Uncertainty propagation, addition, multiplication, power
Independently measured quantities combined, with the absolute and relative uncertainty propagated on the result and the applied formula.
Result
Awaiting the quantities required for the calculation.
Calculation, assumptions and sources
OverviewThe values and absolute uncertainties of independently measured quantities give, for a sum, difference, product, quotient or power, the result, its absolute uncertainty, its relative uncertainty as a percentage and the propagation formula applied.
The values and absolute uncertainties of independently measured quantities give, for a sum, difference, product, quotient or power, the result, its absolute uncertainty, its relative uncertainty as a percentage and the propagation formula applied.
Method & formulasAddition / subtraction: Δf = √(Δx₁² + Δx₂² + …)
For independent quantities (uncorrelated errors), contributions combine by quadratic sum; the arithmetic sum of uncertainties would overestimate the uncertainty of the result.
- x, Δx
- measured value and its absolute uncertainty, expressed in the same unit.
- Absolute uncertainty
- the uncertainty of the result, in the unit of the quantity, denoted Δf.
- Relative uncertainty
- the ratio Δf / |f|, as a percentage. Dimensionless, it compares the precision of measurements of different kinds.
- Independent quantities
- measurements whose errors are uncorrelated, the assumption behind the quadratic sum.
Validity domainIf two measurements share a source of error (the same poorly calibrated instrument), their uncertainties are correlated and the quadratic sum underestimates the actual uncertainty.
If two measurements share a source of error (the same poorly calibrated instrument), their uncertainties are correlated and the quadratic sum underestimates the actual uncertainty. The relative uncertainty of a zero quantity is undefined; the case is flagged.
ExampleFor L = 12.0 cm ± 0.1 cm and l = 5.0 cm ± 0.2 cm, the area A = L × l equals 60.0 cm².
For L = 12.0 cm ± 0.1 cm and l = 5.0 cm ± 0.2 cm, the area A = L × l equals 60.0 cm². The relative uncertainty on A equals √((0.1/12.0)² + (0.2/5.0)²) ≈ 4.1%, hence an absolute uncertainty of approximately 2.5 cm²: A ≈ 60.0 ± 2.5 cm².