\[b = \frac{\sum(x_i - \bar x)(y_i - \bar y)}{\sum(x_i - \bar x)^2},\quad a = \bar y - b\bar x,\quad r = \frac{\sum(x_i - \bar x)(y_i - \bar y)}{\sqrt{\sum(x_i - \bar x)^2\sum(y_i - \bar y)^2}}\]
由校准曲线求未知样浓度及其不确定度unknown concentration and its uncertainty
\[x_0 = \frac{y_0 - a}{b},\qquad s_{x_0} = \frac{s_{y/x}}{b}\sqrt{\frac1m + \frac1n + \frac{(y_0 - \bar y)^2}{b^2\sum(x_i - \bar x)^2}}\]含义Meaning
用标准溶液建立响应 y 与浓度 x 的直线关系，使残差平方和最小。r 接近 1 表示线性好，但不能替代残差分析。
Fits the response y of standards to concentration x by minimising squared residuals. r near 1 indicates linearity but does not replace residual analysis.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(a, b\) | 截距、斜率（灵敏度）intercept and slope (sensitivity) | |
| \(r\) | 相关系数correlation coefficient | 1 |
| \(s_{y/x}\) | 回归残差标准差residual standard deviation | |
| \(m\) | 未知样平行测定次数replicate readings of the unknown | 1 |
推导Derivation
目标函数 \(S = \sum(y_i - a - bx_i)^2\)。
Minimise \(S = \sum(y_i - a - bx_i)^2\).
令 \(\partial S/\partial a = 0\)、\(\partial S/\partial b = 0\) 得正规方程，解出 a、b。
Set both partial derivatives to zero and solve the normal equations.
假设只有 y 有误差且方差相等（同方差）。
Assumes error only in y with constant variance.
适用条件与局限Assumptions & limitations
异方差（误差随浓度增大）时用加权回归；x 也有误差时用正交回归。
Heteroscedastic data need weighted regression; errors in x need orthogonal regression.
只在校准范围内内插可靠，外推风险大。
Interpolate only within the calibrated range.