Calculating propagation of uncertainty

Question

I need to compute the propagation of uncertainty $u(N')$.

It is known that $N' = \frac{N}{t}$ and:

• $u(N) = \sqrt{N}$
• $u(t) = 0.1$

My attempt at the task: $$u(N') = \sqrt{\Bigg[ \frac{\partial N'}{\partial t} u(t) \Bigg]^2 + \Bigg[\frac{\partial N'}{\partial N} u(N)\Bigg]^2} = \sqrt{\Bigg[ -\frac{N}{t^2} u(t)\Bigg]^2 + \Bigg[\frac{1}{t} u(N)\Bigg]^2}$$

Does this look correct?

Answer 1

Yes. Errors can be computed that way, i.e. if $N'=N'(x_1,x_2,\dots,x_n)$ the uncertainty of $N'$, $u(N')$ is given by $$ u(N')=\sqrt{\sum_k^n\Big((\partial_{x_k}N')u(x_k)\Big)^2} $$ where $\partial_{x_k}=\frac{\partial}{\partial x_k}$ is the partial derivative with respect to $x_k$. It is easy to check that $$ \frac{u(N')}{N'}=\sqrt{\sum_k^n\Big(\frac{u(x_k)}{x_k}\Big)^2}. $$