What are the independent and identically distributed random variables that are being averaged in this case?
"Identically distributed" is a not a requirement in CLT, it's just the simplest variant to prove.
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.
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
"The distribution of the measured IQ very closely follows a bell curve." would have been the correct way to formulate.
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.
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.
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'll outgrow that attitude eventually, you should revisit that last post when you do.
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).
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.