0

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:

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

ladca
  • 11
  • 1
  • 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. – whuber Jan 04 '22 at 00:22
  • @whuber I actually need the sample formula for the last formula. Any, ideia how I can derive it ? – ladca Jan 04 '22 at 12:40
  • I don't understand your comment, because your last formula evidently *is* for a sample. – whuber Jan 04 '22 at 15:27
  • @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))? – ladca Jan 04 '22 at 20:24

0 Answers0