As a rigorous statistical approach, statistical Taylor expansion extends the conventional Taylor expansion by replacing precise input variables with random variables of known distributions and sample counts to compute the mean, the deviation, and the reliable factor of each result. It tracks the propagation of the input uncertainties through intermediate steps, so that the final analytic result becomes path independent. Therefore, it differs fundamentally from common approaches in applied mathematics that optimize computational path for each calculation. Statistical Taylor expansion may standardize numerical computations for analytic expressions. This study also introduces the implementation of statistical Taylor expansion termed variance arithmetic and presents corresponding test results across a wide range of mathematical applications. Another important conclusion of this study is that numerical errors in library functions can significantly affect results. It is desirable that each value from library functions be accomplished by an uncertainty deviation. The possible link between statistical Taylor expansion and quantum physics is discussed as well.