Sensitivity coefficients: how input uncertainties reach the result

The GUM's law of propagation of uncertainty multiplies each input's standard uncertainty by a sensitivity coefficient — the partial derivative of the measurement model with respect to that input — before combining. The coefficient answers: if this input moves by one unit, how much does the result move?

Examples

• Correction = reference − indicated: both coefficients are ±1 in temperature units.
• A resistance thermometer: converting an uncertainty in ohms to kelvin needs the sensor's sensitivity, about 0.39 Ω/K for a Pt100 near 0 °C, so c ≈ 2.6 K/Ω.
• Relative humidity from a temperature error: the coefficient can be several %rh per kelvin near saturation.

Writing the measurement model down explicitly is the step most budgets skip — and the step that reveals the coefficients.

Worked coefficients for temperature work

A sensitivity coefficient is the partial derivative of the model with respect to one input — how much the result moves per unit of that input. Most of the ones you need in temperature work are either 1 or a number you can look up.

The two thermocouple rows are worth pausing on. They are the reason a microvolt of readout noise is a minor nuisance on a type K and a serious problem on a type S: the same electrical uncertainty reaches the temperature seven times larger, because the Seebeck coefficient of a noble-metal couple is that much smaller.

The step most budgets skip

You cannot obtain a coefficient without a model, and writing the measurement model down explicitly is the step that gets left out. A budget that lists components with no equation behind them has assumed every coefficient is 1, which is true for a direct temperature comparison and false for everything else. Write the model first — even in a single line — and the coefficients fall out of it.

When you cannot differentiate the model

The analytical route assumes you can write the measurement as a function and differentiate it. Often you cannot: the model is a lookup table, a manufacturer’s polynomial, an iterative solution, or a spreadsheet nobody wants to unpick. The coefficient is still obtainable, numerically, and the method is the one most practitioners end up using without quite naming it.

• Compute the result at the nominal input values and record it.
• Perturb one input by a small amount — a convenient choice is its own standard uncertainty — leaving the others alone.
• Recompute. The change in the result divided by the change in the input is the sensitivity coefficient for that input, in the units the budget needs.
• Repeat for each input. Perturb in both directions where the model might be asymmetric, and use the larger magnitude.

The step size is a judgement rather than a formula. Too large and the finite difference stops approximating the derivative wherever the model curves; too small and it is swamped by the rounding of whatever computed it. A perturbation of one standard uncertainty is usually a sound compromise, and it has the useful property that the resulting change in the output IS that component’s contribution, so you can read it straight into the uncertainty budget without multiplying anything.

Coefficients that are not constant

• Almost all of the values above vary with temperature. A Pt100's sensitivity falls from about 0.391 Ω/K at −200 °C to 0.296 Ω/K at 800 °C, so the coefficient must be evaluated near the point of interest, not once for the whole range.
• Where a model is strongly non-linear over the range of an input's uncertainty, the linear approximation behind the law of propagation starts to fail. That is one of the standard triggers for Monte Carlo evaluation under JCGM 101.
• A coefficient of zero is a real and useful result: it says the input does not reach the output through this model, and the component can be dropped with a note rather than carried forward.
• Signs matter only where inputs are correlated. For independent inputs the terms are squared and the sign vanishes — which is exactly why correlated inputs need separate treatment.