Inaccuracy of the X²Test of Goodness of Fit When Expected Frequencies Are Small

In applied statistics there is an increasing tendency toward leniency regarding minimum expectation prerequisite to Pearson's X 2 test for goodness of fit, based on various empirical studies of the distribution of X2 in sampling from multinomial populations with given parameters. In this study it is shown that the chi-square probabilities of X 2 may differ markedly from the exact cumulative multinomial probabilities. Contrary to prevailing opinion, the approximation did not improve, when samples were small, as expectation increased nor as the number of categories or degrees of freedom increased.