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I was wondering about the way we refer to the term "moments". In statistics and probability the first moments is the mean $$E[X^1] $$ The second moments is the variance: $$E[(X-E[X])^2]$$ I notice that there's a difference in which we define moments. In the first case the $k'$th moment is $E(X^k)$ and the second case it is $E[(X-E[X])^k]$. Why is there a difference? In Common scientific language when people refer to the third moment is it then $E[(X-E[X])^3]$ or $E[X^3]$

KinkyLaura
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    There is a distinction between "moments" and "central moments". The second moment is $E(X^2)$ and the second central moment is the variance. – Michael R. Chernick Jan 28 '18 at 08:59
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    As @Michael Chernick hints, moments are always defined in terms of means of powers of differences or distances or deviations from some origin. For the mean, the origin is 0. It's the first moment about or around zero. For so-called central moments. the origin is the mean. The first moment about the mean is identically zero whenever it's defined, so it's useless. – Nick Cox Jan 28 '18 at 12:42

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