EngineerBenchCombining measurement uncertainty contributions
An uncertainty budget describes how several sources of variation affect a reported measurement. Each contribution must first be expressed as a standard uncertainty and translated into the output's units using its sensitivity coefficient. Independent contributions can then be combined by root-sum-square.
Use the right distribution and divisor
A stated bound, a standard deviation, and an expanded uncertainty are not interchangeable inputs. Select the representation that matches the evidence for each component. A sensitivity coefficient describes how much the result changes per unit change in that input; it is not a confidence level.
Explain the coverage factor
Expanded uncertainty is the combined standard uncertainty multiplied by the chosen coverage factor. That factor alone does not establish an exact confidence probability. The calculation assumes independent components and does not include covariance; correlated contributions need a method that accounts for their relationship.
Converting contributions to a common basis
An input uncertainty must be multiplied by its sensitivity coefficient to express its effect in the result's units. A small input uncertainty can dominate if the result is highly sensitive to that input. Conversely, a large raw uncertainty in another unit may contribute little after scaling. Compare the output-equivalent contributions rather than ranking the unconverted input numbers.
For independent standard contributions of 0.03 mm and 0.04 mm, root-sum-square gives 0.05 mm combined standard uncertainty. An entered coverage factor of two gives 0.10 mm expanded uncertainty. The arithmetic does not by itself establish an exact 95 percent interval; that interpretation depends on the model, distributions, and available information. Avoid counting the same source twice under different names, such as a specification allowance that already includes a calibration term. Record whether a component came from repeated measurements, a certificate, resolution, or another bound. If inputs share a source of variation, the independence assumption needs review before interpreting the combined result.
Formula
u꜀ = √Σ(cᵢuᵢ)²; expanded uncertainty U = k u꜀.