Estimate combined and expanded uncertainty

Define the estimate before calculating

Measurement uncertainty describes the dispersion of values reasonably attributed to a measured quantity under stated conditions. It is not a correction for an identified error, a patient reference interval or a guarantee that an individual result is correct. This calculator provides a simplified relative estimate using an imprecision component and an uncertainty component associated with bias. It cannot determine whether those two components adequately represent your examination procedure.

Both inputs must be standard uncertainties expressed as percentages. The imprecision input is a CV selected to represent the variability relevant to routine reporting. The bias input is u(bias), the standard uncertainty associated with the bias estimate or its treatment in your procedure. It is not automatically the numerical bias itself. Entering a bias magnitude in that field without justification can misrepresent the uncertainty budget.

Choose defensible components

A long-term internal QC series may provide an imprecision estimate that includes variation across days, operators, calibration events and reagent changes. Review whether the control material behaves sufficiently like patient samples and whether the estimate applies at the concentration being reported. One percentage may not describe the full measuring interval, particularly near the lower end where absolute variability can dominate.

The uncertainty associated with bias may depend on reference values, comparison data, the number of observations and other elements specified in your local approach. Keep the supporting calculations and traceability information. Avoid counting the same source of variation twice if it is already captured in the long-term imprecision estimate. Conversely, do not omit a significant contribution merely because this calculator has only two input fields.

Combine and expand

The calculation is u = √(u_imprecision² + u_bias²), followed by U = k × u. Root-sum-of-squares combination assumes that the two input components can be treated as uncorrelated. If important correlations exist, covariance terms or another justified method may be needed. Inputs must be nonnegative and the coverage factor k must be positive. Zero inputs are mathematically permitted, but claiming zero uncertainty requires much stronger justification than an empty data set.

A coverage factor of 2 is commonly used to obtain an interval with approximately 95 % coverage when the distributional assumptions and effective degrees of freedom support that interpretation. It is not a universal guarantee of exactly 95 % coverage. The laboratory must select and justify the factor for its intended application.

Report with appropriate context

The outputs remain percentages. Converting expanded relative uncertainty to an absolute value requires multiplying the reported concentration by U / 100, with suitable attention to sign conventions and rounding. Retain enough precision during calculation and round only for reporting. State the coverage factor, the relevant concentration range and any important limitations alongside the estimate. Review the budget after material changes to the procedure or its performance. The laboratory’s own approved procedure governs component selection, calculation, review and use of the result.

CV = 2.5 %, u(bias) = 1.2 %, k = 2.

u = √(2.5² + 1.2²) = √7.69 = 2.77 %. U = 2 × √7.69 = 5.55 %. The expanded result uses the unrounded combined uncertainty, not 2 × the rounded display value.