '13 at 18:42
    
But if a child's score is compared to a population strictly of the same age, definitions used in psychometrics like exceptionally gifted or profoundly gifted become useless. I don't really know how IQ is assessed in children, but since gifted education programs request such unbelievably rare scores, I suppose that the age factor is taken somewhere into account. Question is wether the score obtained as a child will remain stable even into adulthood, at the very extreme of the distribution, or not. –  quark1245 Apr 8 '13 at 19:07
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exceptionally gifted or profoundly gifted Not at all! These are completely relevant within the age group. Later, after development, the comparison group becomes 'all adults', which amounts to reducing/eliminating the bias created by the age of the test-taker. However, all of the people tested before are still able to be tested now, meaning that the scores, in relation to each other will remain relatively the same. Unfortunately, I cannot definitively answer your last question (if that's the important question, I'd either change this StackExchange question or create a new one). –  BenCole Apr 8 '13 at 21:04
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1 Answer

General points about IQ measurement

IQ scores are standardised scores typically with a mean of 100 and a standard deviation of 15. As Ben Cole notes, IQ scores are typically normed relative to age groups in children, and then normed relative to the adult population for adults.

Can the IQ scores of exceptionally and profoundly gifted children be extrapolated into adulthood?

IQ scores correlate very highly over time. That said, correlations decrease as (a) the period of time between testing sessions increases, and (b) the earlier the age of initial testing (e.g., testing at 2,3, 4, 5 years of age, etc.). Presumably greater periods of cognitive change have the potential for greater individual differences in that change, and such greater potential for reductions in the correlation over time.

Here's an extract from a major review of IQ testing and intelligence relevant to stability that reiterates these points with sources (Neisser et al, 1996):

Stability. Intelligence test scores are fairly stable during development. When Jones and Bayley (1941) tested a sample of children annually throughout childhood and adolescence, for example, scores obtained at age 18 were correlated Y = .77 with scores that had been obtained at age 6 and r = .89 with scores from age 12. When scores were averaged across several successive tests to remove short-term fluctuations, the correlations were even higher. The mean for ages 17 and 18 was correlated r = .86 with the mean for ages 5, 6, and 7, and r = .96 with the mean for ages 11, 12, and 13. (For comparable findings in a more recent study, see Moffitt, Caspi, Harkness, & Silva, 1993.) Nevertheless, IQ scores do change over time. In the same study (Jones & Bayley, 1941), the average change between age 12 and age 17 was 7.1 IQ points; some individuals changed as much as 18 points.

So, in summary exceptionally gifted children will tend generally to be highly intelligent adults. However, there are a few notes of caution:

  1. Any IQ measurement has error. Thus, the IQ measurement is only an estimate of the individual's true IQ. For standard IQ tests, the size of this measurement error is greater as you move towards the tails of the distribution of IQ scores, such as is the case with the very gifted.
  2. There is an effect known as regression toward the mean. This says, among other things, that when two measures are imperfectly correlated, extreme scores on one variable will tend to be relatively less extreme on the other. This can be explained both in terms of measurement error (i.e., those identified as having a 160 IQ might just have got luck on the test) and in imperfect correlation of latent IQ over time. Thus, in general if you identify a sample of very high IQ children and select them, you may find that there average IQ drops at a second time point. They'll still of course be measured as very intelligent on average.

Is it true that that a profoundly gifted child with an IQ of 170 would score better than 99.9998% of the general population?

A child's IQ is defined relative to a population of children of the same age as that child, and not the general population.

Based on a definition of IQ as a normative score, you get the following percentiles using R:

> iq <- c(100, 115, 130, 145, 160, 170)
> percentile <- 100 * pnorm(iq, 100, 15)
> cbind(iq, percentile=round(percentile, 6))
      iq percentile
[1,] 100   50.00000
[2,] 115   84.13447
[3,] 130   97.72499
[4,] 145   99.86501
[5,] 160   99.99683
[6,] 170   99.99985

Thus, yes, you are correct in a theoretical sense that a true IQ of 170 by definition implies that that individual has an IQ greater than 99.99985% of the relevant normative population.

However, the reality is that you never measure true IQ. You only get responses to an individual on a test. And this measurement is made with error, and this error in measurement is typically greater at the high end. Thus, you get a measure with confidence intervals or some form of bayesian estimate of IQ. Furthermore, there is also error in measuring population norms as well. Thus, if you want to be more precise in your description of what a test says about an individual's IQ, then you need to specify a range on that individual's IQ, and this range would be larger at the high end, because intelligence tests are not typically optimised to the high end and normative estimates of IQ at the high end would be less precise.

References

  • Neisser, U., Boodoo, G., Bouchard Jr, T. J., Boykin, A. W., Brody, N., Ceci, S. J., ... & Urbina, S. (1996). Intelligence: Knowns and unknowns. American psychologist, 51(2), 77. PDF
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+1 This is a very meticulous answer. I am curious, what about high-range IQ tests? Are these tests considered to be valid by professionals? How do they correlate with more standard tests? And what are their findings - e.g. do the resulting distribution of exceptionally high scores agree with the gaussian model even in the extreme range? –  quark1245 Apr 9 '13 at 15:08
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