Calculating Percentiles of T-Distribution Using Gaussian Integration Method - Tzu-Li Tien (Engineering Proceedings)
Statistical inference is used to estimate population parameters based on sample information and to quantify the sampling error based on the probability narrative. The population mean is inferred by its sample mean, but when using sample variance, the population variance is needed. In the quantitative analysis of the sampling error, the t-distribution is used. To determine the percentiles of the t-distribution, the cumulative probability density function is necessary. However, the analytic expression does not exist for the cumulative probability density function of the t-distribution. Its values are obtained using numerical integration. However, the percentiles of the t-distribution are not listed for degrees of freedom over 30, while only listed for every 10 data points in probability theory or mathematical statistics. This is inconvenient for research. Therefore, the cumulative probability density function of t-distribution was calculated using the Gaussian integration method in this study. The results show that the percentiles of the t-distribution are accurately estimated using the algorithm developed in this study.