Uncertainty budgets and negative results

For a transparent Gaussian example, let every reading equal one constant true force plus a batch-wide calibration error plus independent reading noise. Both errors have zero mean in the stipulated repeated-measurement model; their known standard deviations are given. The calibration error is shared within a batch, while new independent calibration and noise errors are drawn across hypothetical repetitions of the full procedure. Averaging divides only the independent-noise variance by the number of readings.

This is stronger than merely calling an unknown offset systematic. A completely unmodeled fixed bias has no automatically justified Gaussian distribution or coverage probability. Calibration information must support the adopted uncertainty model. NIST’s Type A and Type B labels classify evaluation methods, not a universal split into random and systematic physics. Components obtained by different methods can still be correlated, and a known correction should be applied rather than hidden inside an enlarged error bar.

For a linear combination, uncertainty follows from the covariance matrix, not from indiscriminately adding absolute errors or dividing every component by a square root. Positive correlation increases uncertainty in a sum but decreases uncertainty in a difference with opposite coefficients. Two detectors with shared calibration can therefore agree unusually well while both remain wrong relative to an external standard. A difference test and an absolute calibration test answer different physical questions.

For a nonlinear derived quantity, first-order propagation uses sensitivities evaluated at the estimated inputs. It requires that linearization represent the input uncertainty adequately; near singularities or hard boundaries, numerical propagation or a fuller likelihood may be necessary. Uncertainty in a fitted model is also distinct from an omitted physical effect. A narrow statistical interval cannot absorb an unexamined change in temperature, background coupling or instrument response simply by being labeled a precision result.

If the estimated force is small compared with its standard uncertainty, the data may be consistent with zero without establishing that zero is exactly true. An upper confidence construction states how an analysis procedure covers the parameter over hypothetical repetitions under its model. It is not, by itself, a posterior probability that the force lies below the observed limit. A Bayesian probability statement would additionally require a prior and a specified likelihood.

The chosen alternative matters as well. A bound on a constant positive force does not constrain an oscillating force with unknown phase in the same way, and a search over many frequencies needs its search procedure included in its significance calculation. Physical parameter boundaries can require another interval construction rather than clipping an unconstrained Gaussian interval. Report the estimator, uncertainty components, correlation assumptions and tested signal form so a negative result can be compared or combined without silently changing its meaning.