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.

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.

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

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.

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.

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.

> 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.

    I could say race is defined by genetics, and while that's technically
    correct, it adds nothing to a discussion about race relations.

Whether a statement does or does not add to a discussion is an independent to whether it's true or not.

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.

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.

> 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 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.