误差传播

设有一般函数（线性函数和非线性函数）

Z=${\displaystyle f(x_{1},x_{2},\dots ,x_{n})}$

式中${\displaystyle x_{1},x_{2},\dots ,x_{n}}$ 为可直接观测的相互独立的未知量，z为不便于直接观测的未知量。已知${\displaystyle x_{1},x_{2},\dots ,x_{n}}$ 的標準差分别为${\displaystyle m_{1},m_{2},\dots ,m_{n}}$ ,现在要求z的標準差${\displaystyle m_{z}}$ 。已知函数z的中误差关系式为${\displaystyle m_{z}^{2}}$ =${\displaystyle k_{1}^{2}m_{1}^{2}+k_{2}^{2}m_{2}^{2}+\dots +k_{n}^{2}m_{n}^{2}}$（其中${\displaystyle k_{1},k_{2},\dots ,k_{n}}$为任意常数）。由数学分析可知，变量的误差与函数的误差之间的关系，可以近似的用函数的全微分来表达，为此对上式求全微分，并以真误差的符号“Δ”替代微分的符号“d”得 ${\displaystyle \Delta z={\frac {\partial f}{\partial x_{1}}}\cdot \Delta x_{1}+{\frac {\partial f}{\partial x_{2}}}\cdot \Delta x_{2}+\cdots +{\frac {\partial f}{\partial x_{n}}}\cdot \Delta x_{n}}$

式中${\displaystyle {\frac {\partial f}{\partial x_{i}}}}$ （i=1，2，，…，n）是函数对各个变量变量所取得偏导数，对上式以標準差平方代替真误差,由函数z的中误差关系式可得

${\displaystyle m_{z}^{2}}$ =${\displaystyle \left({\frac {\partial f}{\partial x_{1}}}\right)^{2}m_{1}^{2}+\left({\frac {\partial f}{\partial x_{2}}}\right)^{2}m_{2}^{2}+\dots +\left({\frac {\partial f}{\partial x_{n}}}\right)^{2}m_{n}^{2}}$

将上式开根号可得误差传播定律的一般形式

${\displaystyle m_{z}}$=±${\displaystyle {\sqrt {\left({\frac {\partial f}{\partial x_{1}}}\right)^{2}m_{1}^{2}+\left({\frac {\partial f}{\partial x_{2}}}\right)^{2}m_{2}^{2}+\dots +\left({\frac {\partial f}{\partial x_{n}}}\right)^{2}m_{n}^{2}}}}$

• Uncertainties and Error Propagation, Appendix V from the Mechanics Lab Manual, Case Western Reserve University.
• Mathieu Rouaud, 2013: Probability, Statistics and Estimation 页面存档备份，存于 Propagation of Uncertainties in Experimental Measurement.
• A detailed discussion of measurements and the propagation of uncertainty 页面存档备份，存于 explaining the benefits of using error propagation formulas and monte carlo simulations instead of simple significance arithmetic.
• Uncertainties and Error Propagation, Vern Lindberg's Guide to Uncertainties and Error Propagation.