Normal distribution can be parameterized with almost any measures of location and dispersion. The reason why standard deviation (which is just variance in hiding) is used is because it makes the equations (that non-statisticians practically never need to look at) a bit nicer given the other (mostly arbitrary) conventions used in the maths.

Put simply, standard deviation is arbitrary for most people.

I don't want to keep going on because it sounds like I'm trolling, but when you tell me 'it tells us what the final normal distribution would look like if you averaged a lot of independent distributions' I just want to say 'why do I care about that? where did that requirement come from?'

Economics of stocks is a good example, when you put a lot of stocks in one portfolio you can calculate its risk (standard deviation) and returns (average value), and see how much each piece of stock contributed to that. This is because the results of many variables looks like a normal distribution with the combined average and standard deviation. Now it gets a bit more complicated in the real world since stocks are not independent from each other, but it shouldn't be hard to understand why knowing the standard deviation of something is useful.

> Now it gets a bit more complicated in the real world since stocks are not independent from each other

This is also where I get frustrated.

I'm taught about the CLT but every place I think I might use it, the samples aren't really IDD when you look hard enough, so it doesn't apply.

Computer benchmarking (which is where I'm usually trying to apply statistics) is the classic example. One iteration changes the state of the computer for the next iteration. They aren't remotely IDD, but people still try to claim the CLT. Seems like all their stats beyond that point is broken.

Right, that is how statistics usually is done. Note however that you don't need IDD, independence is enough. So you assume independence, from that CLT follows which means that standard deviation is all that matters. Then the real world deviates from this based on how much your distributions fails to be independent.

I'd assume that computer benchmarks have very low correlation between runs, unless you leak ram or your components don't keep a stable temperature. So in that case the CLT seems to apply just fine.

I can't understand this. Surely either samples are independent, or they are not?

And they are not in a computer program. The state of, to just give one example, the processor cache depends on what happened in the previous run.

We have all this rigour in how statistical techniques are applied, and we make black or white decisions based on the numbers that come out of them, but at their very foundation we're saying 'whatever I'll just pretend this rule applies it's probably fine'. How can we build on that flawed foundation? I don't get it!

Statistical techniques are a model of reality: Reality is not rigorous, but when our models are rigorous, imprecision enters the computation at one place only. If our models aren't rigorous, imprecision occurs at more places, blurring the answer even more.

"All models are wrong, but some models are useful." Just remember that there's some imprecision in the answer a model gives you, and keep that fact in mind when applying that answer to make real-world decisions.

I think it begins with binomial expansion and coin tossing.

e.g. https://socratic.org/questions/what-is-the-difference-betwee...