Tools › Statistics › Uncertainty and Monte Carlo › Uncertainty Propagation › Example
JCGM 101 §9.2.2 (Table 2): additive model with four normal inputs, GUM against Monte Carlo
| Tool | Uncertainty Propagation |
| Source | Joint Committee for Guides in Metrology (2008). Evaluation of Measurement Data: Supplement 1 to the "Guide to the Expression of Uncertainty in Measurement": Propagation of Distributions Using a Monte Carlo Method. JCGM 101:2008, BIPM, Sèvres. Section 9.2.2 and Table 2. |
| Level | Graduate |
Problem
Y = X₁ + X₂ + X₃ + X₄ where each input is a standard normal variable N(0, 1). Compare the GUM uncertainty framework (estimate, standard uncertainty, 95 % coverage interval with k = 1.96) with the Monte Carlo propagation of distributions and test whether the GUM interval is validated at the numerical tolerance of one significant digit in u(y).
Inputs to Enter
- x1, x2, x3, x4: normal, mean 0, u = 1 (ν blank)
- Formula y = x1 + x2 + x3 + x4; coverage 95 %; trials 100,000 or more (the test uses 400,000, seed 1)
Answers
- GUM: y = 0, u(y) = 2.00, 95 % interval −3.92 to 3.92
- MCM: y = 0.00, u(y) = 2.00, probabilistically symmetric 95 % interval −3.92 to 3.92 (the handbook's own 10⁵-trial run printed −3.94)
- Numerical tolerance δ = 0.05 for u(y) = 2.0 (7.9.2); the GUM framework is validated (Table 2: Yes)
Notes
Enter four normal rows and the sum as the model. The tool's Monte Carlo endpoints agree within 0.025 and u(y) within 0.5 % at 400,000 trials (test_monte_carlo.php); the adaptive procedure of 7.9.4 is available as a checkbox.