Propagation of uncertainty
For the covariance, refer to page:
Given a function of
$n$
non-correlated variables
$f(x_1, x_2, \ldots, x_n)$
, such that each variable
$x_i$
has its own uncertainty
$\Delta x_i$
, we can compute the uncertainty on
$f$
as
$\Delta f(x_1, x_2, \ldots, x_n, \Delta x_1, \ldots, \Delta x_n) = \left[\sum_{i=1}^{i=n} \frac{\partial f}{\partial x_i} \Delta x_i\right]^{\frac{1}{2}} \ .$
If the variables are correlated, then there is the covariance term:
$\Delta f(x_1, x_2, \ldots, x_n, \Delta x_1, \ldots, \Delta x_n) = \left[\sum_{i=1}^{n} \sum_{k=1}^n \frac{\partial f}{\partial x_i} \frac{\partial f}{\partial x_k} C_{ik}\right]^{1/2} \ ,$
where
$C_{ik}$
is the covariance matrix.
In the most common cases:
• $f = x \pm y \Rightarrow (\Delta f)^2 = (\Delta x)^2 + (\Delta y)^2 \pm 2 C_{xy}$
• $f = cx \Rightarrow \Delta f = c \Delta x$
• $f = xy \Rightarrow (\Delta f)^2 = y^2 (\Delta x)^2 + x^2 (\Delta y)^2 + 2 xy C_{xy}$
• $f = \frac{x}{y} \Rightarrow (\Delta f)^2 = \frac{(\Delta x)^2}{y^2} + \frac{x^2}{y^4} (\Delta y)^2 - 2 \frac{x}{y^3} C_{xy}$