Measurement Models and Uncertainty Contributions

The international standard for evaluating and expressing measurement uncertainty is JCGM 100:2008, commonly known as the GUM (Guide to the Expression of Uncertainty in Measurement). Under the GUM, evaluating uncertainty is not a subjective guessing game; it is a structured, mathematically rigorous engineering process. An uncertainty budget is the formal document that itemizes all known input components affecting a measurement, quantifies their standard uncertainties, translates them through sensitivity coefficients, and synthesizes them into a single, defensible combined expanded uncertainty.

---

The Measurement Model Equation: $Y = f(X_1, X_2, \dots, X_n)$

In metrology, a measurement result is rarely obtained from a single, isolated instrument reading. Instead, the measurand (output quantity $Y$) is determined from several other input quantities ($X_1, X_2, \dots, X_n$) through a functional relationship:

$$Y = f(X_1, X_2, \dots, X_n)$$

Input Estimates and the Output Estimate:

• Each input quantity $X_i$ has an estimated value denoted by $x_i$.
• The output estimate of the measurand $y$ is obtained by evaluating the functional relationship using the input estimates:

$$y = f(x_1, x_2, \dots, x_n)$$

Direct vs. Indirect Measurement Models:

1. Direct Measurement Model (Additive Comparison): When an instrument reading is directly compared to a reference standard, the functional relationship is typically additive:

$$Y = X_{\text{STD}} + \delta X_{\text{res}} + \delta X_{\text{rep}} + \delta X_{\text{temp}} + \delta X_{\text{drift}}$$

Each residual influence term has an estimated value, often zero when the model justifies that estimate, and associated uncertainty. Apply significant estimated nonzero corrections in the model rather than hiding them inside zero-centered terms. Unknown residual effects and uncertainties in corrections remain after compensation. 2. Indirect Measurement Model (Non-linear Physics): When the measurand is calculated from disparate physical dimensions (e.g., electrical power $P = V \cdot I$, pressure $P = F / A$, or torque $\tau = F \cdot L$), the functional relationship is multiplicative or non-linear.

Sensitivity Coefficients ($c_i$):

To determine how an uncertainty in an input quantity $X_i$ affects the output measurand $Y$, metrologists calculate the partial derivative of the model function with respect to $X_i$, evaluated at the input estimates:

$$c_i = \frac{\partial f}{\partial X_i} \Bigg|_{X_1 = x_1, \dots, X_n = x_n}$$

The sensitivity coefficient $c_i$ acts as a scale factor or conversion multiplier. It converts the units of the input uncertainty $u(x_i)$ into the units of the output measurand:

• For an additive model $Y = X_1 - X_2$: $c_1 = \frac{\partial Y}{\partial X_1} = +1$, $c_2 = \frac{\partial Y}{\partial X_2} = -1$.
• For a multiplicative model $P = V \cdot I$: $c_V = \frac{\partial P}{\partial V} = I$, and $c_I = \frac{\partial P}{\partial I} = V$.

The uncertainty contribution of the $i$-th component to the output is:

$$u_i(y) = |c_i| \cdot u(x_i)$$

---

Classification of Uncertainty: Type A vs. Type B Evaluations

A cornerstone principle of the GUM is the classification of evaluation methods into Type A and Type B.

Type A Evaluation (Statistical Analysis)

A Type A evaluation of standard uncertainty is obtained by the statistical analysis of a series of repeated independent observations under repeatable or reproducible conditions:

• Sample Mean ($\bar{q}$): The best estimate of the quantity from $n$ repeated readings:

$$\bar{q} = \frac{1}{n} \sum_{k=1}^n q_k$$

• Sample Standard Deviation ($s(q_k)$): Characterizes the repeatability dispersion of individual readings:

$$s(q_k) = \sqrt{\frac{1}{n - 1} \sum_{k=1}^n (q_k - \bar{q})^2}$$

• Standard Uncertainty of the Mean ($u(\bar{q})$): If the reported calibration result is the average of the $n$ readings, the standard uncertainty is the standard deviation of the mean (standard error):

$$u(\bar{q}) = \frac{s(q_k)}{\sqrt{n}}$$

(Note: If a single reading is used in routine service, the standard uncertainty is $s(q_k)$, not divided by $\sqrt{n}$).

• Degrees of Freedom ($\nu$): For a Type A evaluation of $n$ observations, the degrees of freedom are:

$$\nu = n - 1$$

Type B Evaluation (Non-Statistical Analysis)

A Type B evaluation of standard uncertainty is obtained by methods other than the statistical analysis of current observations. Type B evaluations draw upon scientific judgment, engineering principles, and available documentation:

• Calibration certificates and reports of higher-echelon reference standards.
• Manufacturer technical specifications, instrument manuals, and tolerance limits.
• Reference data from certified standard reference materials (SRMs) or physical handbooks.
• Historical drift data from laboratory control charts.
• Knowledge of the physical behavior of materials, thermal coefficients, and fixturing mechanics.

---

The Six Primary Sources of Calibration Uncertainty

Consider relevant contributions from the following six categories. They are a useful checklist, not an exhaustive or universally mandatory six-part budget; include effects that matter for the actual measurement model:

Reference Standard Uncertainty and Drift

• Calibration Uncertainty: The reference standard's calibration certificate provides an expanded uncertainty $U_{\text{STD}}$ at a stated coverage factor $k$ (typically $k = 2$). The standard uncertainty is $u(x_{\text{STD}}) = U_{\text{STD}} / k$.
• Reference drift: Evaluate stability evidence, elapsed time, and the applicable model. A reliable symmetric drift bound with no better information can justify a rectangular component $u_{drift}=a/\sqrt{3}$; observed drift may support a different distribution or an explicit correction. Include uncertainty in predictions and do not assume every drift process is uniform.

Unit Under Test (UUT) Resolution and Repeatability

• Digital Display Resolution ($\delta x$): A digital display can only resolve values to within its least significant digit $\delta x$. Under a nearest-rounding quantization model, the rounding error is assumed equally likely within the interval $[-\delta x/2, +\delta x/2]$. The semi-range (half-width) is $a = \delta x / 2$. Modeled as a rectangular distribution:

$$u_{\text{res}} = \frac{\delta x / 2}{\sqrt{3}} = \frac{\delta x}{2\sqrt{3}} \approx 0.289 \cdot \delta x$$

• Analog Scale Resolution: For an analog pointer and dial scale, the half-width is typically taken as half of the smallest scale graduation (or $1/5$ to $1/10$ of a division if optical interpolation is verified), divided by $\sqrt{3}$ or $\sqrt{6}$.
• UUT Repeatability: Short-term repeatability observed during the calibration run, evaluated as Type A.

Environmental Influence Quantities

Environmental parameters alter both the standard and the UUT:

• Temperature Fluctuation ($\Delta T$): Deviations from the $20.0^\circ\text{C}$ reference baseline cause differential thermal expansion in dimensional metrology.
• Barometric Pressure ($P_{\text{amb}}$): Atmospheric pressure changes affect air density ($\rho_{\text{air}}$), altering the buoyant force in precision mass calibration and deadweight pressure balances.
• Relative Humidity ($RH$): Variations influence surface moisture adsorption on mass standards, dielectric constants in electrical capacitance standards, and dimensional stability of polymers.

Measurement Method and Fixturing Artifacts

• Electrical Calibrations: Lead resistance in 2-wire vs. 4-wire connections, thermal electromotive forces (thermal EMFs / Seebeck effect) generated at bimetallic junctions, AC common-mode leakage, and electromagnetic interference (EMI).
• Dimensional Calibrations: Cosine error caused by angular misalignment between the measuring axis and the artifact axis ($\Delta L = L(1 - \cos\theta) \approx L\theta^2 / 2$), Abbe offset errors ($E = d \cdot \tan\theta$), and elastic Hertzian contact deformation under measuring probe forces.
• Pressure Calibrations: Head height fluid column offsets ($\Delta P = \rho g \Delta h$) and fluid meniscus surface tension.

Operator Technique and Interaction

• Human parallax error when reading analog scale pointers from an oblique angle.
• Clamping torque variability on mechanical micrometers lacking calibrated friction thimbles.
• Hand-heat transfer warming precision steel gauge blocks during prolonged manual manipulation.

Material Thermo-Physical Properties

• In dimensional comparisons between a standard and a UUT made of different materials, the differential thermal expansion equation is:

$$\Delta L = L \cdot [(\alpha_{\text{UUT}} - \alpha_{\text{STD}}) \cdot \Delta T + \alpha_{\text{UUT}} \cdot \delta T_{\text{diff}}]$$

where $\alpha$ is the coefficient of thermal expansion (CTE), $\Delta T$ is the deviation of the standard temperature from $20^\circ\text{C}$, and $\delta T_{\text{diff}}$ is the temperature difference between the two artifacts.

• Even if temperatures are measured, uncertainty in the literature values of CTE ($\pm 10\%$ to $\pm 15\%$ on $\alpha$) introduces a non-negligible Type B uncertainty component.