Radiative temperature uncertainty after background subtraction (${\sigma }_T/T={\sigma }_{P_{\mathrm{total}}-P_{\mathrm{back}}}/(4P)$)

If emitted power obeys $P=A{T}^4$ with known constant $A$, then small relative temperature uncertainty is one quarter of relative net-power uncertainty. For measured quantities $X,Y$ with covariance $C_{XY}$, $\mathrm{Var}(X-Y)={\sigma }_X^2+{\sigma }_Y^2-2C_{XY}$. A weak net signal obtained by subtracting a large background can therefore have poor relative precision even when both original powers are measured accurately. Correlated standard uncertainties and worst-case error bounds require different propagation rules.