Sensitivity coefficient ($c_i=\partial f/\partial x_i$)

A sensitivity coefficient describes how an estimated output of a measurement model changes when one input changes slightly. For $y=f(x_1,\dots ,x_n)$, the first-order coefficients, evaluated at the input estimates, areA sensitivity coefficient can have a sign and a unit of measurement: for an output in ohms and an input in volts, its unit is ohms per volt. The GUM also describes estimating sensitivity coefficients experimentally by varying inputs.^[ref]

In a measurement uncertainty budget, propagation of uncertainty uses these coefficients to transform input variances and covariances:Here $u_i$ denotes an input standard uncertainty. The derivative's sign disappears from its independent variance term but still matters in a covariance term.^[ref]

For an original electrical resistance example, $R=V/I$, $V=10\mathrm{V}$ and $I=2\mathrm{A}$ give $R=5\Omega$ andIndependent input standard uncertainties of0.1volt and0.02ampere each contribute0.05ohm on the output scale, giving $u_R=\sqrt{0.05^2+0.05^2}\Omega \approx 0.0707\Omega$. A larger sensitivity coefficient alone does not identify the dominant uncertainty: the input standard uncertainty and any correlations also matter.

The derivative is a local approximation. If the input spread spans a strongly nonlinear part of the measurement model, first-order propagation of uncertainty can be inadequate even when every derivative has been computed correctly.