Chi-Square Percentiles: Old and New Approximations, with Applications to Sample Size Determination

SYNOPTIC ABSTRACT

Percentile points of the chi-square distribution can be approximated by functions of the corresponding percentiles of the standard normal distribution. We show that the Wilson-Hilferty approximation gives accuracy to at least one decimal place even when the degrees of freedom are ≤ 10. Also, we provide a new and simple approximation which is nearly as accurate as the Wilson-Hilferty approximation for most percentiles commonly used by statisticians, and which is slightly better than the Wilson-Hilferty approximation when the percentile p is .95 (which corresponds to significance level .05). These approximations are particularly useful for sample size determination since they give simple analytic expressions for the percentiles as a function of the sample size and a single standard normal percentile.

Key Words and Phrases:

• Chi-square distribution
• sample size
• determination
• approximations