Tools › Statistics › Uncertainty and Monte Carlo › Uncertainty Propagation › Example
GUM Annex H.1: calibration of a 50 mm end gauge by comparison (uncertainty budget, ν_eff, k₉₉, U₉₉)
| Tool | Uncertainty Propagation |
| Source | Joint Committee for Guides in Metrology (2008). Evaluation of Measurement Data: Guide to the Expression of Uncertainty in Measurement. JCGM 100:2008, BIPM, Sèvres. Annex H.1 (Table H.1, H.1.4 to H.1.6) and Table G.2. |
| Level | Professional |
Problem
A 50 mm end gauge is calibrated by comparison with a standard of the same nominal length: l = l_S + d − l_S (δα θ + α_S δθ), where l_S is the standard's length, d the measured difference, α_S the standard's expansion coefficient, θ the deviation of the gauge's temperature from 20 °C, δα the difference of the two expansion coefficients and δθ the difference of the two temperatures. From the Type A and Type B uncertainties of the six inputs, find the gauge length, its combined standard uncertainty, the effective degrees of freedom and the 99 % expanded uncertainty.
Inputs to Enter
- l_s: normal, 50.000623 mm, u = 0.000025 mm (25 nm), ν = 18
- d: normal, 0.000215 mm (215 nm), u = 0.0000097 mm (9.7 nm), ν = 25.6
- alpha_s: rectangular, 11.5 × 10⁻⁶ /°C, half-width 2 × 10⁻⁶ /°C
- theta: normal, −0.1 °C, u = 0.41 °C
- dalpha: rectangular, 0, half-width 1 × 10⁻⁶ /°C, ν = 50; dtheta: rectangular, 0, half-width 0.05 °C, ν = 2
- Formula l = l_s + d - l_s * (dalpha * theta + alpha_s * dtheta); coverage 99 %
Answers
- l = 50.000838 mm (H.1.5)
- Budget: |c_i| u(x_i) for δα = 2.9 nm and for δθ = 16.6 nm (Table H.1)
- u_c(l) = 32 nm (H.1.6, first-order terms)
- ν_eff = 16.7 → 16 (Welch-Satterthwaite, truncated per G.4.1), k₉₉ = 2.92 (Table G.2), U₉₉ = 93 nm
Notes
Enter the six rows in mm and °C and type the model; set the coverage to 99 %. The tool is first order, so the second-order terms the GUM adds in H.1.7 (u_c = 34 nm) are not included; the GUM Table G.2 factors t₉₇.₅(13) = 2.16, t₉₇.₅(9) = 2.26 and t₉₉.₅(16) = 2.92 are reproduced to 1e-3 (test_monte_carlo.php).