双语化学辞典Chem Lexicon
本科Undergraduate 误差统计、校准与色谱 Statistics, calibration & chromatography

误差传递Propagation of uncertainty

\[\sigma_y^2 = \sum_i\left(\frac{\partial y}{\partial x_i}\right)^2\sigma_{x_i}^2\]
常用规则common rules
\[y = a \pm b:\ \sigma_y^2 = \sigma_a^2 + \sigma_b^2;\qquad y = ab\ \text{或}\ a/b:\ \left(\frac{\sigma_y}{y}\right)^2 = \left(\frac{\sigma_a}{a}\right)^2 + \left(\frac{\sigma_b}{b}\right)^2;\qquad y = \lg a:\ \sigma_y = 0.434\frac{\sigma_a}{a}\]\[y = a \pm b:\ \sigma_y^2 = \sigma_a^2 + \sigma_b^2;\qquad y = ab\ \text{or}\ a/b:\ \left(\frac{\sigma_y}{y}\right)^2 = \left(\frac{\sigma_a}{a}\right)^2 + \left(\frac{\sigma_b}{b}\right)^2;\qquad y = \lg a:\ \sigma_y = 0.434\frac{\sigma_a}{a}\]

含义Meaning

由各直接测量量的不确定度计算结果的不确定度（独立随机误差）。加减法合并绝对误差，乘除法合并相对误差。

Combines uncertainties of independent measured quantities into that of the result: absolute errors add in quadrature for sums, relative errors for products.

符号说明Symbols

符号Symbol含义 Meaning单位 Unit
\(\sigma_{x_i}\)输入量的标准不确定度standard uncertainty of input
\(\partial y/\partial x_i\)灵敏系数sensitivity coefficient

推导Derivation

  1. 把 y 在平均值处一阶泰勒展开：\(\delta y \approx \sum_i(\partial y/\partial x_i)\delta x_i\)。

    Linearise y with a first-order Taylor expansion.

  2. 平方取期望；各 \(x_i\) 独立时交叉项（协方差）为零。

    Square and average; cross terms vanish for independent inputs.

  3. 对乘积 \(y = ab\)：\(\partial y/\partial a = b\)，除以 \(y^2\) 得相对误差形式。

    For y = ab, divide by y² to get relative errors.

适用条件与局限Assumptions & limitations

  • 线性化近似：相对误差大或函数强非线性时不准（可用蒙特卡罗）；相关输入需加协方差项。

    Linearisation fails for large errors or strong non-linearity; correlated inputs need covariance terms.

  • 系统误差不按此规则相互抵消。

    Systematic errors do not combine this way.

例题Worked example

c = m/(MV)：m = 0.2000 ± 0.0002 g（0.1%），V = 100.0 ± 0.1 mL（0.1%）：c 的相对不确定度 √(0.1² + 0.1²) = 0.14%。

c = m/(MV) with 0.1 % on m and V: 0.14 % on c.