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I think you're being downvoted a bit unfairly. What most non-statisticians don't realize is that some of the most "unnormal" distributions actually look pretty
by jackpirate 9y ago
I think you're being downvoted a bit unfairly. What most non-statisticians don't realize is that some of the most "unnormal" distributions actually look pretty normally distributed, and some of the most ugly looking distributions actually behave like a normal distribution. When you look at a distribution's pdf, you're only looking at the "center" of the distribution. But what statisticians (usually) care about are the tails of a distribution.
Consider the following example: The [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution https://en.wikipedia.org/wiki/Cauchy_distribution) is a pathological example of a distribution with very heavy tails. Heavier than the power law distributions meantioned in the GP. So heavy, the distribution doesn't even have a mean value. But if you just look at the pdf, it looks almost exactly like a gaussian.
- logicallee 9y agothanks! Very interesting. Is the standard deviation still useful for some of the other distributions you mentioned, that look vaguely Gaussian?
- thanatropism 9y agoThe standard deviation is defined as E[(X-EX)^2] If EX = infinity but X is finite with prob > 0, the standard deviation will be infinite too.
- R_haterade 9y agoYou're referring to the square root of the second moment about the mean. The short answer is yes, it's always useful, provided you have an estimator. Check out Chebyshev's inequality. Just don't start talking about it to your pointy-haired boss. You'll get fired.
- logicallee 9y agothanks. but what does your last line, about the boss mean?
- R_haterade 9y agoThat this sort of knowledge is often uninteresting to non-technical types. I'm pretty jaded at this point.
- dbaupp 9y agoFor the specific example of the Cauchy distribution, the standard deviation isn't defined.
- CuriouslyC 9y agoIt is pretty easy to tell a Cauchy or a low k Student's T apart from a Gaussian. Just look at your outliers. One thing a lot of people miss is that a normal distribution with outliers _IS_ a long tail distribution. Too bad inference with long tailed distributions is so hard.