General case

Let us assume there are $n$ independently measured values $x_i$ and $m$ calibration constants $c_j$ with their covariance matrix ${\bf V}_c$. The latter can also be theoretical parameters influencing the data, and moreover they may be correlated, as usually happens if, for example, they are parameters of a calibration fit. We can then include the $c_j$ in the vector that contains the measurements and ${\bf V}_c$ in the covariance matrix ${\bf V}_X$:

$$\underline{x} = \left( \begin{array}{c} x_1 \\ \vdots \\ x_n \\ c... ...cdots & \sigma_n^2 & \\ \hline & & {\bf0} & & {\bf V}_c \end{array} \right)\, .$$ (6.44)

The corrected quantities are obtained from the most general function

$$Y_i = Y_i(X_i,\underline{c}) \hspace{2.0 cm} (i=1,2, \ldots, n)\, ,$$ (6.45)

and the covariance matrix ${\bf V}_Y$ from the covariance propagation ${\bf V}_Y = {\bf M}{\bf V}_X{\bf M}^T$.

As a frequently encountered example, we can think of several normalization constants, each affecting a subsample of the data - as is the case where each of several detectors measures a set of physical quantities. Let us consider just three quantities ($X_i$) and three uncorrelated normalization standard uncertainties ( $\sigma_{f_j}$), the first common to $X_1$ and $X_2$, the second to $X_2$ and $X_3$ and the third to all three. We get the following covariance matrix:

$$\left( \begin{array}{ccc} \sigma_1^2 + \left(\sigma_{f_1}^2 + \si... ... \left( \sigma_{f_2}^2 + \sigma_{f_3}^2\right) \, x_3^2 \end{array} \right)\, .$$ (6.46)