Coverage Factors and Expanded-Uncertainty Reporting

Determining the Coverage Factor $k$ and Expanded Uncertainty $U$

Combined standard uncertainty $u_c(y)$ represents a dispersion of one standard deviation (approximately $68\%$ confidence for a normal distribution). In commerce and regulatory compliance, customers require a higher level of confidence—often specified as approximately $95\%$, with other coverage levels possible.

The expanded uncertainty ($U$) is obtained by multiplying $u_c(y)$ by a numerical coverage factor ($k$):

$$U = k \cdot u_c(y)$$

Coverage selection depends on the output distribution, reliability of component uncertainties, model approximation, and requested coverage. With a nearly normal output and adequate effective degrees of freedom, $k\approx2$ often gives approximately 95% coverage. Large degrees of freedom alone do not make a single dominant rectangular or arcsine contribution normal.

For a normally distributed, small-sample dominated budget meeting the Welch–Satterthwaite assumptions, use the appropriate Student t factor. There is no universal ISO rule that makes 30 a sharp cutoff. More complex or nonlinear models may require another propagation and coverage method.

Student's $t$ Reference Table for $95\%$ Confidence

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Compliant Uncertainty Reporting under ISO/IEC 17025

A clear GUM-based expanded-uncertainty statement includes the following information. Applicable accreditation reporting policies can add requirements; ISO/IEC 17025 permits uncertainty in absolute or relative form:

1. The measured numerical value of the measurand $y$
2. The expanded uncertainty $U$
3. Units for the result and uncertainty, or a clearly identified relative uncertainty with its basis
4. The coverage factor $k$ used
5. The approximate coverage probability (level of confidence, typically stated as "approximately 95%")

Rounding Conventions for Calibration Certificates

1. Rounding the Expanded Uncertainty ($U$): GUM 7.2.6 normally considers at most two significant digits adequate for uncertainty, while allowing more digits when needed for subsequent calculations. For two significant digits, $0.01634\text{ mm}$ becomes $0.016\text{ mm}$, and $4.782\ \Omega$ becomes $4.8\ \Omega$. State and apply any required conservative rounding policy; rounding upward is not a universal mandatory rule.
2. Rounding the Reported Calibrated Value ($y$): For an absolute uncertainty, round the reported result consistently with the uncertainty’s final decimal place. Retain the unrounded values internally for calculations and decisions. Relative-uncertainty reports require their own clearly stated precision and units.

Compliant Reporting Template

Compliant Statement: "The measured resistance is $100.0248\ \Omega \pm 0.0038\ \Omega$. The reported expanded uncertainty is based on a combined standard uncertainty multiplied by a coverage factor of $k = 2.23$, which for an effective degrees of freedom $\nu_{\text{eff}} = 10$ defines a coverage probability of approximately $95\%$."

Step-by-Step Worked Calibration Uncertainty Budget

A calibration laboratory calibrates a precision $100.0000\ \Omega$ standard resistor against a master reference standard using an automated direct-reading ratio set. The functional equation is $R_{\text{uut}} = R_{\text{std}} + \Delta R$.

Identifying the Uncertainty Components

1. Type A Repeatability ($u_1$): Technician performs $n = 5$ repeated observations: $s = 0.00300\ \Omega$.

$$u_1 = u(\bar{x}) = \frac{s}{\sqrt{n}} = \frac{0.00300\ \Omega}{\sqrt{5}} = 0.001342\ \Omega, \quad \nu_1 = 5 - 1 = 4$$

1. Reference Standard Certificate ($u_2$): The hypothetical certificate states $U=0.00200\ \Omega$ with $k=2.00$. This example treats the reliability of the Type B uncertainty estimates as sufficient to use infinite degrees of freedom; the provider’s name alone would not justify that assumption.

$$u_2 = \frac{0.00200\ \Omega}{2.00} = 0.001000\ \Omega, \quad \nu_2 = \infty$$

1. Digital Multimeter Resolution ($u_3$): Digital readout displays $0.001\ \Omega$ ($1\text{ m}\Omega$). Rectangular semi-range $a = 0.0005\ \Omega$:

$$u_3 = \frac{0.000500\ \Omega}{\sqrt{3}} = 0.000289\ \Omega, \quad \nu_3 = \infty$$

1. Oil Bath Temperature Variation ($u_4$): Bath temperature stable within $\pm 0.10^\circ\text{C}$ (rectangular distribution). Resistor temperature coefficient $\alpha = 10\text{ ppm}/^\circ\text{C} = 10 \times 10^{-6}/^\circ\text{C}$. Maximum resistance shift: $a = 100\ \Omega \times (10 \times 10^{-6}/^\circ\text{C}) \times 0.10^\circ\text{C} = 0.000100\ \Omega$.

$$u_4 = \frac{0.000100\ \Omega}{\sqrt{3}} = 0.000058\ \Omega, \quad \nu_4 = \infty$$

Master Uncertainty Budget Table

Combine unrounded contributions

The contributions in milliohms are $3/\sqrt{5}$, $1$, $0.5/\sqrt{3}$, and $0.1/\sqrt{3}$. Their squared sum is $1.8+1+0.0833333+0.0033333=2.8866667\ (\mathrm{m}\Omega)^2$. Thus $u_c=1.6990193\mathrm{m}\Omega=0.0016990193\ \Omega$.

Effective degrees of freedom

With only the repeatability contribution assigned finite degrees of freedom, $\nu_{eff}=(2.8866667)^2/(1.8^2/4)=10.28746$. Using the integer t table conservatively, use 10 degrees of freedom. This calculation assumes the conditions for the independent-component Welch–Satterthwaite approximation; do not use it unchanged for correlated inputs.

Determine Coverage Factor ($k$) and Expanded Uncertainty ($U$)

For the conservative integer-table choice of 10 effective degrees of freedom, the approximate two-sided 95% Student t factor is 2.23. This choice relies on the stated model assumptions; k = 2 is not automatically a 95% factor for every small-sample budget.

$$U = k \cdot u_c(y) = 2.23 \times 0.0016990\ \Omega = 0.003789\ \Omega$$

Rounding $U$ to two significant figures yields $U = 0.0038\ \Omega$ ($3.8\text{ m}\Omega$).

If the measured mean was $100.00423\ \Omega$, it is rounded to match $U$ at four decimal places:

$$R_{\text{uut}} = 100.0042\ \Omega \pm 0.0038\ \Omega \quad (k = 2.23, \nu_{\text{eff}} = 10, \approx 95\% \text{ confidence})$$

Interpreting the reported interval

A reported expanded uncertainty is not the instrument’s allowable error, the customer’s tolerance, or a promise that every future indication will lie in the reported interval. It describes the stated measurement result under the conditions and coverage model used for that evaluation. Transfer to a different range, environment, time, or measurement use may require additional contributions.

Keep the result and conformity decision separate. A resistor can have a well-characterized value and uncertainty yet fail the customer’s tolerance under the agreed rule. Conversely, a result may lie numerically inside tolerance while uncertainty prevents acceptance under a guarded rule. Reporting the uncertainty supports that decision; it does not choose the rule.

For a relative report, identify the basis unambiguously. An absolute expanded uncertainty of 0.0038 ohm at a result near 100 ohms is approximately 38 ppm of that result. It is not 38 ppm of an arbitrary full-scale range. State the quantity, relevant conditions, coverage information, and any limitations so the recipient can use the report without guessing the denominator.