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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

by 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

by rawnlq 11y ago

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