Correlated inputs in uncertainty evaluation

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.

The mapping case, worked

Twelve loggers are calibrated against the same reference thermometer, which carries U = 0.04 °C. Each logger has its own repeatability and resolution contributing u = 0.03 °C independently. Two questions get different answers, and the difference is entirely correlation.

Combining naïvely would get the second row wrong by including the reference twice, once for each logger, when its error is common to both and subtracts out. This is the useful direction of correlation: a shared reference makes differences between sensors better known than their absolute values, which is precisely what a mapping study is measuring.

When it works against you

The dangerous direction is a shared source appearing in several terms of a sum. If the same reference certificate is used to correct both the bath temperature and the unit under test in a model where the two add rather than subtract, the errors reinforce and the naïve root-sum-of-squares understates the uncertainty. Positive correlation between terms that add always understates; ignoring it produces a confident, too-small number.

Practical handling

• Identify shared sources first: one reference used twice, one readout, one ambient condition affecting several components, one certificate applied to several instruments.
• Where a shared source cancels — a difference between two things measured against it — drop it from the combination and say so in the budget.
• Where it reinforces, either add the covariance term from the GUM expression, or take the simple conservative route and add the correlated contributions linearly before combining the rest in quadrature.
• Restructure the model where you can. Writing the measurand as a difference rather than as two separate absolute values often makes the correlation disappear into the algebra instead of needing a covariance term.
• Monte Carlo handles correlation naturally: draw one value for the shared input per trial and use it everywhere it appears. That is often easier than the analytical covariance treatment.

The rule of thumb: if two inputs share a physical cause, they are correlated, and the question is only whether it matters. If they share a certificate, they certainly do.