IQ difference and jokes.

Factual Questions
181

You know, thinking about the confusion that rose before with this, and expanding on the explanation Wendell already gave, I think it’s worth explaining in simple terms what IQ measurement is, and why the result should not be viewed in nearly the same sort of way as, say, a measurement of height or how much weight one can benchpress or such things.

Here’s the basic definition of IQ: you give everyone (ideally) in the world a gradeable test of some sort which you think more intelligent people will do better on. Perhaps the test consists of a series of questions, and you assign as a grade the number of questions answered in the manner considered correct. THIS GRADE IS NOT THE IQ! THE MAGNITUDE OF DIFFERENCES IN THIS GRADE DOES NOT CORRESPOND TO THE MAGNITUDE OF DIFFERENCES IN IQ! THE ONLY CONNECTION BETWEEN THIS GRADE AND IQ IS THAT BETTER GRADES CORRESPOND TO HIGHER IQs; HOWEVER, ALL THE MAGNITUDE INFORMATION IS (DELIBERATELY) DESTROYED!

What happens next is, as the capitalized letters begin to indicate, we forget everything about people’s grades except their relative ranking; who scored better than who (with no ties, ideally). Your “unnormalized” IQ score is given by the percentage of the rest of the world which scored below you (aka, your percentile rank). Thus, the person in the world with the very highest grade gets an unnormalized score of 100% = 1, while the person in the world with the very lowest grade gets an unnormalized score of 0% = 0, and everyone else’s unnormalized scores are distributed at equal intervals inbetween these. (Note how all the information about how much better the top scorer did than the second-top scorer, and so on, is gone now. The only information retained is the relative ordering.)

Since the unnormalized scores are (by definition) distributed at equal intervals on a finite range, they do not form anything like a “bell curve” distribution (rather, they form a flat boxtop). So, the last step, which is entirely superfluous, is to take the unnormalized scores and manually stretch them into a bell curve. Specifically, if your unnormalized score is p, then your normalized score will be n, where n is the unique value on a bell curve with total area 1, center 100, and standard deviation 15 such that the area to the left of n is p. (Note that all this does is change the way you present the score. It does not recover the magnitude information that was already lost. All the meaningful information an IQ score carries was already present in the unnormalized score (your percentile rank on the test))

Note also that all of this could be done just as well for any test. We could measure height this way, even, and report people’s HQs, with this being 100 if exactly half the rest of the world was shorter than you, this being 115 if just over 84% of the world was shorter than you, this being 130 if just under 98% of the world was shorter than you, and so on. But we don’t report height that way, because we’re much more interested in the actual length, an intrinsic, directly measured quantity which carries magnitude information beyond simply the information of who is taller than who. And similarly for weight you can benchpress and all the rest of it. But not for IQ. The only information IQ carries is what percentage of the world you scored better than on the test; this number just happens to be reported in a skewed way which makes it look like it carries more information than it does.

182

You know I don’t think that there is a single person around here who does not understand how IQs are reported by S.D. variation.

The question is whether it is qualitatively that much easier for the “brighter” to tell that someone 2 S.D. less bright not as bright as they are when the two S.D.s mean that it is 84.1% compared to 15.9% (115 to 85), than when those two S.D.s mean that it is 99.9% to 84.1% (145 to 115). Some here claim that the first is easy but that the second is almost impossible; some believe that they are likely each just as likely.

183

The difference between 115 and 85 will show up in the lack of certain everyday knowledge in the person with an 85 I.Q. For example, the person with the 85 I.Q. might have difficulty preparing a common income tax return, or will not know where to find additional information about a health condition, or how to fix common computer problems.

The difference between 145 and 115 involves esoteric knowledge less necessary for everyday activities. In everyday life, the people with 145 and 115 will less often find themselves in a situation where their differences are apparent than the people with 115 and 85.

184

Does anyone have a reference to any more or less scientific studies showing how difficult or easy it is for someone of a given I.Q. to tell what the I.Q. of someone else is in a conversation of a few minutes, or when knowing them for a few days, or for a few years? If not, there’s no point to any further discussion of the issue. All we have is people stating their own opinions on the matter, despite the fact that they sometimes don’t even appear to understand exactly how the measurement of I.Q. works. Even the people who do understand what a standard deviation is are treating the measurement of I.Q. as if it were something as well established as the measurement of length, mass, frequency, time, or some other standard physical object. It’s not.

The argument for it is actually rather complicated. You have to start by assuming that there is a reasonably constant quality in each person called intelligence. You have to assume that it can be measured by a single number. You have to assume that certain sorts of tests can measure it. You have to assume that what is measured on one of those tests is equivalent to what is measured on another one of them. You have to assume that the way in which it is reported (mental age, ratio I.Q., or deviance I.Q.) is the best or even a useful way to report it. These aren’t absurd assumptions, but they have all been questioned. And, even if all these assumptions are correct, in threads on this subject here we have to deal with the problem of misreported I.Q.'s. It seems like every time this matter comes up, someone has to chime in with a dubious report of their I.Q. or the I.Q. of someone they know.

185

First off remember that populations do self-segregate to some degree. Odds are that the individuals with 115 and 145 are in a more challenging professional environment together or in a social circumstance in which conversations may be a bit more than about who they like on American Idol. In their everyday lives they may encounter each other dealing with issues in which more information or ability than that needed for the average person’s everyday activities require.

Secondly, it sounds like you are saying that our 145 IQ individual being able to tell the difference between him and herself and the 115 IQ individual is a more difficult problem that requires greater problem solving skills, memory, and observation skills than the problem our 115 individual faces telling him/herself apart from the 85 individual … things that we assume the 145 individual has in greater amount than the 115 individual does.

On edit: The above is to Walloon; yeah WW I’ve long ago made the point that without such studies we are just making WAGs about what makes sense to each of us.

186

I’m sure everyone understands standard deviations are involved. At the same time, it also seems apparent that at least some people think the way it works is something like “Give everyone a special test on which each question is worth a certain number of points towards their grade; the mean grade is then considered to be an IQ of 100, and each standard deviation in this grade is considered to be 15 IQ points”, which is not at all correct. I think it’s important to point that out, and to particularly stress how meaningless the idea is that every one standard deviation corresponds to some particular measured magnitude of difference. I think it’s easy to lose sight of the fact that IQ scores are distributed along a bell curve not as an empirical fact, but by number-massaged fiat; frankly, it’s all rather obfuscatory compared to simply reporting percentile rank.

I didn’t go into all the concerns raised by Wendell (stability of one’s score on this test (without which there is no reason to consider the score a measurement of some intrinsic characteristic of the testee), and, even granted that, reason to believe relative ordering of scores on this test in particular should be considered the correct relative ordering of “intelligence”, as opposed to all other potentially conflicting tests or sources of intelligence judgements (without which there is no reason to ascribe the result such importance), and so on), but I share all of them.

187

Basically, the most important thing to point out is that, intrinsically, IQ scores are not a cardinal measurement, but rather only an ordinal measurement. Ordinal quantities don’t have means or standard deviations; they just have medians. The whole problem is that we happen to purposely misleadingly report this ordinal quantity in such a way as to make it look like it measures/tracks a Gaussian-distributed cardinal quantity (“intelligence”), but unless we actually had some direct ability to measure this latter purported quantity cardinally and to empirically verify its Gaussian-distribution, it makes no sense at all to speak of the magnitude of said quantity contained in a standard deviation (or its mean level or any such thing). This is precisely the confusion that was contained in the comment I was originally replying to.

188

Put another way, the problem with “Of course the differences are the same! A standard deviation is a standard deviation!” is this:
Consider people with IQs of 100, 115, and 130. What this means, and all this means, is that the first scored better than 50% of everyone else, the second scored better than about 84.1%, and the third scored better than about 97.7%.

“Yeah. That means the second person scored one standard deviation above the first person, and the third person scored two standard deviations above the first person.”

No! No, not at all. It is meaningless to speak of the standard deviations of an ordinal measurement. Rather, what we have is if we arbitrarily choose to map this data onto a Gaussian (i.e., bell curve) distribution, the second person will land at one standard deviation above the first person, and the third person will land at two standard deviations above the first person. But just as well, if we choose to map this data onto a uniform (i.e., flat boxtop) distribution, then the second person lands at about 1.18 standard deviations above the first person, and the third person lands at about 1.65 standard deviations above the first person. And with other distributions, it will come out even differently; moreover, many distributions don’t even have the median coinciding with the mean, and thus don’t even have the first person landing at 0 standard deviations above the mean.

The idea that the standard deviations here mean anything is bunk, for the same reason that doing anything with IQ scores other than seeing whose is higher than whose is bunk; it’s all just an artifact of the presentation. An ordinal quantity carries no standard deviation information; such information is only created afterwards (and thus with garbage values) when one pretends the ordinal quantity is a cardinal quantity, one way or another.

189

Agreed that this discussion is using the results on IQ tests as a meaningful proxy for “general intelligence”, and that both the value of the test as that proxy, and the concept that there is a meaningful concept that can be called general intelligence are, minimally, simplistic concepts that are very much open for debate. Nevertheless, the discussion is predicated upon those simplified assumptions, so we can leave questions regarding IQ test validity as a measure of whatever general intelligence is for another thread, leaving as accepted that there are good reasons to question them.

That said, no I doubt anyone suffers from the confusion you think they do. In fact, IMHO, believing that any other metric than that statistical one of S.D.s can mean anything regarding relative “intelligence” would be bunk. The confusion resides in your believing that the raw scores are what is meaningful.

The meaning you believe people believe would be pretty meaningless. What we care about in this discussion is how we relatively compare and contrast with each other. That is a sense captured by S.D.s, not raw scores. In that sense a S.D. is a S.D and is all that matters.* Statistically* they are as far away from each other, whether it is 85 to to 100 or 100 to 115 or 115 to 130.

190

What is it that you think “standard deviation” means? Do you know? And if you do, can you explain the details of how it applies to this situation? And why it is a relevant notion to invoke? Because it’s not clear to me that you see the subtleties here.

I’m not saying the raw scores are meaningful. I’m saying the so-called “standard deviations” you are referring to aren’t meaningful either. There’s no more reason to say the difference between a percentile rank of 50% and 97.7% is two standard deviations, than to say it’s 1.65 standard deviations, or π + 7 standard deviations, or what have you. It’s just that, on a bell curve distribution, the difference between percentile ranks of 50% and 97.7% is two standard deviations; but on a uniform distribution, the difference between percentiles ranks of 50% and 97.7% is 1.65 standard deviations, on a Laplace distribution, the difference between percentile ranks of 50% and 97.7% is just under 2.18 standard deviations, on infinitely many other distributions, it’s π + 7 standard deviations, and in general it can be anything. Moreover, given a percentile-span A (say, from 84.1% to 97.7%) and a disjoint percentile-span B (say, from 50% to 84.1%), there will be some distributions where the number of standard deviations contained in A is larger than that in B, some where the number of standard deviations contained in B is larger than that in A, and some where they’re precisely the same. Sure, on the bell curve distribution, it comes out one way, but on other distributions, it comes out another way.

So why should “standard deviations, according to what locations on the bell curve distribution would reach this percentile rank” be considered a useful notion of how to compare differences in percentile rank (e.g., differences in IQ) to one another, rather than according to the uniform, or Laplace, or any other distribution? If you cannot answer this question, then there is no grounds for considering the difference between an IQ of 100 and an IQ of 115 to be somehow “equal” to the difference between an IQ of 115 and an IQ of 130. We can all agree that 130 > 115 > 100, but to say that both steps are by the same amount is to give undue attention to a mere arbitrary artifact of the presentation.

191

Also, I suppose, do you know what exactly a bell curve distribution is? What is, and, just as importantly, what is not special about it? Because, as I said, to make the claim that “a 15 point IQ difference” is a meaningful concept, it’s not enough to invoke “Yeah, that’s one standard deviation (because it isn’t)”. You have to invoke “Yeah, that’s one standard deviation along a bell curve distribution”, and then furthermore say something about why bell curve distributions are special and why we should be looking at standard deviations along them. Without doing that, one cannot give an argument for the meaningfulness of “a 15 point IQ difference” as a unit of measurement.

All of this is entirely orthogonal to any other concerns about IQ testing; it’s a simple matter of concern over the nature of the statement “Statistically they are as far away from each other, whether it is 85 to to 100 or 100 to 115 or 115 to 130”. My point is precisely that when the only information is “how we relatively compare and contrast with each other”, then discussion of standard deviations is meaningless.

192

I just reported it as received when I was 14. I have learned (and mentioned it previously) that there are different tests that score differently. You have to know the names of the specific tests to go see how the scores compare. There is more than one IQ test. It’s probably a non-mensa-approved test or something. So, my 145 which you acknowledge as true, probably equals the 160 I posted before.

So not a big deal really. I’m seriously not trying to say how smart I am, just that IQ was in the title as part of the question and I happened to know mine and debunked it in answering the question.

193

Also, missed the window:

It was a class of the smartest people in the school. Yes, really. We knew what we were taking, and the teacher only announced the lowest and highest score. My friend and myself shared our scores but we didn’t know anyone else’s.

194

Yes. But.

I did tech support for College Professors. These were all eggheads with multiple PhD and living buried on bunkers lined with books they had actually read and annotated and could pull out stuff from on call. I won’t call them geniuses because I just don’t know but they were certainly not idiots. Most of them couldn’t make a printer work to save their lives. Also, most of them saw the internet as a scary monster they seldom entered except for chosen bookmarked sites.

They were all over 60 and in Philosophy, History, Theology, Sociology, and similar fields. They were smart but just didn’t have the technical chops to survive a wired world.

This doesn’t disprove your point but I just wanted to point out that it is easy to assume lack of intelligence when we see people limp on areas of knowledge where we are strong. They might just have focused their development in a different area.

195

I should, of course, have punctuated this with the parenthetical outside the quotation marks.

Anyway, to illustrate the point, suppose off in Letterland, the 26 letters began to quibble about their earliness. “I bet I’m earlier than you”, “Maybe, but not by nearly as much as I’m earlier than him”, “No way, he’s easily ten times earlier than the both of us combined”, and so on, they say boastfully amongst themselves. Finally, one day, a test is designed, to measure Earliness Quotient (EQ). Perhaps unsurprisingly, A scores better than B, who in turns scores better than C, and so on down to poor Z. This puts the percentile rankings at: A = 100%, B = 96%, C = 92%, …, X = 8%, Y = 4%, and Z = 0%.

But EQ calculation doesn’t stop there. Rather, as the last step, we convert the percentile rankings into z-scores (and then take the EQ to be 100 + 15 * this z-score). Carrying out the calculations, we find that the z-scores are approximately: A = positive infinity (off the charts because A was revealed to be undoubtedly the earliest letter in all of Letterland), B = 1.75, C = 1.405, …, M = 0.05, N = -0.05, …, X = -1.405, Y = -1.75, and Z = negative infinity.

“Wow”, says Y. “I sort of knew B was earlier than me. But I didn’t realize the extent to which B was earlier than me was a whole whopping 35 times greater than the extent to which M is earlier than N.”

“That’s cause you’re too late to possibly understand statistics”, sneers B. “But I can assure you, it’s all true. After all, numbers don’t lie”

Is this reasonable? Should we really say that the letter Y is 1.75 standard deviations later in the alphabet than the mean, the letter M is 0.05 standard deviations earlier in the alphabet than the mean, and so on, so that the difference between B and Y is 35 times as significant as the difference between M and N? If so, why, and if not, why not?

196

First time posting, but this thread has outed me from the closet of lurkdom.

Although I don’t know of any studies specifically seeking the ability of exceptionally gifted individuals to gauge the IQs of brighter than average peers, there is a lot to suggest that the answer is yes. IQ tests in children estimate the degree to which a child’s cognitive development is in synch with his chronological development. The Stanford Binet was developed with the (mental age/chronological age * 100) formula. If a child is 9 years old both chronologically and mentally, therefore, the IQ is 100…average.

Morelock’s (2000) and Morelock and Feldman’s (2003) research supported that premise, and with it the concept of “asynchronous development”–a hot buzzword in gifted ed issues, as it describes the disconnect between a gifted child’s physical age, and his intellectual function. Using these suppositions, then, and since IQ changes little throughout a person’s lifetime (the mental/chronological * 100 formula has demonstrated validity in adults as well as children), then a 20 year old with an IQ of 130 has a mental age of about 26. On the other hand, the 20 year old with an IQ of 115 has a mental age of approximately 22. Not a huge difference, perhaps, but one that a person of high IQ would likely recognize.

The difference is extremely pronounced in the exceptionally (IQ of 160-179) and profoundly gifted (IQ > 180). Studies have repeatedly found that these individuals have significant socio-emotional barriers, and greater adjustment problems in that they are often unable to find true peers. Janos (1983) found that isolation was a problem for these children even in schools with age peers of moderate giftedness (IQ 130-144).

That may be useful for extrapolation. Here’s my own anecdotal evidence: I teach gifted children in a district whose minimum IQ for gifted programming is 120. In these small classes (about 10 middle school and high school students), the kids absolutely know which peers are in the mildly gifted range and which are highly and exceptionally gifted. In Cecil-esque fashion, I’d offer to conduct a little informal research of my own, but I think the ethics are sticky. “Hey kids, how smart do you think Elmer is over there? 120? 140?” :stuck_out_tongue:

Interestingly, an IQ in the 125-155 range has been described as being the “socially optimal intelligence.” People in this range have been found to be well-balanced socially and emotionally, and are able to gain the cooperation and respect of same-age peers. One researcher studying leadership (Simonton, 1999) believes that the optimal IQ is 119, especially for politicians and leaders; any higher and the person may be perceived as “talking over the heads” of others. This is also known as being one of the “liberal elite.” (Ok, I threw that one in there…:D)

Sorry for all the study authors and dates, but I’ll be damned if my first post was going to go down as a “Where’s the cite?” failure. :cool:

197

Sister Vigilante, Mensa does not approve I.Q. tests. Mensa is not the official professional society of I.Q. test makers that decides whose test is allowed to get its seal of approval. It’s a club that has decided to only allow people into it who score above a certain I.Q. It’s like someone had decided to create a club that only allowed people who were above a certain height. If feet and inches were so sloppily defined that a manufacturer could make yardsticks using their own definitions for feet and inches, it would make the whole point of measurement pointless. The same is true for I.Q. scores. If the test manufacturer has defined I.Q. scores in his own idiosyncratic fashion, it would make the whole point of measurement pointless.

A teacher who announces the I.Q. scores to anyone other than the student himself and his parents is clearly violating the student’s rights.

Was this just an ordinary high school or was it one that took students from all over the state based on a test or something? The problem is that it’s too improbable for three such students to be in a single grade in a single high school. Let’s say that your high school was huge, with 2,000 students in a single grade. Then we would have to believe that a random group of 2,000 people happened to contain one person who would be the smartest person in a random group of 662,000, another who would be the smartest person in a random group of 134,000, and a third person who would be the smartest person in a ranom group of 31,500. (And you’ve only told us the scores of yourself, your friend, and the highest scorer. Were there other people in the class who supposed had scores of 160 or greater?) This is just too improbable to be believed. This could only be based on a hopelessly nonstandard way of scoring I.Q. tests.

198

Creole nice first post, with some actual data that bears on both the question that this hijack is bickering over (and our bicker has gotten a bit silly IMHO) and on the original op. Stay out of the closet!

Indistinguishable, your tone gives me the sense that this has hit some kind of nerve. I think you must believe I insulted you and if so I apologize. In any case, I have already stated that this conversation accepts a whole set of assumptions, both explicitly and implicitly, some of which may debatable in actuality. Yes, since we have assumed that IQ is a meaningful measure of “general intelligence”, we have also implicitly accepted the assumptions that underlay the validity of that measure, including the normal distribution of the ability. Yes, I know that curves of human ability are rarely true normal distributions, but we simplify our discussions by approximating many of them as such. My understanding is that modeling them as normal usually approximates reality better than modeling them as uniform distributions or as Laplace distributions.

Obviously your letter distribution example is not a normal distribution, or even imaginably approximated as such, so it is a bit silly to use it as a comparison. If you want an example that makes both our points to some degree I can give you one, though. Let us use a different human ability: the ability to run 100 meters quickly. Like most human abilities not really a normal distribution either, but more similar to normal than it is to the letter distribution. The world record for that distance is 9.58 s. The average young adult male can run it in about 12 s. Most humans are likely clustered within a second of that 12 s mark. (Again, not really in a true normal distribution and the spread in each direction is highly unlikely to be symmetric.) Two individuals differ by 0.5 seconds on their typical times to run that race. Is that difference as meaningful to two individuals whose scores are 12.25 and 11.75 s as it would be to two whose scores are 9.75 and 10.25 s? Or does the difference of even a hundredth of a second become more meaningful the rarer individuals who can perform the skill to that level becomes? Bolt’s 9.58 s performance being 0.11 s faster than anyone else’s ever recorded means a lot more than Joe Schlubb running 0.11 s (say 12.01s) than does Mike Grabowski in another class. Implicit in that is an appreciation that Bolt is a true sigma outlier … hmmm sigma … Trying to say that Bolt is twice as fast or 35 times as fast as someone else is pretty meaningless whether the metric is absolute time or S.D., what matters is how rare it is to be able to run that fast.

In matters of comparing human performance we appreciate variation most meaningfully by understanding how relatively rare or common it is; given the nature of human variablity S.D.s are at least a less meaningless metric to use than any other scale we have available for many items of potential variation, especially one in like “intelligence” that we define in relative terms and attempt to impose a normal distribution on.

Again, Creole’s post actually brings more than pontification to bear. The rest of us relatively are just giving humble opinions.

199

Eh, no, it’s my fault for giving that tone. Just the typical “I’m on a message board and I have to win this argument” situation of getting worked up over nothing. Don’t worry about it. :slight_smile:

200

Hmmm… I was also thinking of the running speed analogy. There has to be a high-end limit to genius; and simple math (as found in idiot-savant math prodigies) may not be a good example?

I recall the early arguments over cultural bias in tests; for example - “the hedge goes all around Dr. Jones’ house; the lot is 50 feet wide, 70 feet deep, with a 10 foot wide driveway and a front gate 4 fet wide” - the point made was that inner city kids may not have a clue what a hedge was; one reading comprehension test talked about the ranks in the army - sargeant, corporal, lieutenant, major etc. then asked questions for reading comprehension. Who will do better on that test, based on prior accumulated knowledge - boys or girls?

So what exactly can an intelligence test check for, and what can it really measure quantifiably without having to measure absorbed cultural trivia? Do you know who Descartes is, or which particles are bosons? Or is IQ like blood pressure, something that can vary several points depending on age, mood, and the circumstances of the test?

As for teachers keeping things confidential… what a novel concept. That sort of privacy is only a few years old. I recall in the 60’s not only the teacher announcing everyone’s mark to the class - average down to one decimal point - but lining up the class 3 semesters a year from smartet to dumbest (rather, highest to lowest mark) when report cards were handed out. Not that this was being mercenary, often some would not mention the lowest few marks in the class other than to say “failed”; but in the days before the “poor dears and their self esteem” trend a decade later, this was considered normal. What you did was what you did. If you didn’t like it, try to do better.