Direct Approximations for Chi-Squared Percentage Points

In contrast to the customary approximations based on normalizing transformations, direct approximations involve only simple functions of parameters (such as degrees of freedom and tail area in the chi-squared distributions). This article uses techniques of exploratory data analysis to develop direct approximations for percentage points in the commonly used portions of both tails of the chi-squared distributions with small to moderate numbers of degrees of freedom. These approximations are convenient to use, and they compare favorably in accuracy with the popular approximations based on normalizing transformations.