normal distribution 0.20
uniform distribution 0.19Error propagation also works for systematic uncertainties. I discuss along a simple example.
We want to error propagate to variable \(c\) with \[ c = \frac a b, \] where \(a\) is constant and \(b\) has uncertainty \(\sigma[b]\). All variables shall be positive.
Error propagation yields
\[ \sigma^2[c] = \big(\frac{\partial c}{\partial b}\big)^2 \, \sigma^2[b] = \frac {a^2} {b^4} \, \sigma^2[b] = \frac{c^2}{b^2} \, \sigma^2[b] \]
Or equivalent
\[ \frac{\sigma[c]}{c} = \frac{\sigma[b]}{b} \]
The \(\sigma\) here means (as it always does) square-root of variance. Error propagation is correct for any distribution where the variance is defined.
Numerical check:
The numerical results of \(\pm 20\,\%\) and \(\pm 19\,\%\) differ a little from the analytical result \(\pm 18\,\%\) because error propagation is approximate for non-linear functions. For small deviations, all functions are linear, but \(\pm 18\,\%\) is not that small. In general, errors are usually only known up to 10 %, so this is not a shocking deviation.
Since we don’t know the probability density of systematic offsets, we need to use a distribution-independent framework for the calculation to propagate errors, where systematic uncertainties are the square roots of variances of unknown distributions. The error propagation framework provides just that.