Today’s post was supposed to be about Ann Coulter, but I thought I’d write a quick post about a commenter who is currently using the pseudoname “Jesse Watters”  who claims to have obtained some freakishly high scores on the WAIS-IV.

Keep in mind, I’m not a professional, but I do know a bit about psychometrics.

The WAIS-IV is an IQ test that actually consists of 15 mini IQ tests (subtests), however to differentiate the subtest scores from overall IQ scores, the subtest scores are expressed as scaled scores.  Unlike the overall IQ scores, which have a mean of 100 and a standard deviation of 15 in the U.S. population, the scaled scores have a mean of 10 and an SD of 3, and as commenter Animekitty noted, can be converted to IQ equivalents by multiplying by 5 and then adding 50.

So commenter “Jesse Watters” claims to have obtained the following scaled scores on the 15 WAIS-IV subtests:

Similarities: 19
Vocabulary: 17
Information: 19
(Comprehension: 19)

Block Design: 19
Matrix Reasoning; 18
Visual Puzzles: 19
(Figure Weights: 19)
(Picture Completion: 19)

Digit Span: 19
Arithmetic Reasoning: 19
(Letter-Number Sequencing: 19)

Symbol Search: 19
Coding: 18
(Cancellation: 19)

Now the subtests in brackets are supplementary tests, which are not supposed to be used to calculate the full-scale IQ unless one of the other subtests gets “spoiled” or is deemed inappropriate for that subject a priori, however since he took all 15 subtests, I am going to sum them all, and then prorate to estimate his sum of scaled score if only the 10 core subtests were given.  The prorated sum is 187.3, remarkably close to the 186 he actually obtained on the 10 core subtests.

Prorating works!

Unfortunately, the WAIS-IV assigns all sum of scaled scores of 181 or higher, an IQ of 160. however we can estimate, from the roughly linear relationship between sum of scaled scores and IQ, that he is above 160.

For example:

sum of scaled scores           IQ

10                                        40

33                                        55

55                                        70

79                                         85

100                                       100

122                                       115

141                                       130

162                                      145

182                                      160

From here we can deduce that full-scale IQ = 0.697057(sum of scaled scores) + 31.533486.

Plugging in “Jesse Watters’s” prorated sum of scaled scores of 187.3 into this formula gives a full-scale IQ of 162, however even this might be an underestimate, because look at his scaled scores, when listed from lowest to highest:

17,18,18,19,19,19,19,19,19,19,19,19,19,19,19

Notice how his lowest scaled scored (17) is below his median scaled score (19), but his highest scaled score (19) is equal to his median scaled score (19).  That’s not a symmetrical distribution of scaled scores, but one that is truncated by the fact that the WAIS-IV does not give scaled scores beyond 19.  Now if I knew “Jesse Watters’s” raw scores and age, I might be able to extrapolate some of his scaled scores beyond 19, but without that information, all I can do is calculate that a scaled score of 17 is at the 7th  percentile of Jesse’s distribution of scaled scores (normalized Z = -1.5) , and that a scaled score of 18 is at the 20th percentile of his distribution (normalized Z = -0.87). [NOTE: these Z scores are with reference to Jesse’s other scaled scores, not with reference to other people]

Assuming his distribution of scaled scores would have been Gaussian had the artificial ceiling of 19 not been imposed on each subtest, we can extrapolate linearly from those two Z scores to guess that a Z score of 0 in his distribution of scores (the median score) would have been 19.385.  Since the median score (50 percentile) is the average score in a normal distribution, and since the sum of scaled scores is just the average scaled score multiplied by the number of scaled scores, we can guess that if not for the artificial ceiling, Jesse’s sum of 10 scaled scores (actually he has 15, but we will prorate) would be:

Sum of scaled scores = (Median scaled score)(10)

Sum of scaled scores = (19.385)(10)

Sum of scaled scores = 194

Now converting sum of scaled scores to full-scale IQ using the formula I suggested above:

full-scale IQ = 0.697057(sum of scaled scores) + 31.533486

full-scale IQ = 0.697057(194) + 31.533486

full-scale IQ = 167 (U.S. norms)

If his self-reported scores are true, that would make him almost certainly the smartest person to ever post on this blog, or any other blog you frequently visit.  Because only one in 251,515, Americans have a deviation IQ of 167+.

An IQ of 167 is 4.47 standard deviations above the U.S. mean.  To put that in perspective, the average American young adult (men and women combined) is about 5’7″ with a standard deviation of about 4 inches.  Thus being 4.47 standard deviations above average in height (the vertical equivalent of a 167 IQ) would be a height of 7’1″!

I imagine some readers might object to me calculating his sum of scaled scores on a theoretically ceiling-less WAIS-IV and then applying a formula that equates summed scaled scores to IQ on the actual WAIS-IV (which does have subtest ceilings), however because that formula is based on a linear relationship across the full range of WAIS-IV scores (the vast majority of which are unaffected by ceiling bumping) it’s unlikely to make much if any difference.

But I’m just an amateur so don’t take my calculations too seriously!

And “Jesse Watters” is just a pseudonymous commenter.  His scores have not been verified.

It should also be noted that although the Wechsler scales are probably the most accurate IQ tests in existence, not even they correlate perfectly with the putative general mental ability (the famous g factor) or mental ability in general, so anyone who scores this freakishly high, likely got a bit lucky, either in not making mistakes, or in the selection of talents the test sampled.  He should not be surprised if he regresses to the mean on other standardized tests, especially low quality ones.

[Update March 5, 2016:  The WAIS-IV was normed in 2006, a decade ago, so depending on how recently “Jesse Watters” was tested, he might have to deduct as much as 3 IQ points from his full-scale IQ since old norms inflate IQs at a rate of 0.3 points per year.  Of course we don’t know if the Flynn effect has still be in full force in the last decade, and because the WAIS-IV expresses IQs in U.S. norms, instead of U.S. white norms,  as the original Wechsler scales did, it’s even possible that environmental gains in IQ, have been negated by demographic shifts]

[Update March 16, 2020: I am grateful to commenter A Very Purple & Skeptical Entity for pointing out a couple math errors that I have just now corrected]