6 ms·

On Mathematical Intelligence and How It Grows

43 points by __Joker 11y ago

rawnlq 11y ago
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?
- gohrt 11y ago
  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.
- waps 11y ago
  > 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).
  - panic 11y ago
    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.
    - waps 11y ago
      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.
- panic 11y ago
  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.
  - thaumasiotes 11y ago
    > 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.
    - rawnlq 11y ago
      I am really impressed by wolfram alpha's ability to parse my butchered syntax: https://www.wolframalpha.com/input/?i=probability+that+gaussian%28100%2C+15%29+%3C+0 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
  - Hobotron1 11y ago
    So define 'average' as 0?
  - hackinthebochs 11y ago
    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.
    - myg204 11y ago
TerryADavis 11y ago
I think by definition, intelligence is a bell curve. I think the difference between 140 IQ and 141 IQ, for example, is a difference in percentile of the population.
waps 11y ago
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 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_approximation http://en.wikipedia.org/wiki/Binomial_distribution#Normal_ap...
- gohrt 11y ago
  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
  - mturmon 11y ago
    "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.
david927 11y ago
[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.
- myg204 11y ago
  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]
  - leni536 11y ago
    "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.
    - gwern 11y ago
      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.
      - leni536 11y ago
        >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").
        deleted 11y ago
davidgerard 11y ago
> 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.
Synaesthesia 11y ago
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.
j2kun 11y ago
While I don't care much for IQ as a statistic, I do believe that discipline and hard work are important. 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-to-be-a-genius-to-do-maths/ https://terrytao.wordpress.com/career-advice/does-one-have-t...
gwern 11y ago
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 https://en.wikipedia.org/wiki/SAT#Preparation and reviews like http://libgen.org/scimag/get.php?doi=10.1111%2Fj.1468-2397.2011.00812.x http://libgen.org/scimag/get.php?doi=10.1111%2Fj.1468-2397.2... ).
cwhy 11y ago
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.

43 points by __Joker 11y ago

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?

gohrt 11y ago
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.
waps 11y ago
> 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).
- panic 11y ago
  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.
  - waps 11y ago
    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.
panic 11y ago
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.
- thaumasiotes 11y ago
  > 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.
  - rawnlq 11y ago
    I am really impressed by wolfram alpha's ability to parse my butchered syntax: https://www.wolframalpha.com/input/?i=probability+that+gaussian%28100%2C+15%29+%3C+0 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
- Hobotron1 11y ago
  So define 'average' as 0?
- hackinthebochs 11y ago
  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.
  - myg204 11y ago

TerryADavis 11y ago

I think by definition, intelligence is a bell curve. I think the difference between 140 IQ and 141 IQ, for example, is a difference in percentile of the population.

waps 11y ago

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 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_approximation http://en.wikipedia.org/wiki/Binomial_distribution#Normal_ap...

gohrt 11y ago
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
- mturmon 11y ago
  "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.

david927 11y ago

[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.

myg204 11y ago
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]
- leni536 11y ago
  "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.
  - gwern 11y ago
    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.
    - leni536 11y ago
      >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").
      - deleted 11y ago

davidgerard 11y ago

> 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.

Synaesthesia 11y ago

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.

j2kun 11y ago

While I don't care much for IQ as a statistic, I do believe that discipline and hard work are important. 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-to-be-a-genius-to-do-maths/ https://terrytao.wordpress.com/career-advice/does-one-have-t...

gwern 11y ago

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 https://en.wikipedia.org/wiki/SAT#Preparation and reviews like http://libgen.org/scimag/get.php?doi=10.1111%2Fj.1468-2397.2011.00812.x http://libgen.org/scimag/get.php?doi=10.1111%2Fj.1468-2397.2... ).

cwhy 11y ago

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.