I am having trouble understanding how the sample formulas for distribution moments are derived, for example, the third standardized central moment is:

$$ \frac{1}{n}\frac{\sum{(x - \mu)^3}}{\sigma^3} $$

However, if we are using the sample mean and standard deviation it becomes:

$$ \frac{n}{(n-1)(n-2)}\frac{\sum{(x - \hat{x})^3}}{s^3} $$

I can´t understand why this makes sense and neither a place (book, website, …) that explains this well. If someone could explain this or just point to good resources on this I would really appreciate.

Also, how this would change if we were using the non standardize formula:

$$ \frac{\sum{(x - \hat{x})^3}}{n} $$

  • $\begingroup$ The first formula defines a property of a distribution. The second is an estimator of that property. You can find a great deal about this by searching our site for threads that use "estimator." The third is the (empirical) third moment (about the mean) of the data. It rarely is used by itself--it is incorporated in other estimators or descriptive statistics. $\endgroup$
    – whuber
    Jan 4, 2022 at 0:22
  • $\begingroup$ @whuber I actually need the sample formula for the last formula. Any, ideia how I can derive it ? $\endgroup$
    – ladca
    Jan 4, 2022 at 12:40
  • $\begingroup$ I don't understand your comment, because your last formula evidently is for a sample. $\endgroup$
    – whuber
    Jan 4, 2022 at 15:27
  • $\begingroup$ @whuber it uses the sample mean instead of the mean but it is a bias estimator. What I meant was, if I want an unbias estimator, what normalization should I apply? should I also multiply by n/((n-1)(n-2))? $\endgroup$
    – ladca
    Jan 4, 2022 at 20:24