Propagation of uncertainty through functions
For a general function of measured quantities, the uncertainty in the result is found from the function's partial derivatives with respect to each input.
Practise this conceptThe addition and multiplication rules for combining uncertainties are special cases of a more general idea: for any function of measured quantities, the uncertainty in the result comes from how sensitive that function is to small changes in each input — its partial derivatives.
For z = xⁿ, that sensitivity works out to a clean rule of thumb: the relative uncertainty in z is n times the relative uncertainty in x. Squaring a quantity doubles its relative uncertainty; square-rooting it halves that uncertainty — the exponent tells you directly how the uncertainty scales.
This general approach isn't limited to simple powers, though. It applies just as well to trigonometric functions, logarithms, or any messy combination of measured quantities a real experiment might produce — the formula is more work to apply, but the underlying idea is the same one already at work in the simpler addition and multiplication rules.
Key ideas
Requires: Combining uncertainties
Common misconceptions
- For z = x^n, the absolute uncertainty in z equals n times the absolute uncertainty in x.
- The general uncertainty propagation formula only applies to linear functions.
- Rounding intermediate uncertainty values does not affect the final reported uncertainty.