Two-Input Measurement Uncertainty

Method & assumptions

1. Propagate two input standard uncertainties through addition, subtraction, multiplication or division. Standard uncertainty is expressed as a standard deviation; do not enter a tolerance bound or expanded uncertainty unchanged.
2. Use the same unit for A and B in addition/subtraction. For multiplication the output unit is A-unit × B-unit; for division it is A-unit / B-unit. Enter each absolute standard uncertainty in the same unit as its corresponding estimate. No unit conversion is performed.
3. Let cA = ∂f/∂A and cB = ∂f/∂B. Combined variance uc² = (cA uA)² + (cB uB)² + 2r cA cB uA uB, where r is the correlation coefficient. A value of zero specifies uncorrelated inputs; it is not inferred from the data.
4. Sensitivities are (1, 1) for a sum, (1, −1) for a difference, (B, A) for a product and (1/B, −A/B²) for a quotient. This is first-order propagation; nonlinear effects can matter for products and ratios, especially with a denominator near zero. A zero first-order result does not prove zero uncertainty for a nonlinear model. No Monte Carlo propagation, distribution or degrees-of-freedom estimate is provided.
5. Expanded uncertainty U = k uc. The displayed estimate ± U does not have an automatically assigned confidence level. Choosing k = 2 alone does not establish 95% coverage. Relative standard uncertainty is 100 uc/|estimate| and is undefined at a zero estimate.

Sources

• NIST TN 1297 Appendix A: Law of Propagation of Uncertainty ↗
• NIST TN 1297 Section 6: Expanded Uncertainty ↗