Two measurements can share a reference standard, calibration correction or other input. Their uncertainties may therefore be correlated. When the question concerns the difference between the results, a shared effect can behave differently from separate random effects, so combining the two total uncertainties as if everything were independent can give the wrong answer.
Identify the common input
The GUM explains that using the same instrument, measurement standard or reference datum can introduce correlation. Its uncertainty-propagation framework includes covariance when significant correlated inputs are present.JCGM 100:2008 — Guide to the expression of uncertainty in measurement (opens in a new tab)
For an original simplified model, let both measurements include exactly the same additive calibration offset c. Their difference is (x + c) − (y + c) = x − y. The common offset cancels algebraically.
This cancellation concerns one specified additive term. It does not remove separate preparation errors, sample variation or changes in the calibration between measurements.
Compare independent and common contributions
Suppose two hypothetical estimates are 10.0 mg and 9.0 mg. Each has an independent standard-uncertainty component of 0.10 mg and a common additive-offset component of 0.20 mg, entering with the same sign and coefficient.
Each individual result has total standard uncertainty √(0.10² + 0.20²), approximately 0.224 mg. If those totals were incorrectly treated as independent, the uncertainty of the difference would be approximately 0.316 mg.
| Quantity | Illustrative calculation |
|---|---|
| Difference | 10.0 − 9.0 = 1.0 mg |
| Independent contributions | √(0.10² + 0.10²) ≈ 0.141 mg |
| Shared additive contribution | Cancels because it enters both results equally |
| Standard uncertainty of difference | Approximately 0.141 mg |
The difference is more tightly determined than either absolute amount in this particular model. That is possible because uncertainty about the common offset affects the absolute values but not their separation.
Check cancellation rather than assuming it
For a difference d = x − y, the variance expression is u(d)² = u(x)² + u(y)² − 2 cov(x,y). In the example, the shared-offset covariance is 0.20² = 0.04 mg², giving 0.05 + 0.05 − 0.08 = 0.02 mg².
The square root is approximately 0.141 mg, matching the direct cancellation calculation. This is original arithmetic under the stated model, not a claim about a supplied certificate.
A shared multiplicative factor behaves differently. If results are a × x and a × y, their difference is a × (x − y). Uncertainty in a can still affect that difference. It would cancel in their ratio under this idealised model, provided the ratio is defined.
Changes over time or unequal sensitivities can also prevent complete cancellation. A common standard used on different days does not automatically justify assigning perfect correlation to every component.
Ask for the information needed for comparison
A pair of certificates may report only total expanded uncertainties. Those totals do not reveal the common and independent components. Recovering standard uncertainties from coverage factors is useful but does not reconstruct the covariance.
Ask which standards, calibration runs and correction inputs the results share and whether the laboratory has evaluated uncertainty for their difference. Preserve the result definitions and sample identities as part of that question.
Do not infer that a statistically distinguishable difference is practically important, or that an uncertain difference proves equality. The decision also needs a stated purpose and a meaningful scale for the attribute being compared.
Sources and further detail
- JCGM 100:2008 — Guide to the expression of uncertainty in measurement (opens in a new tab)
Section 5.2, including common-input correlation and covariance terms, read. The additive-offset and multiplicative-factor examples are original algebraic illustrations with explicit assumptions.
Sources checked 19 September 2026. Worked examples are illustrative unless a supplied report is explicitly identified. This article has not undergone independent scientific peer review.