Combined Uncertainty ($u_c$) is the total standard uncertainty of a measurement result obtained by combining all individual Type A and Type B uncertainty components — calculated using the RSS (root sum of squares) method following the GUM (Guide to the Expression of Uncertainty in Measurement).

Combining Uncertainties

• RSS: $u_c = sqrt{u_1^2 + u_2^2 + u_3^2 + cdots}$ — for independent, uncorrelated uncertainty sources.
• Sensitivity Coefficients: $u_c = sqrt{sum_i (c_i u_i)^2}$ where $c_i = partial f / partial x_i$ — for indirect measurements.
• Correlated Sources: Add covariance terms: $2 c_i c_j u_i u_j r_{ij}$ where $r_{ij}$ is the correlation coefficient.
• Dominant Source: Often one uncertainty component dominates — reducing the dominant source has the most impact.

Why It Matters

• GUM Standard: The internationally accepted methodology for uncertainty reporting — ISO/BIPM standard.
• Traceability: Combined uncertainty is essential for establishing metrological traceability to SI standards.
• Decision: Combined uncertainty determines the reliability of measurement-based decisions — pass/fail, process control.

Combined Uncertainty is the total measurement doubt — the RSS combination of all uncertainty contributors into a single number representing overall measurement reliability.