Correlated inputs in uncertainty evaluation
Root-sum-of-squares assumes independence. When two inputs move together — the same reference used twice, for example — the combination must include a correlation term.
The familiar u_c² = Σ(c_i u_i)² holds only for uncorrelated inputs. When inputs share a source of variation, cross terms appear. Ignoring them can understate the uncertainty (positive correlation) or overstate it (negative correlation, as in a difference measurement with a common reference).
Where it shows up in temperature work
- Mapping studies: all loggers calibrated against the same reference share its error; the uncertainty of a difference between two loggers is smaller than naïve combination suggests, while the uncertainty of their common offset is not.
- Uncertainty of a mean of many sensors: the reference's contribution does not shrink with √n.
- Two measurements made with the same instrument minutes apart: drift and environment are largely common.
The GUM gives the full expression with covariance terms; in practice, identify shared sources and treat them separately from the independent ones.
Frequently asked questions
- Do I always need to compute covariances?
- No. Restructure the model so shared components appear once, then combine the rest as independent. That handles most practical cases.
References
- [1]JCGM 100:2008 — Guide to the expression of uncertainty in measurement (GUM)
General technical guidance written against the cited sources. It is not regulatory or legal advice and does not replace the applicable standard, guideline or a qualified reviewer's judgement.
Have a question on this topic?
Ask ValiTrac AI and see the evidence and calculation behind the answer.
Ask ValiTrac AIFree calculator: Uncertainty budget
Combine Type A and Type B components per the GUM: divisors by distribution, sensitivity coefficients, u_c, effective degrees of freedom and U at k = 2.
Open the calculatorMetrology on ValiTrac
Measurement science, traceability, uncertainty and technical evidence.
Ask a metrology questionRelated articles
Building an uncertainty budget step by step
Write the measurement model, list every input, evaluate each as a standard uncertainty, apply sensitivity coefficients, combine in quadrature, expand with k. Eight steps, one table.
Sensitivity coefficients: how input uncertainties reach the result
A sensitivity coefficient converts an input's uncertainty into its effect on the output. For a direct temperature comparison it is 1; for derived quantities it comes from the model.