On Mathematical Intelligence and How It Grows(appliedsentience.com)
appliedsentience.com
On Mathematical Intelligence and How It Grows
http://appliedsentience.com/2015/02/10/on-mathematical-intelligence-and-how-it-grows
9 comments
[The bell curve] crops up a lot when talking about traits in a population, such as intelligence. The reason for this is well understood (see central limit theorem) and won’t concern us here.
No! The Central Limit Theorem states that the average of the values in the distribution creates a bell curve -- not that it starts out that way. Traits in a population do not follow a bell curve and the CLT doesn't give any reason why they should.
No! The Central Limit Theorem states that the average of the values in the distribution creates a bell curve -- not that it starts out that way. Traits in a population do not follow a bell curve and the CLT doesn't give any reason why they should.
If one considers intelligence (or other features) as the sum of many random sources, then it makes sense again to invoke the CLT. Maybe that's what the article implied.
[edit: see other replies saying the same thing below]
[edit: see other replies saying the same thing below]
"Intelligence" itself is not a well defined quantity like height or weight. IQ is well defined but it is defined to follow a bell curve, it has nothing to do with CLT.
Intelligence is a well-defined quantity like temperature, and it follows a genetic architecture like height: thousands of genetic variants of small effect size. Which setup does indeed give you a binomial or normal sort of distribution, justifying the norming as more than a mathematical convenience.
>Which setup does indeed give you a binomial or normal sort of distribution
distribution of what? Still no quantity defined.
>Intelligence is a well-defined quantity like temperature
As a physicist it sounds about right. Temperature is tricky to define in non-stationary systems (aka. "real world").
distribution of what? Still no quantity defined.
>Intelligence is a well-defined quantity like temperature
As a physicist it sounds about right. Temperature is tricky to define in non-stationary systems (aka. "real world").
[deleted]
As per wikipedia:
"An intelligence quotient, or IQ, is a score derived from one of several standardized tests designed to assess human intelligence. "
I fail to see how IQ is "defined to follow a bell curve" as you wrote. The tests were designed so the answers distribution is a bell curve?
In my statement above, just replace "Intelligence" by "IQ", I believe that's what being discussed.
In any case, "Intelligence" can takes a thousand forms that I don't think IQ tests or even our vocabulary can really pin down.
I fail to see how IQ is "defined to follow a bell curve" as you wrote. The tests were designed so the answers distribution is a bell curve?
In my statement above, just replace "Intelligence" by "IQ", I believe that's what being discussed.
In any case, "Intelligence" can takes a thousand forms that I don't think IQ tests or even our vocabulary can really pin down.
As per wikipedia:
"When current IQ tests are developed, the median raw score of the norming sample is defined as IQ 100 and scores each standard deviation (SD) up or down are defined as 15 IQ points greater or less"
Raw test scores are transformed so the test results distribute on a bell curve. Can't find a more precise definition though.
"When current IQ tests are developed, the median raw score of the norming sample is defined as IQ 100 and scores each standard deviation (SD) up or down are defined as 15 IQ points greater or less"
Raw test scores are transformed so the test results distribute on a bell curve. Can't find a more precise definition though.
> Otherwise known as the standard normal distribution. It crops up a lot when talking about traits in a population, such as intelligence. The reason for this is well understood (see central limit theorem) and won’t concern us here.
No, the reason IQ follows a bell curve is that IQ is defined as following a bell curve. Results are adjusted to make a bell curve.
No, the reason IQ follows a bell curve is that IQ is defined as following a bell curve. Results are adjusted to make a bell curve.
So the gist of the article is that advanced mathematical abilities are attainable to anyone! It just takes some clerical work. This is an important point, many people still believe in the myth that a mathematician, physicist or programmer is born, not made. In reality it is both, but if you are not as naturally endowed you can still compensate for that.
SAT is a bad example because SAT prep works very poorly. People like to cite anecdotes (which are driven by the noise of retesting until an extreme score is reached and selection effects) and prep companies like to cite how 'students who took our course scored 200 points higher than those who didn't', but that's driven by the obvious confounding of who goes through with prep courses; estimates from the randomized experiments tend to look more like 20 points (see https://en.wikipedia.org/wiki/SAT#Preparation and reviews like http://libgen.org/scimag/get.php?doi=10.1111%2Fj.1468-2397.2... ).
Can someone with more stats background explain how to invoke the central limit theorem to see that intelligence should be distributed as a bell curve?
What are the independent and identically distributed random variables that are being averaged in this case?
What are the independent and identically distributed random variables that are being averaged in this case?
> What are the independent and identically distributed random variables that are being averaged in this case?
The random variable is simply whether you answered question N on the test correctly or not. For every question there is a probability Pn that any given test taker answers it correctly, and that probability is relatively uniform per-question. (so scores on any test, where the same test is administered to a large numbers of test takers, will be normally distributed. Yes, even if the questions are nonsensical and the answers random)
So an IQ score simply indicates the probability, relative to the general population, whether a test taker will answer simple abstract mathematical questions correctly (where correctly should be understood to have cultural biases).
The random variable is simply whether you answered question N on the test correctly or not. For every question there is a probability Pn that any given test taker answers it correctly, and that probability is relatively uniform per-question. (so scores on any test, where the same test is administered to a large numbers of test takers, will be normally distributed. Yes, even if the questions are nonsensical and the answers random)
So an IQ score simply indicates the probability, relative to the general population, whether a test taker will answer simple abstract mathematical questions correctly (where correctly should be understood to have cultural biases).
The most important condition of the central limit theorem is that the variables are independent. In an IQ test, you'd hope the probability of answering a question correctly depends on your intelligence. Therefore these variables can't be independent from one another.
The thing about the central limit theorem is that violating it's assumptions doesn't really make it no longer apply. There's simply a few pathological cases where it totally breaks down, which violate that assumption and are very hard to state otherwise.
But taking 2 variables that correlate 80% or 20% or even more will not change the outcome. It's just very near the edge that it totally breaks down, or if the number of variables that correlate is really too high, or if they're really chaotic variables, or ...
So one the one hand you're right. But it's not a point that a statistician would make, only a logicist or a rather pedantic mathematician. In the graphs, even if things correlate it will "look okay". Therefore the statement that intelligence is normally distributed along a bell curve is technically correct, but good look finding a large piece of data that doesn't look like a bell curve.
But taking 2 variables that correlate 80% or 20% or even more will not change the outcome. It's just very near the edge that it totally breaks down, or if the number of variables that correlate is really too high, or if they're really chaotic variables, or ...
So one the one hand you're right. But it's not a point that a statistician would make, only a logicist or a rather pedantic mathematician. In the graphs, even if things correlate it will "look okay". Therefore the statement that intelligence is normally distributed along a bell curve is technically correct, but good look finding a large piece of data that doesn't look like a bell curve.
One theory is that there are many genes that lead to intellectual skills. There are many environmental factors that influence the brain (nutrition, toxins, parenting behavior, etc). Each of these advantages makes you a bit faster or more likely to be able to solve an mental challenge.
"Identically distributed" is a not a requirement in CLT, it's just the simplest variant to prove.
"Identically distributed" is a not a requirement in CLT, it's just the simplest variant to prove.
Intelligence (at least as measured by an IQ test) isn't distributed as a bell curve. Bell curves always include negative values, and intelligence can't be negative.
> intelligence can't be negative
Asserted without proof. To show this, you'd need to interpret intelligence as something that physically couldn't be negative. I feel safe in saying you haven't done that.
In particular, "intelligence (as measured by an IQ test)" can easily be negative, in the sense of yielding a negative IQ score. It would have to be an unrealistically long test, and you'd need way more people than exist in the world, but those problems apply to the tails of actual normal curves too.
Asserted without proof. To show this, you'd need to interpret intelligence as something that physically couldn't be negative. I feel safe in saying you haven't done that.
In particular, "intelligence (as measured by an IQ test)" can easily be negative, in the sense of yielding a negative IQ score. It would have to be an unrealistically long test, and you'd need way more people than exist in the world, but those problems apply to the tails of actual normal curves too.
I am really impressed by wolfram alpha's ability to parse my butchered syntax: https://www.wolframalpha.com/input/?i=probability+that+gauss...
Probability of finding someone with an IQ (as defined as normal distribution with mean 100 and std 15) less than 0 is 1.30839×10^-11
Probability of finding someone with an IQ (as defined as normal distribution with mean 100 and std 15) less than 0 is 1.30839×10^-11
Technically you are correct - the best kind of correct.
"The distribution of the measured IQ very closely follows a bell curve." would have been the correct way to formulate.
"The distribution of the measured IQ very closely follows a bell curve." would have been the correct way to formulate.
To be technically correct is the only way for a statement to be correct.
If it's "non-technically correct" it is ill-frased.
This debates started by people stating wrong things but still wanting others to understand them.
If it's "non-technically correct" it is ill-frased.
This debates started by people stating wrong things but still wanting others to understand them.
While mathematics has for the most part been built up using 2-valued logic, that isn't how humans perceive the world, which is why someone will say something like "you are technically correct", or they'll say "technically that's, but...".
It is true that binary logic is not the ultimate logic. However, "technically correct" vs "correct" is not a debate of whether one value of true is enough.
The word "technically" is used with "correct" as a rhetoric tool. It seems like they are attacking the truthness of the statement, but in fact they are trying to put the statement into a bigger context. So many people misuse it as such that some even understand it as an attack of the truthness of the statement.
The word "technically" is used with "correct" as a rhetoric tool. It seems like they are attacking the truthness of the statement, but in fact they are trying to put the statement into a bigger context. So many people misuse it as such that some even understand it as an attack of the truthness of the statement.
> So many people misuse it as such that some even understand it as an attack of the truthness of the statement.
No, they use it as a device to point out that there is more context around the discussion than whether or not something is true or false (hence my point about 2-valued logic).
I could say race is defined by genetics, and while that's technically correct, it adds nothing to a discussion about race relations.
This is the context in which people use this term.
When people say things like "you're technically correct, the best kind of correct", it's a tongue in cheek way of pointing out the observation is true, but lacks contextual information in order to be useful. This is why it tends to be a joke, in general everyone implicitly understands this.
No, they use it as a device to point out that there is more context around the discussion than whether or not something is true or false (hence my point about 2-valued logic).
I could say race is defined by genetics, and while that's technically correct, it adds nothing to a discussion about race relations.
This is the context in which people use this term.
When people say things like "you're technically correct, the best kind of correct", it's a tongue in cheek way of pointing out the observation is true, but lacks contextual information in order to be useful. This is why it tends to be a joke, in general everyone implicitly understands this.
I could say race is defined by genetics, and while that's technically
correct, it adds nothing to a discussion about race relations.
Genetics define physical race. That is technically correct, hence it's plain correct and it's true.
If a statement doesn't add to a discussion, then it's misusing of language to say that the statement is "technically correct" instead of clearly saying that the statement is not relevant.
It could be conventional to use "technically correct" to mean "true but important context is missing", so everybody understood it. Some define a language to be correct the way it is used commonly. However, I don't agree and I'd still consider it misusing language.
If context is missing, then why talking about the correctness instead of supplying missing context?
People using "technically correct" like you described indirectly devalue correctness. That follows the lines of valuing more the consequences of a statement than whether it's true or not.
I'd rather see the world like it is than how I'd like it to be. Therefore, correctness is very important to me and I think it shouldn't be subtly devalued.
How about you?
you must be fun at parties.
You'll outgrow that attitude eventually, you should revisit that last post when you do.
You'll outgrow that attitude eventually, you should revisit that last post when you do.
The bell curve distribution is a model. That our metric doesn't include negative values does not contradict its effectiveness. Also someone downstream calculated the probability of seeing a negative value for a normal distribution with 100 mean and 15 sd, the probability is about 1 in 100 billion. And so we don't expect the bell curve model to break down until we reach a population size on the order of 100 billion.
Right. Also, in theory, intelligence (or height) can be infinite it if is said it follows a bell curve, but that's sort of impossible too.
IQ is defined to be distributed as a bell curve. IQ below 0 is just extremely improbable.
It depends, Beta(3,3) is also bell-shaped.
Just look at the SAT maths scores of different races. IMHO that the high scores of Asian students was due to the Asian cultures which promotes hard work.
Here's the thing about IQ curves. Let's assume, for the moment, that we create a new "WQ" (waps quotient). It is 1000 random questions that you either know or don't know, which have absolutely nothing to do with eachother (who is the beatles' lead singer, what is the 7th prime number, did Mr. Garrison ever kill Kenny, is Spam searching related to Bayes' rule ? (nope, it's Bayes theorem), ...
For every question you have right you get +0.1, for every wrong answer you get 0. Obviously this proves nothing about you, except that you've at some point looked up the beatles, or actually looked up history of Thomas Bayes, or ... In other words it's a long list of "do you know trick X" bits.
What would the division of WQ in the population look like ? Exactly like the division of IQ in the population. [1] [2]
Here's a supposition : the IQ stat is exactly that. It's measuring of how many of the "little tricks" known by a test maker you know. The better you match the test maker, the better your score (think mostly cultural, but also whether your parents are academics or not, ...)
Although I must say, teaching kids "tricks" with numbers (e.g. how to tell if a number is divisible by 9 by looking at individual digits) and symbols is a good way to make them good at math over time.
[1] http://en.wikipedia.org/wiki/Binomial_distribution (read top paragraph, note that n will be a relatively large 1000) [2] http://en.wikipedia.org/wiki/Binomial_distribution#Normal_ap...
For every question you have right you get +0.1, for every wrong answer you get 0. Obviously this proves nothing about you, except that you've at some point looked up the beatles, or actually looked up history of Thomas Bayes, or ... In other words it's a long list of "do you know trick X" bits.
What would the division of WQ in the population look like ? Exactly like the division of IQ in the population. [1] [2]
Here's a supposition : the IQ stat is exactly that. It's measuring of how many of the "little tricks" known by a test maker you know. The better you match the test maker, the better your score (think mostly cultural, but also whether your parents are academics or not, ...)
Although I must say, teaching kids "tricks" with numbers (e.g. how to tell if a number is divisible by 9 by looking at individual digits) and symbols is a good way to make them good at math over time.
[1] http://en.wikipedia.org/wiki/Binomial_distribution (read top paragraph, note that n will be a relatively large 1000) [2] http://en.wikipedia.org/wiki/Binomial_distribution#Normal_ap...
Your theory would also need to explain correlations between scores on different IQ tests taken by the same test-taker.
And knowing the answers to more or less questions than someone else is suggestive of something, if the questions are broad enough. Especially if someone else grew up in the same culture as you
And knowing the answers to more or less questions than someone else is suggestive of something, if the questions are broad enough. Especially if someone else grew up in the same culture as you
Better to explain the correlation with job performance, and why the correlation improves with more complex jobs. http://www.udel.edu/educ/gottfredson/reprints/1997whygmatter...
"Your theory would also need to explain correlations between scores on different IQ tests taken by the same test-taker."
I think that one is easy:
In fact, @waps has a bag of M questions, where M > 1000. To compose a new WQ test, he draws 1000 questions, without replacement, from his bag. Then the series of tests has some similar or even same questions. In such a case the scores must be correlated.
This is not too far off-base. Especially when you consider that many questions are superficially different, but really the same. E.g., those "name the next number in the series" questions.
I think that one is easy:
In fact, @waps has a bag of M questions, where M > 1000. To compose a new WQ test, he draws 1000 questions, without replacement, from his bag. Then the series of tests has some similar or even same questions. In such a case the scores must be correlated.
This is not too far off-base. Especially when you consider that many questions are superficially different, but really the same. E.g., those "name the next number in the series" questions.
Hmm, is it still binomial if each individual question has different p of being correct? Also you will have to shift and rescale the binomial distribution to get it to have a mean of 100 and a std dev of 15 (which iq is defined as).
But you did answer my question on why IQ is normally distributed: the IIDs being averaged are the test questions themselves! (although I don't fully understand if those questions are bernoullis?)
But you did answer my question on why IQ is normally distributed: the IIDs being averaged are the test questions themselves! (although I don't fully understand if those questions are bernoullis?)
Thinking about it more I think it makes more sense to have N as the number of test takers. That way you can model it as a question x having a probability p_x of being correct. Then the total grade on problem x will be a binomial(N, p_x), which will converge to a normal distribution for large number of test takers. Then you are simply summing up a small number of normal approximations to get the final grade (and sum of normals is normal).
Did I model that correctly?
Did I model that correctly?
If intelligence is about seeing the invisible, Rorschach tests would fare better. I'd love to see 'pioneering quotient' tests, when someone envision something out of a blurry mess; just like a pioneer stumbling on a new idea.
The author gives a quote from Euclid and two physicists, but for the aspiring mathematician, or anyone who is bewitched by the myth of a genius mathematician, I think a more appropriate quote comes from Terry Tao [1]:
> Actually, I find the reality of mathematical research today – in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck – to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of “geniuses”. This “cult of genius” in fact causes a number of problems, since nobody is able to produce these (very rare) inspirations on anything approaching a regular basis, and with reliably consistent correctness.
He goes on to describe how some of the most valuable traits in mathematics include asking dumb questions, pateince, maturity, etc. I would add to that the skill of being wrong. If you're wrong all the time and comfortable with being wrong, you can identify more easily when you're wrong or right and work to fix the wrong stuff and identify the keys to the right stuff.
[1]: https://terrytao.wordpress.com/career-advice/does-one-have-t...