Magnitude uncertainty of a series RL impedance (${\sigma }_z^2=(R^2{\sigma }_R^2+X^2{\sigma }_X^2)/z^2$)

For $z=\sqrt{R^2+X^2}$ with reactance $X=\omega L$, linear differentiation gives $dz=(RdR+XdX)/z$. Independent standard uncertainties give the title formula; correlated inputs add $2RX\mathrm{Cov}(R,X)/z^2$. Hard error bounds instead add absolute linear contributions. The meaning of an error bar is an additional measurement assumption, not a consequence of the impedance formula.