L'approssimazione della distribuzione della statistica X² di K. Pearson con la distribuzione chi-quadrato

The approximation of the Pearson Chi-squared statistics with the Chi-squared distribution. The aim of the present paper is to analyze the distribution of the Pearson Chi-squared statistics, when performing a chi-squared test in which the null hypothesis states that the k observed frequencies come from a symmetric multinomial r.v. with k alternatives. Following a common rule of thumb, in many textbooks of Statistics it is recommended to employ the chi-squared approximation only if the expected frequencies are greater than 5. The purpose here is to compare the exact distribution and some recently tabulated values of the Chi-squared statistics with the chi-squared distribution, to provide a sound methodology for the approximation. From the obtained results it appears that the minimum value for the expected frequencies which assures a good approximation has to be chosen with relation to k, because as k increases the required minimum value decreases.