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The normal distribution is important because it arises naturally when the preconditions of the central limit theorem hold. But you still have to use your brain

The normal distribution is important because it arises naturally when the preconditions of the central limit theorem hold. But you still have to use your brain - you can't unquestioningly assume that any random variable (or sample or whatever) you will stumble upon will be approximately normally distributed.

lordnacho 11y ago
This is the right answer. Quite a lot of things are not normally distributed. Often this is because there's more than one generating mechanism. For instance you may have a financial market where normally things go up and down smoothly, people are reacting to news as normal. But then there are also panics, where people are not behaving normally. There's also cases where the past influences the future, and drawings are not identically and independently distributed (though Wikipedia tells me there's weak variants of the CLT).
- FabHK 11y ago
  The conditions under which the Central Limit Theorem hold are complicated (in particular, weaker and weaker conditions have been shown to be sufficient even fairly recently). But intuitively, they boil down to two conditions that are fairly sensible: 1. The individual observations can't be too extreme. Bounded variance, for example, is sufficient, but weaker conditions suffice. 2. The observations must be somewhat independent. Independence is a strong condition and sufficient, but again weaker conditions suffice. If 1) doesn't hold, a massive outliers could swamp all other observations. If 2) doesn't hold, for example if one underlying factor influences all observations, all bets are off: the observations could "conspire" to be fairly large (or fairly small), thus giving rise to fat tails inconsistent with the normal distribution. (Having IID (independent identically distributed) observations, btw, is not sufficient, if the distribution itself is too fat tailed.)
  - catnaroek 11y ago
    Nice. I was aware that you could weaken independence, but not bounded variance. Without bounded variance, I thought you would get something like Student's t.

lordnacho 11y ago

This is the right answer. Quite a lot of things are not normally distributed. Often this is because there's more than one generating mechanism. For instance you may have a financial market where normally things go up and down smoothly, people are reacting to news as normal. But then there are also panics, where people are not behaving normally. There's also cases where the past influences the future, and drawings are not identically and independently distributed (though Wikipedia tells me there's weak variants of the CLT).

FabHK 11y ago
The conditions under which the Central Limit Theorem hold are complicated (in particular, weaker and weaker conditions have been shown to be sufficient even fairly recently). But intuitively, they boil down to two conditions that are fairly sensible: 1. The individual observations can't be too extreme. Bounded variance, for example, is sufficient, but weaker conditions suffice. 2. The observations must be somewhat independent. Independence is a strong condition and sufficient, but again weaker conditions suffice. If 1) doesn't hold, a massive outliers could swamp all other observations. If 2) doesn't hold, for example if one underlying factor influences all observations, all bets are off: the observations could "conspire" to be fairly large (or fairly small), thus giving rise to fat tails inconsistent with the normal distribution. (Having IID (independent identically distributed) observations, btw, is not sufficient, if the distribution itself is too fat tailed.)
- catnaroek 11y ago
  Nice. I was aware that you could weaken independence, but not bounded variance. Without bounded variance, I thought you would get something like Student's t.