FigureCheck
How to propagate uncertainty through calculations
A practical guide to uncertainty propagation for sums, products, powers, and functions, with worked examples and assumptions.
For independent uncertainties, combine absolute uncertainties in quadrature for addition or subtraction, and combine relative uncertainties in quadrature for multiplication or division. Derivative-based propagation provides the general rule.
A defensible general rule
If a result f depends on measured inputs, approximate its variance by summing each input variance multiplied by the square of the corresponding partial derivative. This first-order method assumes uncertainties are reasonably small. Correlated inputs require covariance terms and should not be treated as independent.
Common classroom rules
For z = x plus or minus y, combine independent absolute uncertainties as the square root of ux squared plus uy squared. For z = xy or x/y, apply the same quadrature rule to relative uncertainties. For z = x to the power n, the relative uncertainty is approximately the magnitude of n times the relative uncertainty in x.
Worked example: area from two lengths
Let L = 5.00 plus or minus 0.05 cm and W = 2.00 plus or minus 0.02 cm. A = 10.00 cm squared. The relative uncertainty is sqrt((0.05/5.00)^2 + (0.02/2.00)^2) = 0.0141, so uA is about 0.14 cm squared. Report A = (10.00 plus or minus 0.14) cm squared.
Common mistakes to avoid
- Adding percentage uncertainties for independent measurements without stating that a worst-case rule is intended
- Ignoring correlations between derived quantities
- Applying a linear approximation across very large uncertainties
Open an editable example and apply the idea to a real graph.
Sources and further reading
These links support the scientific principles summarized above. Always follow the reporting rules required by your course, laboratory, journal, or field.