Tools › Statistics › Uncertainty and Monte Carlo › Uncertainty Propagation › Example
JCGM 101 §9.2.3 (Table 3, Annex E): additive model with four rectangular inputs
| 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 9.2.3, Table 3 and Annex E. |
| Level | Graduate |
Problem
Y = X₁ + X₂ + X₃ + X₄ where each input is rectangular with zero mean and standard uncertainty 1 (half-width √3). The GUM framework gives the same u(y) as in the normal case, but the output distribution is the four-fold convolution of rectangles, whose exact 95 % interval is given in Annex E. Compare the two approaches.
Inputs to Enter
- x1, x2, x3, x4: rectangular, mean 0, half-width 1.7321 (√3, so u = 1)
- Formula y = x1 + x2 + x3 + x4; coverage 95 %; trials 100,000 or more (the test uses 400,000, seed 2)
Answers
- GUM: u(y) = 2.00, interval −3.92 to 3.92 (unchanged)
- Exact 95 % interval (Annex E): ±2√3 [2 − (3/5)^¼] = ±3.88
- MCM: u(y) = 2.00, 95 % symmetric interval −3.88 to 3.88; the output's excess kurtosis is −1.2/4 = −0.30 (the sum of four rectangles is flatter than a normal)
Notes
Enter four rectangular rows with half-width √3. The tool's endpoints agree with the analytic ±3.88 within 0.025 (test_monte_carlo.php).