Few things illustrate this better than getting a handle on what intelligence is. Here we have a subject that has been seriously studied for over a century. Yet researchers have not achieved agreement on whether they should focus on a single measure of intelligence called g, or multiple measures, or if multiple measures then how many there should be and how they should be divided. Note here that if you talk to individual researchers this fact does not worry them, because each researcher works in a community of other researchers who have achieved agreement on this. Each school of thought therefore churns out a stream of research, but they have failed to convert each other. And after a century the debate shows no sign of ending any time soon.

Please note that I am far from expert on any of this, so I'll try to summarize in broad terms my understanding, but my understanding could be incorrect. Also be aware that a lot of experts disagree with each other strongly, and therefore it is easy to find experts who will disagree with anything interesting that I can say. Please keep those caveats in mind as I try to flounder my way through the complications.

Let's start with the simplest approach, namely Spearman's g. What Spearman did in 1904 is take a number of things that one would think correlate to intelligence, such as grades, and found that they seemed correlated well with each other. He then looked for some linear combination which correlated better with each of the initial factors than any initial factor did. He found a very combination that did so, and found that it successfully predicted a very large part of the variation. In modern mathematical terms he did something called a Principal Component Analysis and looked for the most significant component. He then found that the most significant component really did capture a lot of the variability.

Spearman called this factor

*g*for

*general intelligence*.

Not long after the IQ test was invented. Before long people added IQ tests to the mix of factors, and found that a well-designed IQ test served as a fairly good predictor of g. Then g in turn served as a decent predictor for many things, including academic performance, and success in the workplace. Since IQ tests were fast and easy to administer, IQ tests and ability tests were on. (Ability tests such as the SAT and GRE exams actually are fairly directly comparable to IQ tests, and high-IQ societies such as MENSA are happy to accept them as proof that your IQ is high enough to belong.)

All of this begs many important questions. We got g out of a mathematical construct. But is it real? Is there really such a thing as general intelligence? Also even if there is such a thing as general intelligence, is it a stable attribute of a person? Does your general intelligence change over time? And the whole principal components analysis admits of the discovery of

*multiple*components, how important are the lesser components? (The existence of a split between mathematical and verbal ability is obvious enough that most ability tests split those into separate sections.) Are there important factors which have been missed? Also how much of this is affected by heredity versus environment? How does race play into this? Can training change how you score on the test? Can it change your actual intelligence?

Believe you me, those questions have been researched. Extensively. And argued. More extensively. The only ones that there seems to be a decent consensus on is that to the extent that general intelligence makes sense, it seems to be fairly stable, and both hereditary and environmental factors affect it. Every other one is still under debate, though it is easy to find experts that make definitive claims that the matter has been completely settled one way or the other.

As one example of the ongoing debate, ETS, which puts out various tests including the SAT exams, claims that training cannot improve your performance. At the same time Kaplan makes money selling the claim that they

*can*significantly improve your performance. Kaplan has enough evidence of effectiveness that I accept that they are right, but the debate continues. I suspect not the least because ability tests are just IQ tests under a different name, and so acceptance that Kaplan can teach an ability test is evidence that a key tool underlying a lot of research into psychology is fundamentally flawed, which opens up a can of worms that many don't want to look into.

For another example, I strongly suggest that mathematically inclined readers read

**g**a Statistical Myth. It presents a toy statistical model which demonstrates how readily a large number of independent random factors with a small amount of interaction could give rise to all of the statistical evidence which underlies claims that such a thing as general intelligence exists. Given the fact that any hereditary effect on intelligence is the result of a large number of fairly independent random factors called genes, I think the point is an important one. The idea that we each have a huge number of different intellectual abilities at different degrees of competence accords very well with my understanding of how the brain works, and my experience of interacting with others.

However it is also true that a small number of factors have a major effect on many different kinds of mental activities. Thus some combination of the amount of things you can keep in your mind at the same time and speed of reasoning may be influential enough on enough things to deserve the name "general intelligence".

At this point I'd say that the jury is not just out; they are deadlocked and randomly fighting each other. And have been in this state for the better part of a century.

So what do I have to contribute to the discussion? I'd like to throw out a toy mathematical model for how IQ and intelligence relate. Please do not take this too seriously. After all it is not even agreed on at this point that general intelligence is a particularly well-defined concept. But what I'd like to show is that even if you do accept that general intelligence exists, and that IQ is reasonably well correlated with it, that IQ tests are

**not**a particularly effective way to find the smartest people. (Whether or not any better way exists is a question I'd like to avoid. In fact, please forget this parenthetical point.)

Now suppose you sit down to take a test. You come prepared with many relevant abilities. You have your vocabulary, your polished reasoning ability, and a body of basic knowledge you're expected to have acquired about the world. We are generally prepared to accept these as components of intelligence, and the test will try to measure those. Let's call that INT. However you come prepared with other relevant abilities. There is how well you are feeling, how much sleep you got, any tricks you know about how to handle the types of questions that are likely to be asked, cultural cues you share with the test makers and how stressed you are. We are generally not prepared to accept these as components of intelligence, yet they affect how we do. Let us call these TST.

For my toy model let's say that INT and TST are independent random normals with a mean of 0 and a standard deviation of 10. That's math talk describing a particular bell curve. If you want the odds of it being less than a particular number you just have to divide the number by 10 to get the number of standard deviations you're out, and then look at a standard table and read off the probability.

Now let's say that the measured IQ will be 100+INT+TST. If you're a mathematician you'll know that IQ will have a normal distribution with mean 100 and a standard deviation of 10*sqrt(2) which is about 14.14. The correlation between IQ and INT turns out to be 70.7%. By comparison the Stanford-Binet is designed to have a mean of 100 and a standard deviation of 16, and its correlation with other measures of intelligence like academic performance is (depending on who measures it and how it is done) in the range of 70-80%. So while the model is pretty simple, it is not too far off from how a real IQ test behaves. And it captures the obvious observation that there are factors that affect test performance which we don't think of as intelligence.

Now the fun thing about a model like this is that we can play with it since we know exactly how it is set up. And are free to add any internal details we want. And while we can't necessarily

*believe*the answers we get, we can gain valuable intuition about what kinds of answers we can expect to see if we understood a far more complex system. Such as real people with real brains.

To analyze this I'm going to complicate the model one more time by introducing two more random variables. Let's make them independent normal variables with mean 0 and standard deviation 10/sqrt(2), and call them AVG and DIF. It turns out that (AVG+DIF) and (AVG-DIF) are independent normal variables with mean 0 and standard deviation 10. So let's make (AVG+DIF) be INT, and (AVG-DIF) be TST. High-school algebra shows that AVG = (INT+TST)/2 and DIF = (INT-TST)/2. (Hence the names.)

What was the point of that? Well it is this. At this point IQ = 100+2*AVG. And INT = AVG+DIF. Which means that if you give me an IQ, I can calculate AVG then use the known distribution of DIF to calculate confidence intervals on INT.

For example, in this model how intelligent is a typical member of MENSA? Well MENSA accepts anyone whose IQ is in the top 2% of the population. So a typical member might have an IQ at the boundary of the top 1%. (If you haven't taken statistics these calculations may make you eyes gloss over. You'll have to just trust that I'm coming up with correct numbers.) That IQ is about 2.326 standard deviations out, which after multiplying by the 10*sqrt(2) and adding 100 is an IQ of 132.90. (Remember that our standard deviation is lower than the regular IQ test, so our purported IQ scores will be lower than a regular test.) Which means that AVG is 16.45. On average our INT will be the same as the AVG, which puts you 1.645 standard distributions out, which is right around the 95th percentile.

We can go farther and calculate a 95% confidence interval for what the real intelligence of this individual is. 95% of the time a normally distributed variable will be within 1.960 standard deviations of 0, so that means that DIF will be in the range +- 13.86. Which means that INT will be (2.59, 30.31). Which means that INT is anywhere from 0.0259 to 3.031 standard deviations out, which would put INT anywhere from the 51st percentile to almost the 99.9th percentile. Hmmm, we really haven't nailed down INT very closely, have we? A similar calculation shows that with 80% odds INT is between the 77th and 99.5th percentiles. And you can even do fun things like show that with about 78% odds, this individual is not in the top 2% of the population on INT, and therefore would not qualify for MENSA if they were able to measure INT directly instead of the more complicated IQ.

Let's reverse the question. Let's take someone at the boundary of the 99th percentile in INT, and ask what IQ we should expect on average. Well that INT is 2.326 standard deviations out, which is an INT of 16.44, for an IQ of 116.44, which is 1.1625 standard deviations out, which is at the 87.77th percentile. So sorry, you're probably not going to test as smart enough for MENSA even though you're probably smarter than most of the people

**in**MENSA. How likely is this person to fail to meet MENSA's admission standards? Well the cutoff for MENSA is top 2% which is 2.0537 standard deviations out, which is an IQ of 129.04 (again remember that the toy model's IQs are not as spread out as real IQ tests), so we need TST to be 12.6, which is 1.26 standard deviations out, which gives a probability of 89.6% of not qualifying for MENSA.

The upshot? This model predicts that most members of MENSA are not in the top 2% on intelligence, and most people in the top 2% of intelligence would not qualify for membership in MENSA.

Now let's apply this to a question of more personal interest. According to this model, how intelligent am I likely to be? I bring this up because in my last blog post I used my GRE to discuss my likely IQ, and therefore likely intelligence, then used it as a benchmark to compare with the people I've met at Google. So if I believe that high IQs systemically overestimate INT, how badly is mine likely to be overestimated according to my model?

Well my scores were V: 760, Q: 780 and A: 800. Going to a handy IQ calculator I put in 1540 for my GRE V+Q and find that I am 3.625 standard deviations out and only 1/7143 people have an IQ that high. So on my toy model that is an IQ of 151.265... Which implies that AVG is 25.633. So on average you'd expect my INT to be 25.633, which is 2.5633 standard deviations out, which puts me around the 99.5th percentile, or about 1/200 on INT. (Incidentally the average person with an INT that high would fall in the 96% percentile on IQ, and therefore would not qualify for MENSA.) But let's go farther. 50% of the time a normal distribution falls within 0.67449 standard distributions of the mean, which means that DIF is half the time within the range -4.769 to 4.769. That makes a 50% confidence interval for my INT be the range (20.864, 30.402). Which would put me anywhere from the 98th percentile to the 99.9th percentile in INT. Which is in line with my claim in the previous blog post that I wouldn't be entirely surprised to find that I am at the 99.9th percentile in intelligence. According to this model the odds are that I'm not that smart, but it is possible.

Interestingly there is a symmetry in this model which means that all calculations that I've done for INT apply for TST as well. Which means that I am likely to have a number of characteristics that make me good at taking tests but which are not reflected in my general intelligence. This is actually true.

The vast majority of people when they are faced with a major test or interview get stressed. Now what does stress do to you? Well stress is caused by adrenaline. Adrenaline prepares you for a flight-or-fight reaction. This means decreased blood flow to things like your stomach and neocortex (which is where most of your higher reasoning happens), and increased blood flow to areas that are likely to see action, like your muscles. The result? Improved vision, hearing, faster reflexes, and increased strength. Also significant damage to your ability to carry out complex cognitive tasks. Needless to say this is about the worst possible physiological response to sitting in place for several hours and completing a series of complex cognitive tasks.

Apparently evolution didn't anticipate that millions of us would have our life courses depend on how we do on multiple choice exams. :-)

Very oddly, many members of my family, including me, have the exact opposite response. When I walk into a test the thought process I have is basically, "Well this is it. I've prepared all I'm going to, and now I've got a multiple-guess test which will overstate my abilities. Let's see how I do." And I

**relax**. Comparing me to a normal person based on the resulting test score is like starting with two runners, taking one out back and beating on him for a while, then expecting them to run a fair race. No matter how objectively you measure the result, I have an unfair advantage.

Random note. I originally intended to just present the toy model and use it to provide mathematical support for my impression that a really high IQ score is decent evidence of intelligence, but the true intelligence of most intelligent people isn't actually reflected in their IQ. But to sanity test the model I needed to find the estimated correlation between a standard IQ test and g, the general intelligence factor. And reading up on that I realized how much more complicated the subject was than I realized. I also found myself convinced by

**g**a Statistical Myth that g was probably a statistical artifact with no real meaning. Which is how this turned into 2 long and only tangentially related blog posts mushed into 1. I apologize for the length. I didn't intend it to turn out this way, but I hope it was an interesting read.

## 4 comments:

Thanks for the interesting blogpost!

I suspect (based on what I've heard her say in the past) that the major disagreement that the psychologist I mentioned in my comment on your preceding post would have with your toy model is the symmetrical distribution of TST about zero. She would assert that TST can make your performance on the IQ test underestimate your intelligence, but not overestimate it. For example, the fact that many people choke under pressure often contributes a significant negative value to TST, whereas in your case this contribution is zero.

It may seem that this is just one of those matters of definition which can never really be settled. In such cases, though, it often helps to get back to whatever the matter being debated is being used for. (If it's not useful for anything, then indeed there can be no end to the argument.) In the case of g, the point is to predict performance in other areas. So, will your ability to relax under pressure help you execute in many other areas of life? Intuitively this would appear to be so. Then it seems reasonable to consider this to be an aspect of your intelligence, rather than an extraneous factor inflating your scores.

I personally find Robert Sternberg's triarchic theory of intelligence illuminating. (See e.g. this book http://bit.ly/9avdwk) He divides intelligence into three branches, like our government: creativity, execution, and judgment. He argues that the IQ test measures only the executive function, and I would tend to agree. "Executive function" has since come to be used in cognitive neuroscience to refer to a suite of cognitive abilities. Sternberg also had a chip on his shoulder about the IQ test which led him to argue vehemently in some places for the practical uselessness of executive function, which doesn't make much sense to me (I think his later writings may have become more temperate).

On the other hand, the book _Rationality and Intelligence_ http://bit.ly/9cb7fM argues for the independence of these two, which makes sense in terms of Sternberg's division between executive and judging functions. Sternberg also argues convincingly that the IQ test does not measure and cannot predict creativity. This jives with Dean K. Simonton's finding in _Origins of Genius_ http://bit.ly/bFkmAc that a certain threshold IQ may be necessary for creativity, but otherwise creative geniuses are not usually at the extremes of IQ.

The Google interview process involves all three to a certain extent (certainly more than IQ tests and other standardized tests do). I'd be interested to hear more of your thoughts on that after you've had a decent number under your belt from the other side of the interview table.

I've long said that sadly, my greatest skill may be my ability to do well on multiple-choice tests, in part because of a relaxed attitude similar to what you describe.

There's an old saying (which I can't seem to find a source for at the moment): "if a distinguished expert says something in his field is possible, he's almost certainly right; if he says something in his field is impossible, there's a good chance he's wrong". Consider ETS and Kaplan to be experts in the fields of testing and test preparation; ETS says improving your performance is impossible, and Kaplan says it's possible; who do I believe?

@Ruchira Datta: I'd like it if you pointed said psychologist at this post for feedback. I'm sure we'd hear something interesting.

As for the mean of INT and TST, that is just a normalization factor. After all what does it to mean to say that people have a negative INT? I could trivially fix it by setting the mean of INT at 70 and the mean of TST at 30, then making IQ = INT+TST. That would change all of the numbers, but none of the conclusions.

It

wouldadjust the conclusions if we changed the balance of how much INT and TST matters. In fact I know exactly how to adjust the mathematical model to get any desired balance you want there, so for instance you could see how it would matter if INT mattered twice as much as TST. I could even write a JavaScript calculator to crunch numbers. But that would take more effort than I think the exercise is worth.As to the Google hiring process, the same trait that helps me on tests also helps me during interviews. When you walk in feeling relaxed and confident with your brain ready to go, you're bound to make a better impression than if you're stressed out. Try as they might, I am sure that Google hasn't found a way to compensate for this factor.

@Jeremy Leader: You're quoting Aurthur C. Clarke's first law. And yes, I like your application of it. :-)

On your test note, I've noticed that it is easier for people who do well on tests to admit to the disparity than people who do poorly. Probably in large part because if you do well and say, "That's not really an honest assessment of my abilities" it doesn't come off sounding like sour grapes.

Anybody who thinks that human intelligence can be measured using a linear model is laughing too much at a joke that is not particularly funny. That somebody needs to write a few computer programs to do what humans trivially accomplish by their intelligence. The complexity of these programs will serve as an intellectual kick in the aforementioned's theoretical posterior.

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