Gli indici di asimmetria e di curtosi di K.Pearson per la media campionaria

Abstract

It is known that the distribution of the sample mean tends to a normal distribution as the sample size n increases—under quite general conditions—in case of sampling both with replacement and without replacement. A simple criterion, usually applied, to gauge the approach to such limit distribution, is to examine the behaviour of K.Pearson's asymmetry and kurtosis indexes. First of all we have reformulated Pearson's indexes as ratios between quantities expressed in the same unit measure of the observed variable; next we have examined their behaviour, as n increases, in the case of sampling (a) with replacement, and (b) out replacement. The most interesting case, i.e. sampling without replacement, has been roughly investigated, by sampling from several asymmetric populations; in all cases the approach to " normal " values appears much faster for the kurtosis index than for the asymmetric index. Moreover, some paradoxical results concerning the asymmetry of the distribution of the sample mean have been pointed out.