The essential answer
Two estimates are correlated when some uncertain influence moves them together. For a difference d = a−b, the variance is u(a)² + u(b)² − 2cov(a,b). For a sum, the covariance term has the opposite sign. A shared offset can cancel in a difference, while a shared scale factor generally remains. Treating all measurements as independent can therefore overstate or understate uncertainty. Identify the physical source of covariance before choosing a correlation coefficient or claiming that a ratio eliminates calibration uncertainty.
How to reason about it
Separate common and independent components explicitly. Suppose two readings share the same uncertain zero but have independent detector noise: write each as a true value plus common zero plus individual noise. Subtraction removes the zero algebraically only if the same zero applies at both times. Drift between readings breaks that assumption. For multiplicative calibration, a ratio may cancel a shared gain under a linear model, whereas a difference still inherits gain uncertainty proportional to the difference itself. Correlation should follow the measurement history and model, not be selected because it produces a smaller interval. Paired acquisition can preserve useful common-mode cancellation; interleaving introduces a different covariance structure from isolated runs. When estimating covariance from data, distinguish real shared variation from chance sample correlation and retain uncertainty in poorly constrained estimates. Numerical propagation can preserve the common input directly.
ILLUSTRATIVE · NOT A REPORTED RESULT
An example to work through
Assume two illustrative force estimates each have standard uncertainty 0.50 nN and correlation coefficient 0.8 because of a shared background estimate. The uncertainty of their difference is sqrt(0.25+0.25−2×0.8×0.50×0.50) = sqrt(0.10) = 0.316 nN. If incorrectly treated as independent, it becomes 0.707 nN. Their sum instead has uncertainty sqrt(0.90) = 0.949 nN. These contrasting values use identical marginal uncertainties. They show why publishing only individual error bars can be insufficient when readers need to compare or combine measurements. The assumed correlation must come from a defensible shared-input model.
Where interpretation fails
A common instrument does not imply perfect correlation of all uncertainty. Its gain may be shared, its readout noise independent, and its zero partly correlated over time. One global correlation coefficient can obscure these different behaviors, especially when operating conditions or calibration ranges differ across the two observations.
The next decision
Retain shared input variables in the analysis whenever possible, and provide a covariance matrix for reported quantities that others will combine. Before relying on cancellation, test whether the common contribution is stable across the actual switching interval. If that cannot be demonstrated, include the unresolved differential component explicitly.
Sources & context
These sources support the concepts noted below. Our synthetic examples and research framing are educational interpretation; they are not findings from these organizations or validation of our ventures.
- JCGM 100: Guide to uncertainty ↗
Further reading on uncertainty propagation; PDF text inspected, not a full document review.
Source check: Relevant excerpt or resource metadata checked; see support note. Original illustrative calculation, not source data.
Educational research context, not a validated device specification or clinical protocol. Read our editorial and correction policy.