How this calculator works
First-order propagation combines partial derivatives with standard uncertainties. Correlation contributes a cross-term, so perfectly correlated errors can cancel in a difference. Products use derivatives directly and can handle zero measured factors. Powers treat the exponent as exact. Custom sensitivities combine any number of independent contributions without evaluating an expression. Input uncertainties must use the same units as their measurements; custom products must share the output unit.
Formula
u(f)² ≈ (∂f/∂x)²ux² + (∂f/∂y)²uy² + 2rho(∂f/∂x)(∂f/∂y)uxuy. Independent custom inputs: u(f) = sqrt(Σ(ci ui)²).
Worked example
Adding 10 ± 3 and 20 ± 4 with uncorrelated standard uncertainties gives 30 ± 5. Multiplying 2 ± 0.1 by 3 ± 0.2 gives 6 ± sqrt(0.3² + 0.4²) = 6 ± 0.5.
Advertisement
Frequently asked questions
Is this a worst-case error bound?
No. It combines standard uncertainties through a local linear approximation. Worst-case bounds and expanded uncertainties require different treatment.
When can the approximation fail?
Strong curvature, large uncertainties or values near a singularity can invalidate first-order propagation. A denominator near zero needs particular care.
Can the custom mode parse a formula?
No. Supply partial derivatives evaluated at the measured inputs. This mode assumes all entered uncertainty contributions are independent.