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The Sigma Test intends to be innovative in many aspects, assuming as its main objective to build problems of intrinsically high difficulty, not resorting abusively to questions that require specific mathematical knowledge, not sticking to traditional test models with series of figures or numbers, not artificially forging a high degree of difficulty as it involves combinatorial analysis problems that require counting a large number of possibilities, using exclusively unpublished questions at the highest levels of difficulty.

Furthermore, the Sigma Test seeks to distribute the questions over a broad spectrum of difficulty levels, trying to cover the range of 100 to 200 rIQ (equivalent to about 100 to 235 pIQ). It is unlikely that this objective has been achieved, as it is not achieved in the totality of other intelligence tests with nominal ceilings above 180, 190 and even above 200, but whose actual ceilings are close to 165 to 170.

The weighted count, depending on the difficulty, also contributes to making the results more accurate and more precise, preventing carelessness from unfairly damaging the test score. The acceptance of more than one answer as valid also represents an important advance in this regard. The way in which weights are determined as a function of difficulty and the point fraction for incomplete or sub-optimal answers is described in the article “2003 Sigma Test Standard”. Briefly, these weights are determined by the inverse of the number of correct answers and the number of people who gave incomplete/sub-optimal or equivalent answers.

In addition to varying the degree of difficulty, the questions also vary in the nature of the thought required to find the solution. Most questions 1 through 11 can be solved with convergent thinking, while questions 12 through 20 involve convergent thinking at a higher stage of complexity and also some divergent thinking at an elementary stage. Questions 21 through 28 progressively increase the amount of divergent thinking, until questions 29 onwards require powerful convergent thinking combined with powerful divergent thinking. Only people with remarkable logical thinking and great ingenuity can achieve high scores on this test.

From the preliminary rules, we can estimate that a person with normal intelligence would get 4 or 5 questions right. A mid-level academic with a Bachelor's degree in an arbitrary subject would get 9 or 10 questions right. An academic with a Master's degree would score 11 or 12. A Doctor would score 13 or 14 and could become a subscriber in  Sigma III . Mensa members, on average, would get 16 or 17 questions right and would qualify to join as full members in  Sigma Society . A mid-level Doctor in the Exacts area would get 18 or 19 right. Based on the works of the  Dr. Catherine Cox , we can estimate that:

Men of Remarkable Talent:

Napoleon or George Washington would hit about 20

Rousseau or Lincoln would hit 23 (and would qualify for membership in  Sigma III )

Geniuses (by Wechsler classification):

Swift, Rembrandt, La Fontaine, Cervantes or Balzac would hit 25

Molière, Lamartine, Benjamin Franklin or Copernicus would hit 26 or 27

Beethoven, Darwin, Montaigne, Mendelssohn, Watt or Diderot would hit 28 or 29 ( Sigma IV )

Luthero, Lavoisier, Raphael or Alexandre Dumas would hit 30

Great Geniuses:

Kant, Kepler or Spinoza would hit 31 or 32

Descartes, Michelangelo, Victor Hugo, Dickens, Musset or Byron would hit 33 (and would have the possibility of joining  Sigma V )

Newton, Voltaire or Galileo would hit 34

Universal Geniuses:

Leonardo Da Vinci, Pascal or Leibniz would have a chance of hitting 35. (Note: Leonardo had an IQ estimated by Cox at 180, but it was surely higher than that, perhaps as high as 200)  


It is important to emphasize that the IQ estimates made by Catherine Cox's team are controversial and some are clearly incorrect. Newton's IQ, for example, in Cox's first estimate was 135, later corrected to 190, and it's still an underestimate. Copernicus' IQ was initially estimated to be 105, which was clearly incorrect, then corrected to 165, which is still far from correct.


Understand the concepts of potential-IQ and rarity-IQ.


A little more about Sigma Test:


Clarifying some doubts

1. Is the Sigma Test an IQ test?

Do not. It is an intelligence test, but it cannot be considered an “IQ test”. To be classified as an “IQ test” it is necessary to meet certain criteria, among which it must be applied by a psychologist, with supervision, under controlled conditions, with a time limit.


While the Sigma Test cannot be considered an “IQ test”, the results you get on the Sigma Test correlate strongly with some of the top IQ tests in the 90 to 135 range, as well as two of the best IQ tests compared between itself, therefore, from a psychometric and statistical point of view, the Sigma Test is as good as the best IQ tests for evaluating the intellectual level in the range between 90 and 135. In addition, the Sigma Test continues to correctly measure the above intellectual levels from 135 IQ to around 170, surpassing the best IQ tests in this regard and matching the best home tests. There is also some evidence that the Sigma Test can correctly measure IQs above 180 and perhaps even above 190, which makes it stand out even among the top home tests.


Another differential of Sigma Test in relation to traditional IQ tests is that the absence of a time limit eliminates some distortions caused by unfavorable conditions that could harm the results in cases of tests applied in clinic, in which the person may be anxious, tired , sleepy, etc., with a negative impact on the test score. Eliminating the time limit solves this problem, as the person can solve as many questions as they want in one day, interrupt, continue the next day, unhurriedly review the previous ones, etc. What will determine the final score will be the person's real ability to solve problems, not lack of time or carelessness.


On the other hand, as the tests are carried out without supervision, there is a possibility that the person does not respect the rules established for their score to be valid, that is, the person can ask for help from third parties. This invalidates the results.


Therefore, Sigma Test scores are very similar to traditional IQ test scores. When these results diverge for IQs above 130, Sigma Test usually produces results closer to the correct values. When results diverge for IQs below 110, conventional IQ tests are often more likely to produce correct values. In the range between 110 and 130, sometimes the Sigma Test results are closer to correct, sometimes farther from correct.



2. Which tests provide the most reliable estimates of innate intellectual ability?

IQ tests are good tools for determining IQs between 70 and 130 and are satisfactory in determining IQs between 60 and 140. Therefore, they apply very well to over 97% of the population and serve well over 99% of cases. . For people with an IQ above 140, the Sigma Test provides more reliable results and the same can be said for other home tests.

3. Can a person get very different results on two IQ tests or on two intelligence tests? If so, which of the results should be considered?

Yes, it can and often does. It depends on the priorities of each test and the aptitudes of each person. In general, any pair of tests shows small differences when applied to people with IQs close to the average (between 90 and 110) and large differences when applied to people with very high or very low IQs. It is not uncommon to find people who achieve scores that fluctuate between 140 and 180, due to variations between test priorities, due to variation between test ceilings, etc. In addition, emotional state, stress, sleep, fatigue and several other factors can influence the ability to concentrate and this will inevitably affect the tests.


On Darryl Miyaguchi's website you can find the results of people evaluated by different tests, and the differences between the scores can be as large as 73 to 147 or 94 and 156.


4. If the differences are so great, then are IQ tests useless?


If you are interested in better understanding this phenomenon, it is recommended that you study Measure Theory. All measurements in Physics, Engineering, Chemistry, Astronomy, demographic surveys, election surveys, advertising surveys etc. may show large variations in some anomalous cases. In the 2018 election polls, for example, Professor Sérgio Wechsler, one of Brazil's most renowned statisticians, using some of the best statistical tools available, calculated that there was less than a 2% probability that Bolsonaro would be elected, based on polls by voting intentions that had been disclosed up to that moment. Around the same time, I made estimates by slightly different methods, coming up with about 34%. With very similar statistical methods and with equivalent approaches, we arrive at very different results.


Generally, the differences in results between different IQ tests, if they are well regulated, are less than 10 points and often less than 5 points, but situations can occur in which the differences exceed 50 points between one test and another. It is recommended to read my articles on Measurement Theory, as well as my book “Chess - The 2022 best players of all time, two new rating systems”, in which I do a detailed analysis of the method used to assess IQ, which is essentially the same one used to measure game strength in chess, since both are based on the Rasch or Lord or Birnbaum model.

If all the factors that caused variations in the results were related to individual instabilities of the person undergoing the tests, it would be natural to accept the best result as valid (the other results would indicate that the person was acting below his/her potential). However, there are many other factors extrinsic to the subject that need to be considered, including the luck factor in cases of multiple choice tests, distorted norms, etc. Some of these points are analyzed in articles whose links are presented at the end of this page.

5. Does intelligence stabilize at age 16 in all people? 

Do not. On average, intelligence reaches its limit between the ages of 15 and 20. Some studies indicate that this threshold is reached at age 14, others indicate between 20 and 25. It depends on the abilities measured by the test and the individual's pace of development. On a test with many questions about crystallized intelligence, most people tested tend to keep improving their scores until age 25 or a little more, while on an exclusively fluid intelligence test that prioritizes speed to solve simple questions, most people tested would stop evolving in the results from the age of 15 or even a little earlier. The topic is analyzed in more detail in some of my articles and videos cited at the end of this page.



Instructions for Sigma Test:

Sigma Test has not been available since 2006.
  If you want to hire our services for the development of exclusive tests for your company, for the selection of highly qualified professionals,  get in touch using our form .  


More information:


Video with some clarifications about IQ, in my interview for the CXOL channel:

Article with the concepts of potential-IQ and rarity-IQ.

Other articles:

Click here to learn about the Sigma Test:

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