Statistical Taylor expansion replaces the input precise variables in a conventional Taylor expansion with random variables each with known mean and deviation, to calculate the result mean and deviation. It is based on the uncorrelated uncertainty assumption: Each input variable is measured independently with fine enough statistical precision, so that their uncertainties are independent of each other. Statistical Taylor expansion reviews that the intermediate analytic expressions can no longer be regarded as independent of each other, and the result of analytic expression should be path independent. This conclusion differs fundamentally from the conventional common approach in applied mathematics to find the best execution path for a result. This paper also presents an implementation of statistical Taylor expansion called variance arithmetic, and the tests on variance arithmetic.

翻译：统计泰勒展开将传统泰勒展开中的输入精确变量替换为具有已知均值和偏差的随机变量，以计算结果的均值和偏差。该方法基于不相关不确定性假设：每个输入变量均以足够高的统计精度独立测量，因此它们的不确定性彼此独立。统计泰勒展开指出，中间解析表达式不再被视为相互独立，且解析表达式的结果应具有路径无关性。这一结论与应用数学中寻找最优执行路径的传统通用方法存在根本性差异。本文还提出了一种称为方差运算的统计泰勒展开实现方法，并给出了对方差运算的测试结果。