# 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:

- $f = x \pm y \Rightarrow (\Delta f)^2 = (\Delta x)^2 + (\Delta y)^2 \pm 2 C_{xy}$
- $f = cx \Rightarrow \Delta f = c \Delta x$
- $f = xy \Rightarrow (\Delta f)^2 = y^2 (\Delta x)^2 + x^2 (\Delta y)^2 + 2 xy C_{xy}$
- $f = \frac{x}{y} \Rightarrow (\Delta f)^2 = \frac{(\Delta x)^2}{y^2} + \frac{x^2}{y^4} (\Delta y)^2 - 2 \frac{x}{y^3} C_{xy}$