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 standard 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.

翻译：作为一种严谨的统计方法，统计泰勒展开通过将精确的输入变量替换为已知分布和样本数量的随机变量，扩展了传统的泰勒展开，以计算每个结果的均值、标准差和可靠因子。该方法追踪输入不确定性在中间步骤中的传播，从而使最终解析结果具有路径无关性。因此，它与应用数学中为每次计算优化计算路径的常见方法存在根本区别。统计泰勒展开可为解析表达式实现数值计算的标准化。本研究还介绍了统计泰勒展开的实现方式——方差运算，并展示了在广泛数学应用中的相应测试结果。本研究的另一个重要结论是：库函数的数值误差会显著影响计算结果。理想情况下，库函数输出的每个值都应附带一个不确定性偏差。