Abstract: It is well known that the probability integral and quantile of a standard normal distribution are usually calculated by table look-up. A lot of literature gave also some approximate formulas, which are very complex and inconvenient to use. In this paper, the compact closed forms of the standard normal distribution and quantile are presented by using the two-dimensional circular approximation and inverse function, which make calculation of the probability much casier. Furthermore, we make use of the compensation of primary and remainder integrals to derive more accurately compact closed forms, with the error distributions of the compact closed forms given.

Key words: standard normal distribution, quantile, two-dimensional approximation, compact closed form, primary and remainder integrals compensation