 Gauge-block reference uncertainty floor

A gauge-block reference uncertainty floor is the component of a comparison's uncertainty that remains when repeated observations still rely on the same uncertain reference value. It is conditional on the stated reference and measurement model, rather than a universal lower bound on every method of measuring the block.

For a corrected comparison, write the unknown reference-temperature length as $L_X=L_R+\overline{d}$, where $L_R$ is the assigned reference length and $\overline{d}$ the mean of $n$ comparison differences. Suppose the reference has standard uncertainty $u_R$, the differences have independent repeatability scatter $s$, and these contributions are independent. ThenOnly the repeated-comparison term decreases with $n$. The reference-value contribution is shared by all those observations; it is not newly sampled every time. NIST's comparison uncertainty combines information from the interferometric master history with comparison-process variability.^[ref]

For an original example, take $u_R=20\mathrm{nm}$ and $s=40\mathrm{nm}$. Four comparisons give $u(L_X)=\sqrt{400+1600/4}\approx 28.3\mathrm{nm}$. One hundred give about $20.4\mathrm{nm}$, and the limit in this model is 20 nm. Dividing the entire combined uncertainty by $\sqrt{n}$ would wrongly treat the reference assignment as independent repeatability noise.

The same shared reference can cancel from another measurand. If $L_X=L_R+d_X$ and $L_Y=L_R+d_Y$ use the same additive reference contribution, thenShared gauge-block reference covariance is what makes that cancellation consistent in uncertainty propagation. An absolute length and a length difference have different uncertainty budgets. Other shared terms, including temperature and comparator corrections, must be modeled on their own merits.^[ref]

An independently improved reference assignment or additional independent calibration information can lower the floor; more repetitions with the unchanged reference cannot. State whether a certificate quotes standard or expanded uncertainty before combining values. Improved repeatability is useful, but does not establish improved absolute accuracy by itself.