A reader sent me the following email:

Dear PP, 
I recently stumbled upon your blog and find it quite fascinating. I’ve been interested in IQ and psychometrics for many years and have taken many online assessments and standardized exams to gauge my own IQ. Interestingly enough, however, it seems that the more tests I take, the less conclusive my IQ range becomes due to high variances in scores. 
This is why I’m turning to you, an expert on this topic, to request an estimation. For one, in elementary school I was tested for the gifted program and unfortunately rejected. This tells me that my childhood IQ was beneath 130, which I believe is the cutoff for most school gifted programs.

Yes, but the mere fact that you were tested for gifted suggests you were probably close. Perhaps 125 (U.S. norms).

The reader continues:

However, all the standardized exams that I’ve taken (which you’ve claimed to be valid proxies for IQ) seem to tell a different story.  My scores are as follows: 
SAT (Taken 2017) – Math 800, Verbal 720 
ACT (Taken 2017) – 34 Composite 
I was also ranked 6/945 in my high school class for GPA. I figured this might be relevant given the large sample size and the notion that academic performance correlates with IQ. Since my high school was in a lower-middle class area, we can assume a relatively normally distributed sample with an average akin to the American population average. All the students ranked ahead of me were Asian, and I am South Asian myself (Indian). 

A post-2016 SAT score of 1520 equates to an IQ of about 132. I’ve done very little research on the ACT but apparently a 34 on the SAT is like 1535 on the post-2016 SAT which equates to an IQ of 133.

The reader continues:

I’m also currently an undergraduate (a double major in mathematics and economics) and have been studying for the GRE. My diagnostic score for the GRE, with little to no preparation, was a 157 Verbal and 166 Quantitative. I’ve since improved to a 163 Verbal and 167 Quantitative for my second practice exam. 

Averaging across both sittings, you scored 160 V and 167 Q for a composite of 327. Preliminary research suggests this equates to an IQ of 143.

Averaging the SAT, ACT & GRE together, your mean IQ equivalent on college admission tests is 136. Normally I don’t endorse the averaging of tests but these particular tests are so similar that it’s not worth the trouble of considering them different tests.

Meanwhile, your score on an official IQ test was probably around 125.

Given that official IQ tests correlate no more than 0.7 with college admission tests, your score on a composite of both types of tests would be about 134.

I had a couple follow-up questions which the reader answered in another email:

…

my socioeconomic background is generally lower-middle class. My high school was located in the suburbs but had many students from the nearby urban area…
One caveat to my socioeconomic status, which applies particularly to my situation, was that I grew up in a single-parent household. Both my parents were highly educated (held graduate degrees) but my father unfortunately passed away when I was young.

I’m very sorry to hear that, but at least he lived long enough to pass on his high IQ genes through you.

This obviously influenced my household income. 
As for my ethnic background, I come from a Brahmin ancestry. Although I’m not a practicing Hindu, I recognize that this could have played a role in my generally well-educated familial background. 

Indeed!

I recently got an email from a college age man who was concerned that his VERBAL KNOWLEDGE INDEX (as I call it) was in the genius range (on both the children and adult version of the Wechsler) despite the rest of his cognitive profile being mediocre or low. When he asked his psychiatrist to interpret the results there was no helpful reply, so a friend of his suggested he contact me.

I’m not a psychiatrist so my opinions are for entertainment purposes only.

The first thing I did was correct all his subtest scores for norm inflation because the WISC-IV norms were 11 years old when he was tested and the WAIS-IV norms were a decade old. The sources I used were pg 240 of James Flynn’s Are We Getting SMARTER? and this table found here:

Such corrections are approximate because one can’t always assume that the rate of norm inflation can be extrapolated beyond the dates from which we have data and some subtests are so new, their rate of inflation had to be estimated using similar tests. In some cases there was norm deflation (see Coding in the table above, aka Digit Symbol).

The next thing I did was substitute the four index scores used on the WISC-IV and WAIS-IV with the five index scores used on the WISC-V. Again this gives only approximate results because the WISC-V index scores were built exclusively on WISC-V data and you’re not allowed to just substitute different versions, and in some cases I had to substitute subtests or adjust for not having the right number of subtests.

Nonetheless, the five factor model is so superior to the four-factor model, that for entertainment purposes only, I did it anyway.

The other liberty I took was calculating his overall IQ, by weighting all five indexes equally (the WISC-V gives equal weight to all core subtests, but more core subtests fall under some indexes than others).

	WISC-IV (2013)	WAIS-IV (2016)
VERBAL KNOWLEDGE INDEX	135	142
Similarities	17.23	18.4
Vocabulary	15.67	17.1
(Information)		12.55
(Comprehension)	11
SPATIAL INDEX	78	85
Block Design	5.56	6.73
Visual Puzzles
ABSTRACT INDEX	102	117
Matrix Reasoning	8.4	13.45
( Picture Concepts)	12.4
WORKING MEMORY INDEX	90	84
Digit Span	7.89	7.73
(Arithmetic)		7
(Letter number sequence)	8.89
PROCESSING INDEX	87	96
Digit Symbol	8.66	7.81
Symbol Search	6.66	10.81
OVERALL IQ	98	106

The first thing we notice is remarkable stability as we move from the children’s scale (2013) to the adult scale (2016). In 2013 his overall IQ was slightly below the U.S. mean of 100; in 2016 he scored slightly above, and even this modest increase might be partly explained by practice effect.

In 2013 his profile was verbal > abstract > working memory > processing > spatial. In 2016, verbal > abstract > processing > spatial > working memory. In other words there is a near perfect 0.95 correlation between his cognitive profile in 2013 and 2016, despite the fact that different versions of the Wechsler (with different questions) were used on each date.

Vertical reliability vs horizontal reliability

Reliability (not to be confused with stability) is typically measured by dividing all the items on a test in half in some random way (e.g. odd vs even numbered items). If the total score on all the odd number items correlates well with one’s score on all the even numbered items, this suggests your score was reliable, because it internally self-replicates. The reliability of the Wechsler scales are so high at the full-scale level that they are said to have a standard error of only 2 points, meaning in 2/3rd of all cases, one’s score is within 2 points of one’s “true” score and in 95% of cases, one’s score is within 4 points of one’s true score.

But what is true score? True overall score is the overall score one would get on the Wechsler if we could make every subtest infinitely long, yet factor out fatigue, practice effects, and ageing.

However I propose an alterative definition of true overall score: the overall score one would get on the Wechsler if we could increase the number of subtests to infinity, yet factor out fatigue, practice effects, and ageing. But since many subtests redundantly measure the same functions, what we really want to do is increase the number of index scores to if not infinity, then the maximum number that exist within the human mind. Let’s call this horizontal true score, to distinguish it from the typical definition of true score, which we can call vertical true score.

To measure horizontal true score, imagine we were doing a poll of the average IQ in a given school. If we tested five students, and they had an average IQ of 80 with an SD of 10, then the standard error of our poll would be 10 divided by the square root of our sample size.

Now instead of trying to find the average IQ of different people in a school, we’re instead trying to find the average index score of different talents within the same mind. Once we find the standard error of the average index score, we could convert it to standard error of overall IQ (because index scores are imperfectly correlated, one’s composite score on multiples indexes tends to be more extreme than one’s average index score). Multiplying the standard error by 1.96 and then adding and subtracting it from the overall IQ gives the 95% confidence band.

Based on the amount of scatter, I calculate that all we can say with 95% certainty is that the subject’s true horizontal overall score is anywhere from IQ 73 to IQ 122 (in 2013) and anywhere from IQ 78 to IQ 133 (in 2016).

So the subject is either very smart, or very not-smart but we can’t be more precise than that without running several more tests.

Career advise

Subject will be disapointed that I didn’t devote more content to this section, but there’s only so much career advise to be gained from Wechsler test results. He is a genius at overall verbal knowledge, but only slightly above average at social knowledge (comprehension), slight below average at processing and poor at working memory and arithmetic. His social knowledge is not poor (and given the low reliability of a single subtest, should not be over-interpreted), and when combined with his genius overall verbal knowledge, he would probably have a competitive edge. I suggest he pursue whatever he field he is passionate about, as long as it’s not closely related to law, STEM, clerical or visual technical skills because these require working memory, arithmetic, processing speed and spatial ability respectively.