Distribution Selection and Budget Assembly

Probability Distributions for Type B Evaluations

When evaluating a Type B component, the technician determines the bounding limit (semi-range $a$) and selects an appropriate probability distribution:

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Synthesizing the Uncertainty Budget Table

Once all input components are evaluated, the Law of Propagation of Uncertainty combines them. For uncorrelated input quantities:

$$u_c(y) = \sqrt{\sum_{i=1}^n [c_i \cdot u(x_i)]^2} = \sqrt{\sum_{i=1}^n [u_i(y)]^2}$$

The Expanded Uncertainty ($U$) is obtained by multiplying $u_c(y)$ by the coverage factor $k$ (often $k = 2$ for nearly normal output with adequate degrees of freedom; do not assume this coverage for every distribution):

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

Practical Calibration Budget Example: Calibrating a $100\text{ mm}$ Digital Micrometer

Consider calibrating a $100\text{ mm}$ digital micrometer (resolution $\delta x = 0.001\text{ mm}$) using an calibrated reference gauge block ($L_{\text{ref}} = 100.0000\text{ mm}$) at nominal laboratory conditions ($20 \pm 1^\circ\text{C}$):

Combine unrounded contributions

Using micrometres throughout, the squared contributions are $0.060^2$, $0.050^2/3$, $0.500^2/3$, $0.350^2/5$, $0.575^2/3$, and $0.200^2/3$. Their sum is $0.23580833\ \mu\text{m}^2$, giving $u_c=0.4856010022\ \mu\text{m}$. For this illustrative choice of $k=2$, $U=0.9712020044\ \mu\text{m}$, reported as $0.97\ \mu\text{m}$. Rounded table values are for display; do not recompute from them when unrounded contributions are available.

Notice that UUT resolution ($0.289\ \mu\text{m}$) and differential temperature ($0.332\ \mu\text{m}$) heavily dominate the uncertainty budget, demonstrating why high-accuracy calibration standards cannot overcome poor environment or coarse UUT resolution.

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Common Calibration Traps & CCT Exam Pitfalls

Official references (checked October 10, 2026): BIPM JCGM publications: GUM and VIM.

Reviewing the model before accepting the total

The micrometer budget is hypothetical. Its five-observation repeatability estimate comes from an independent characterization using a finer readout, not five readings rounded to the 1 µm display step; such rounded readings cannot produce the stated 0.350 µm sample deviation. The reported mean and remaining resolution contribution use the stated measurement model. Assume the components represent separate effects and justify that assumption to avoid double counting. A single-reading result instead uses the appropriate single-observation repeatability component. More repetitions do not reduce every certificate, drift, temperature, or quantization contribution by the same square-root rule.

The listed temperature term uses a length of 100 mm, a coefficient of 11.5 ppm per kelvin, and a bounded temperature difference of half a kelvin. That produces a length bound of 0.575 micrometres before distribution conversion. The row expresses the environmental effect in length units, so its displayed sensitivity is one. An alternative table could retain temperature in kelvin with a length-per-kelvin sensitivity. Those are equivalent representations; including both rows would double count the same effect.

Check correlation and overlap among other rows. Repeatability observations can already contain some quantization, contact-force, or short-term environmental variation. A separate contribution is justified only for the effect not adequately represented, or under a clearly stated conservative model. Do not delete a contribution merely because it is large, but do not add the same observed variation twice without rationale.

Finally, identify corrections and their uncertainty separately. Apply an established reference-block correction to the result; include uncertainty in that correction in the budget. The magnitude of the correction is not automatically its standard uncertainty. Keep assumptions, source records, relevant distributions, and unrounded calculation values with the work package so another qualified person can reproduce the result.