Coverage Factors and Expanded-Uncertainty Reporting

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Coverage Factors and Expanded-Uncertainty Reporting

Key Takeaways Explain This Topic AI

Expanded uncertainty equals a selected coverage factor times combined standard uncertainty.

Small effective degrees of freedom can require a Student t factor larger than two for approximately 95% coverage.

Report uncertainty and result consistently, retaining sufficient internal precision before final rounding.

Last updated: October 2026

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

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

The expanded uncertainty ( U U U ) is obtained by multiplying u c ( y ) u_c(y) u c ( y ) by a numerical coverage factor ( k k k ) : U = k ⋅ u c ( y ) U = k \cdot u_c(y) U = k ⋅ 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 ≈ 2 k\approx2 k ≈ 2 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 t t Reference Table for 95 % 95\% 95% Confidence

Effective Degrees of Freedom ( ν eff \nu_{\text{eff}} ν eff ) Coverage Factor k = t 95 ( ν eff ) k = t_{95}(\nu_{\text{eff}}) k = t 95 ( ν eff )

1 12.71
2 4.30
3 3.18
4 2.78
5 2.57
6 2.45
8 2.31
10 2.23
15 2.13
20 2.09
30 2.04
∞ \infty ∞ 1.96
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:

The measured numerical value of the measurand y y y
The expanded uncertainty U U U
Units for the result and uncertainty, or a clearly identified relative uncertainty with its basis
The coverage factor k k k used
The approximate coverage probability (level of confidence, typically stated as "approximately 95%")
Rounding Conventions for Calibration Certificates

Rounding the Expanded Uncertainty ( U U 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 mm 0.01634\text{ mm} 0.01634 mm becomes 0.016 mm 0.016\text{ mm} 0.016 mm , and 4.782 Ω 4.782\ \Omega 4.782 Ω becomes 4.8 Ω 4.8\ \Omega 4.8 Ω . State and apply any required conservative rounding policy; rounding upward is not a universal mandatory rule.

Rounding the Reported Calibrated Value ( y y 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 Ω ± 0.0038 Ω 100.0248\ \Omega \pm 0.0038\ \Omega 100.0248 Ω ± 0.0038 Ω . The reported expanded uncertainty is based on a combined standard uncertainty multiplied by a coverage factor of k = 2.23 k = 2.23 k = 2.23 , which for an effective degrees of freedom ν eff = 10 \nu_{\text{eff}} = 10 ν eff = 10 defines a coverage probability of approximately 95 % 95\% 95% ."

Step-by-Step Worked Calibration Uncertainty Budget

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

Identifying the Uncertainty Components

Type A Repeatability ( u 1 u_1 u 1 ): Technician performs n = 5 n = 5 n = 5 repeated observations: s = 0.00300 Ω s = 0.00300\ \Omega s = 0.00300 Ω . u 1 = u ( x ˉ ) = s n = 0.00300 Ω 5 = 0.001342 Ω , ν 1 = 5 − 1 = 4 u_1 = u(\bar{x}) = \frac{s}{\sqrt{n}} = \frac{0.00300\ \Omega}{\sqrt{5}} = 0.001342\ \Omega, \quad \nu_1 = 5 - 1 = 4 u 1 = u ( x ˉ ) = n s = 5 0.00300 Ω = 0.001342 Ω , ν 1 = 5 − 1 = 4

Reference Standard Certificate ( u 2 u_2 u 2 ): The hypothetical certificate states U = 0.00200 Ω U=0.00200\ \Omega U = 0.00200 Ω with k = 2.00 k=2.00 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 = 0.00200 Ω 2.00 = 0.001000 Ω , ν 2 = ∞ u_2 = \frac{0.00200\ \Omega}{2.00} = 0.001000\ \Omega, \quad \nu_2 = \infty u 2 = 2.00 0.00200 Ω = 0.001000 Ω , ν 2 = ∞

Digital Multimeter Resolution ( u 3 u_3 u 3 ): Digital readout displays 0.001 Ω 0.001\ \Omega 0.001 Ω ( 1 m Ω 1\text{ m}\Omega 1 m Ω ). Rectangular semi-range a = 0.0005 Ω a = 0.0005\ \Omega a = 0.0005 Ω : u 3 = 0.000500 Ω 3 = 0.000289 Ω , ν 3 = ∞ u_3 = \frac{0.000500\ \Omega}{\sqrt{3}} = 0.000289\ \Omega, \quad \nu_3 = \infty u 3 = 3 0.000500 Ω = 0.000289 Ω , ν 3 = ∞

Oil Bath Temperature Variation ( u 4 u_4 u 4 ): Bath temperature stable within ± 0.10 ∘ C \pm 0.10^\circ\text{C} ± 0.1 0 ∘ C (rectangular distribution). Resistor temperature coefficient α = 10 ppm / ∘ C = 10 × 10 − 6 / ∘ C \alpha = 10\text{ ppm}/^\circ\text{C} = 10 \times 10^{-6}/^\circ\text{C} α = 10 ppm / ∘ C = 10 × 1 0 − 6 / ∘ C .

Maximum resistance shift: a = 100 Ω × ( 10 × 10 − 6 / ∘ C ) × 0.10 ∘ C = 0.000100 Ω a = 100\ \Omega \times (10 \times 10^{-6}/^\circ\text{C}) \times 0.10^\circ\text{C} = 0.000100\ \Omega a = 100 Ω × ( 10 × 1 0 − 6 / ∘ C ) × 0.1 0 ∘ C = 0.000100 Ω . u 4 = 0.000100 Ω 3 = 0.000058 Ω , ν 4 = ∞ u_4 = \frac{0.000100\ \Omega}{\sqrt{3}} = 0.000058\ \Omega, \quad \nu_4 = \infty u 4 = 3 0.000100 Ω = 0.000058 Ω , ν 4 = ∞

Master Uncertainty Budget Table

Source of Uncertainty ( X i X_i X i ) Value ( ± a \pm a ± a ) Probability Distribution Divisor Standard Uncertainty u ( x i ) u(x_i) u ( x i ) Sensitivity Coeff ( c i c_i c i ) Uncertainty Contrib u i ( y ) u_i(y) u i ( y ) Degrees of Freedom ( ν i \nu_i ν i )

Repeatability (Type A) s = 3.00 m Ω s = 3.00\text{ m}\Omega s = 3.00 m Ω Normal ( s / 5 s/\sqrt{5} s / 5 ) 5 \sqrt{5} 5 1.342 m Ω 1.342\text{ m}\Omega 1.342 m Ω 1.0 1.0 1.0 1.342 m Ω 1.342\text{ m}\Omega 1.342 m Ω 4 4 4

Master Standard (Type B) 2.00 m Ω 2.00\text{ m}\Omega 2.00 m Ω Normal ( k = 2 k=2 k = 2 ) 2.0 2.0 2.0 1.000 m Ω 1.000\text{ m}\Omega 1.000 m Ω 1.0 1.0 1.0 1.000 m Ω 1.000\text{ m}\Omega 1.000 m Ω ∞ \infty ∞

Digital Resolution (Type B) 0.50 m Ω 0.50\text{ m}\Omega 0.50 m Ω Rectangular 3 \sqrt{3} 3 0.289 m Ω 0.289\text{ m}\Omega 0.289 m Ω 1.0 1.0 1.0 0.289 m Ω 0.289\text{ m}\Omega 0.289 m Ω ∞ \infty ∞

Temperature Bath (Type B) 0.10 m Ω 0.10\text{ m}\Omega 0.10 m Ω Rectangular 3 \sqrt{3} 3 0.058 m Ω 0.058\text{ m}\Omega 0.058 m Ω 1.0 1.0 1.0 0.058 m Ω 0.058\text{ m}\Omega 0.058 m Ω ∞ \infty ∞

Combine unrounded contributions

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

Effective degrees of freedom

With only the repeatability contribution assigned finite degrees of freedom, ν e f f = ( 2.8866667 ) 2 / ( 1.8 2 / 4 ) = 10.28746 \nu_{eff}=(2.8866667)^2/(1.8^2/4)=10.28746 ν e ff = ( 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 k k ) and Expanded Uncertainty ( U U 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 ⋅ u c ( y ) = 2.23 × 0.0016990 Ω = 0.003789 Ω U = k \cdot u_c(y) = 2.23 \times 0.0016990\ \Omega = 0.003789\ \Omega U = k ⋅ u c ( y ) = 2.23 × 0.0016990 Ω = 0.003789 Ω

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

If the measured mean was 100.00423 Ω 100.00423\ \Omega 100.00423 Ω , it is rounded to match U U U at four decimal places: R uut = 100.0042 Ω ± 0.0038 Ω ( k = 2.23 , ν eff = 10 , ≈ 95 % confidence ) R_{\text{uut}} = 100.0042\ \Omega \pm 0.0038\ \Omega \quad (k = 2.23, \nu_{\text{eff}} = 10, \approx 95\% \text{ confidence}) R uut = 100.0042 Ω ± 0.0038 Ω ( k = 2.23 , ν eff = 10 , ≈ 95% 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.