One might wonder how a man like Ron Hoeflin (who had very little income or power) was able to create a test that measured one in a million level U.S. intelligence. After all , wouldn’t you need to administer a test to several million Americans to reliably estimate how well a one a million mind would perform?

But the clever Hoeflin found a shortcut called equipercentile equating, and when I first heard about it a couple decades ago, it blew my mind. Hoeflin didnt need to give the Mega Test to millions of people because millions of people had already taken the SAT, so all Hoeflin had to do was ask people who took his Mega Test to reveal their SAT scores. Assuming they didn’t lie or selectively report, it was simply a matter of equating the distribution of Mega scores with the distribution of SAT scores and then equating a one in a million SAT score (perfect 1600 pre-1995) with its counterpart in the Mega score distribution.

The Prometheus MC Report explained it as follows:

**If one assumes that raw scores on the Mega and the SAT are monotonically related to mental ability, i. e., that a higher raw score on either test correlates with higher mental ability, then there is some function z _{1}(n) that relates raw scores on the Mega to standard intelligence scores z and there is some function z_{2}(m) that relates raw scores on the SAT to standard scores z, where z = (IQ-100)/16. It is plausible to assume that the joint probability distribution of z_{1} and z_{2} is just the bivariate normal distribution p(z_{1},z_{2},r) for some correlation r. This function is symmetric in z_{1} and z_{2}. Thus, for any random sample for which raw scores exist for both the SAT and Mega, if we have n scores with z_{1} > 4, then we would expect n scores with z_{2} > 4. These would not generally be the same n individuals in each case. Thus, if we know the 1-in-30,000 cutoff on the SAT (raw score=1560), and if there are N people in the sample of people taking both the SAT and the Mega scoring at this level or higher on the SAT, then counting down the highest N Mega scores from the sample would give a reasonable estimate of the 4-sigma cutoff on the Mega (raw score=36). Ron Hoeflin showed that, if you do this for several different cutoffs, then the resulting Mega normalization is linear over a range of scores including 36. This linearity feature seems to be standard on IQ tests over their range of applicability.**

**We are aware that there are difficulties in this argument (e.g., with respect to self-reporting of SAT scores, nonrandomness of sampling, small sample sizes, and mathematically allowed but “unphysical” test scores associated with ceiling effects).**

To this day I have no idea what “unphysical” test scores are by they sound fascinating!

In my opinion equipercentile equating only works if (1) both tests being paired are more or less equally correlated with g (or some other factor(s)), and (2) the people in your sample were not selected by one of the tests being equated. So if your sample were Harvard undergrads, you wouldn’t want to equate Mega scores with SAT scores because the undergrads were selected by the SAT and thus would be expected to regress to the mean on any other test they take. Much better to equate their LSAT scores to their Mega scores.

**Equipercentile equating the PATMA & the WAIS**

I noticed that at least six commenters had reported both their PATMA and WAIS scores. The first two columns of table 1 rank the commenters by their PATMA scores. The second two rank them by their WAIS scores. This side by side ranking allows us to equate PATMA scores with their WAIS equivalent. In some cases commenters took a version of the WAIS over a decade after its norms were published so scores might be inflated by a few points because of the Flynn effect.

Name | PATMA score | WAIS IQ | Name |
---|---|---|---|

Ganzir | 10 | 150+ | Teffec P |

Teffec P | 10 | 150 | Ganzir |

Dexter | 9 | 141 | Billy |

Billy | 8 | 133 | Dexter |

Gman | 7 | 120 | Gman |

Illuminaticat | 6 | 112 | Illuminaticat |

Although the sample size is small, preliminary data suggests an absurdly high correlation between the two tests (r = 0.94!). Indeed the line of best fit predicts:

Expected WAIS IQ = 9.025(PATMA score) + 59.125

Instead if we graph the data by score of equivalent rank (rather than scores made by the same person) it seems the PATMA has an incredibly linear relationship to IQ (r = 0.9975) expressed by the equation:

WAIS IQ Equivalent = 9.625(PATMA score) + 54.125:

Note the subtle distinction between formulas. The first is telling us the expected WAIS IQ of someone with a given PATMA score. The second is telling us what WAIS IQ you’d get if you performed as well or as poorly on the WAIS as you did on the PATMA.

Notice that only two people in this extremely bright sample scored a perfect 10 on the PATMA and only two people also scored 150+ on the WAIS (incidentally the same two people). This suggests a perfect 10 on the PATMA equates to about a 150 IQ (+3.33 SD) however this is a conservative estimate because Teffec P suspects his PATMA score might have benefited from reading this blog (teaching to the test) and his WAIS score was likely suppressed by ceiling bumping (his median scaled score was the highest possible scaled score and much higher than his mean scaled score). Correcting for this suggests that on a ceiling-free WAIS he would have scored 164. So if his PATMA score is too high and his WAIS score is too low, than we get a very different equivalents:

Name | PATMA score | WAIS IQ | Name |
---|---|---|---|

Ganzir | 10 | 164 | Teffec P |

Dexter | 9 | 150 | Ganzir |

Teffec P | 8 | 141 | Billy |

Billy | 8 | 133 | Dexter |

Gman | 7 | 120 | Gman |

Illuminaticat | 6 | 112 | Illuminaticat |

However for now I’ll stick to the flawed but real data in table 1 rather than the speculation that is table 2.

Ganzir

said:“To this day I have no idea what “unphysical” test scores are by they sound fascinating!”

I think this means extrapolated “scores” over the ceiling of the test, e.g., Grady Towers’ claim that a perfect score on the Titan Test corresponded to 1780 on the SAT, which has a ceiling of 1600. See the chart at the bottom of the page, where unphysical scores are in parentheses: http://miyaguchi.4sigma.org/hoeflin/titan/gradynorm.html

City Slicker

said:Just to add, my WAIS-IV IQ is 132 and my PATMA 8. My younger brother has PATMA 6 and WAIS-IV 108.

pumpkinperson

said:Both you AND your brother took the WAIS-IV? That’s unusual.

City Slicker

said:I have a student of psychology in the family who has access to the test who performed it on us for free.

pumpkinperson

said:Cool! Thanks for the data!

aronkibly

said:I got an 8 on PATMA. I have never taken an actual IQ test, but in elementary school I took a test called the Cognitive Abilities Test (CogAT) and scored 123.

pumpkinperson

said:Thanks for the data

aronkibly

said:Anytime, Pumpkin.

GondwanaMan

said:Wow i dont know if this is blowing my mind simply because im high or what is going on….im driving to Georgia right now I hope all the regular commenters are still here, you guys have been an inspiration…

pumpkinperson

said:OMG good to hear from you!

kakaak

said:When did Gman take the WAIS?

GondwanaMan

said:It was March 2015. Around the time I started looking at this blog….time flies!!!!

KC Armstrong

said:i got a 900 on the sat because i circled random bubbles and a 7 on da patma

rnfn

said:Did u correspond with all of these ppl via email and have em send u legitimate test results?

pumpkinperson

said:No it’s self-reported data but in most cases they provided extensive details about their WAIS scores which are hard to fake unless they know a lot about test

RuralRuski

said:This approach is a great thing, but there is one thing I dont understand. PP, if you would, can you please bring some light on it. Lets assume first that somebody has taken 10 iq tests in a row, all of the tests are recognized as those having good psychometrical properties etc , etc. Let’s say , he got a range of final scores , averaging, say, on 130. For simplicity, let’s imagen, that a distribution of the scores was roughly normal, moda and the average coincided – I dont know – at IQ=130.But we all know that if we consider the entire experiment as some meta-test, his ‘composite score’ or ‘overall Iq” is higher. We knew, for instance , an intercorellacion between those testes and came up with – ooops – Iq =140/Then, If we put a test number 11 ( we want to norm it up, that test is still under construction ) and the guy got a raw score 90/100, what iq this result would be equivalent to? statistically – Im just guessing – its 130. Is something wrong with the whole picture , PP? thank you

pumpkinperson

said:Equipercentile equating would obviously be extremely unreliable with only one person in the sample but the IQ we’d assign the 11th test would be 130 since that’s what the other 10 tests equated it to (on average)

But why is this lower than the composite IQ? Because as I stated in the article, in my opinion equipercentile equating only works if the tests have similar correlations with g (or whatever caused the correlations). Typically a composite score, especially one based on many tests, will be much more g loaded than the subtests it’s detived from.

Brilliant question btw