Uncertainty Propagation Calculator

Uncertainty propagation answers how the uncertainties of input quantities combine into the uncertainty of a result computed from them. The general law of propagation takes the partial derivative of the model with respect to each input, multiplies it by that input standard uncertainty, and sums the squares. For sums and differences the derivatives are unity, so absolute uncertainties add in quadrature: 25.0 plus or minus 0.3 combined with 10.0 plus or minus 0.2 gives an uncertainty of 0.36, not 0.5. For products and quotients the relative uncertainties add in quadrature instead, which is why a concentration formed from a mass known to 1.0 percent and a volume known to 0.2 percent carries about 1.02 percent. The larger term dominates, so effort spent improving the smaller one is largely wasted. The reported figure is normally an expanded uncertainty, the combined standard uncertainty multiplied by a coverage factor k. Analytical laboratories use k = 2 for an interval of roughly 95 percent coverage and occasionally k = 3 for about 99.7 percent, and accreditation to ISO/IEC 17025 requires the factor to be stated alongside the value. Two assumptions limit the quadrature rules. Inputs must be independent: if the same calibrated pipette or the same balance calibration feeds two terms, a covariance term is needed, and ignoring it can understate or overstate the result substantially. The inputs must also be approximately normal and the model close to linear over the uncertainty range. Where either condition fails, as with strongly non-linear calibration functions, the Monte Carlo propagation described in GUM Supplement 1 is the appropriate alternative.

The mathematics predates analytical chemistry by nearly two centuries. Carl Friedrich Gauss, working on orbit determination and publishing Theoria Motus in 1809, developed the normal error distribution and the least-squares treatment from which the propagation formula follows directly, building on the least-squares method Adrien-Marie Legendre had published in 1805. Nineteenth-century geodesy and astronomy applied these results routinely, and the quadrature rules for sums and products were standard fare in physics laboratory teaching long before they were formalised for measurement science. Harmonisation for metrology came much later. In 1980 the International Committee for Weights and Measures issued Recommendation INC-1, which set out the now familiar distinction between Type A uncertainties evaluated statistically from repeated observations and Type B uncertainties evaluated from other information such as calibration certificates or manufacturer tolerances. That recommendation grew into the Guide to the Expression of Uncertainty in Measurement, published by ISO in 1993 on behalf of seven international organisations and reissued as JCGM 100:2008. Analytical chemistry received its own interpretation through the Eurachem and CITAC guide Quantifying Uncertainty in Analytical Measurement, first issued in 1995 and now in its third edition, which introduced cause-and-effect diagrams for identifying uncertainty sources within a laboratory procedure.