= Finite-perturbation reference sensitivity
\b[Finite-perturbation reference sensitivity] measures the change in a comparison when one shared reference value is changed by a finite amount everywhere it occurs. It checks both local <sensitivity coefficients> and their usefulness over the actual reference-uncertainty range. The <reference-sensitivity cancellation check> is incomplete if it tests only a derivative at one point.
Let $F(R)$ be the final comparison, with all other independent inputs held fixed. A <central finite difference> and a second difference are
$$
C_h=\frac{F(R_0+h)-F(R_0-h)}{2h},\qquad
Q_h=\frac{F(R_0+h)-2F(R_0)+F(R_0-h)}{h^2}.
$$
For sufficiently smooth $F$, these approach $F'(R_0)$ and $F''(R_0)$, with errors of order $h^2$ when the required derivatives exist. Their units differ: comparison per reference unit and comparison per squared reference unit. The GUM defines <sensitivity coefficients> as partial derivatives and also discusses determining them by varying an input experimentally.<Guide to the expression of uncertainty in measurement by JCGM>[§§5.1.3–5.1.4]{ref}
An original model writes $x=R-R_0$ and
$$
F(R)=D_0+\epsilon x+\beta x^2+\kappa x^3.
$$
It gives $C_h=\epsilon+\kappa h^2$ and $Q_h=2\beta$. Comparing $C_h$ with $C_{h/2}$ can expose cubic contamination of the local slope. But if $\epsilon=\kappa=0$, every central slope is exactly zero although $F(R_0+h)-F(R_0)=\beta h^2$ is nonzero. A symmetric slope check alone misses an even-in-perturbation response.
For $\beta=1\,\mathrm{mm^{-1}}$ and $h=0.001\,\mathrm{mm}$, the one-sided change is 1 nm, while $C_h=0$ and $Q_h=2\,\mathrm{mm^{-1}}$. If the shared reference error instead has a zero-mean <normal distribution> with standard deviation $s=h$, the quadratic contribution has mean 1 nm and standard deviation approximately 1.414 nm. These are original model calculations. They show why exact cancellation, zero local slope and negligible finite-range influence are distinct claims.
Use <step-size stability of a reference perturbation> to select a sensible test range. Extremely small steps can magnify numerical subtraction errors; large steps probe a different region of a nonlinear response. Test several scales near the physical standard uncertainty and inspect one-sided changes as well as central slopes. If the model is stochastic, separate simulation noise from the response to the changed reference.
A computational perturbation changes the assigned shared reference in every dependent calculation while keeping its identity shared. Independently perturbing two uses of the same reference changes the covariance model. A physical experiment must additionally hold other influences stable, because changing a standard or its temperature may change contact or instrument behaviour. NIST separates master-block uncertainty from the comparison process in its uncertainty budget.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§4.4.3]{ref} Retain those physical distinctions when interpreting a numerical sensitivity test.
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