Hi i must determine the unbiased sample standard deviation of an unknown probability distribution.I dont have the data of the full population so i must work with a sample.
Now according to Wikipedia this is the best formula which would calculate it:
Wikipedia page of the unbiased estimation of SD
$$\hat{\sigma}=\sqrt{\frac{1}{n-1.5-\frac{1}{4} \gamma_{2}} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}$$
The dilemma with this is that i dont know which formula of the kurtosis to use:
The $\gamma_2$ denotes the "excess kurtosis", but should it be the excess kurtosis of the sample or of the entire population ? According to wikipedia it should be for the entire population, however that doesnt seem logical, as the formula is based on the sample.
Please clarify and help me , thanks :)
Note that the article you reference does not guarantee the unbiasedness of the estimator in your post. A good approach in this situation is to jacknife or bootstrap correct the bias. Unlike the sample variance estimate, the sample standard deviation estimator bias is very sensitive to the actual distribution (or family of distributions).