7 CONCLUSION

The $\chi^{2}$ distribution is often used to model the behavior of the $s^{2}$ statistics. This is obviously valid for normal variates, but this doesn't apply to the general case. The first reason is that, apart from this special case of the normal distribution, statistic $s^{2}$ (the sample variance) is not independent from the sample mean $m$ so that only a joint distribution can make sense when computing confidence intervals. Moreover, it appears that in some circumstances, $s^{2}$ is quite-normally distributed even for small values of $n$, while in some other circumstances $s^{2}$ is distributed in a very different way. Such a behavior must be taken into account when determining the minimal size of a sample for various statistic tests, aggravating the problematic described in citeseppen-1000cite##1##2(##1@tempswa , ##2)##1##2##3##1 ##3internalciteChan08.

By the way, a method for computing and checking the product moments has been developed: the determinant of all the product moments of a given degree must split into a product of simple linear factors. The value of $\Delta_{11}$ -eq:new-result-eleven- is a new result, while values of $\Delta_{d}$ were not explicitly stated in the many research papers devoted to the product moments of degree $d$ less than $11$.

Acknowledgement   We like to thank the Anonymous Referees for their helpful and constructive as well as very extensive comments on the submitted manuscript.