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.

翻译：作为一种严格的统计方法，统计泰勒展开将传统泰勒展开中的精确输入变量替换为具有已知分布和样本数量的随机变量，以计算每个结果的均值、偏差和可靠因子。该方法通过中间步骤追踪输入不确定性的传播，使得最终分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本质区别。统计泰勒展开可规范解析表达式的数值计算。本研究还介绍了统计泰勒展开的实现方法——方差算术，并呈现了其在广泛数学应用中的相应测试结果。另一重要结论是，库函数中的数值误差可能对结果产生显著影响。因此，库函数输出的每个值最好附带不确定偏差。此外，本文还探讨了统计泰勒展开与量子物理学之间的潜在关联。