Estimation of type I error probability from experimental Dixonʹs “Q” parameter on testing for outliers within small size data sets

Common significance tests carried out using statistical software packages usually return to the user the probability p of type I error as the result. Based on p and the preset confidence level the user will decide on the acceptance or the rejection of the associated null hypothesis. Dixonʹs test (Q-test) is commonly used for the detection of an outlier within a set of N observations (typically: N = 3–12). Q-test can only be applied by comparing the experimental value of the statistic Q with tabulated critical Q-values corresponding to some standard values of p. Hence, for a given value of Q and a number of observations, N, the user knows only the range and not the value of the associated probability p of type I error (erroneous rejection). This is due to the lack of explicit expressions of the form p = F(Q,N). In this work, a simple stochastic (Monte Carlo) approach is presented for the estimation of p corresponding to a given experimental value of Q and size N of the data set. In addition, based on Dixonʹs equations, explicit expressions of p are given for N = 3 and 4.