How to run a propagation 01 Type the measurement model into the y = f(x₁, x₂, …) field — 4 * pi^2 * L / T^2, or any expression the scientific calculator accepts. 02 Fill one row per input distribution (normal, rectangular, triangular, scaled-and-shifted t, log-normal, arcsine), its parameters — estimate and u, or centre and half-width a — and degrees of freedom ν where known. Each row prints the u(xᵢ) it implies. 03 Set the sampling controls trials M (default 100 000), seed, coverage 68%–99.5%; leave them at the defaults for the worked example. 04 Read the comparison cards for y and u(y) from both methods, the two coverage-interval definitions, then the method-by-method table with differences. 05 Move to [Sensitivity](/science/uncertainty/sensitivity) for cᵢ and variance shares, or to Adaptive M to let the sequence scatter choose M. Worked example: g from a pendulum The page opens on the pendulum preset; the Pendulum g = 4π²L/T² button restores it. Model 4 * pi^2 * L / T^2; L normal (1.000, 0.001); T normal (2.006, 0.002) with ν = 9; M = 100 000, seed 20250101, coverage 95%.
Readout
GUM first-order
Monte Carlo
y
9.8106522
9.8107275
u(y)
0.0218848
0.0219124
95% interval (symmetric)
[9.763744, 9.8575604]
[9.7677391, 9.8537341]
coverage factor k
2.1434
1.9622
effective dof ν_eff
14.096
—
Two more readouts: the shortest 95% interval is [9.7680546, 9.8540251], and the §8 check compares endpoint differences 4.00e-3 and 3.83e-3 with δ = 5.00e-4 , so it declines to validate the first-order result. Sensitivity gives c_L = 9.8107 , c_T = −9.7813 , |c_L|u_L = 0.009811, |c_T|u_T = 0.01956: T carries 79.9% of the first-order variance, L 20.1%.
GUM first-order versus Monte Carlo The GUM row linearises the model at the input estimates — u_c²(y) = Σ cᵢ²u²(xᵢ) + 2Σ cᵢcⱼu(xᵢ)u(xⱼ)rᵢⱼ — with sensitivity coefficients cᵢ = ∂f/∂xᵢ computed numerically and a coverage factor from the effective degrees of freedom. The Monte Carlo row draws M samples from each input distribution, evaluates the model on every draw, and reads y, u(y) and the interval from the sampled output. Two interval definitions are reported: probabilistically symmetric (equal tails) and shortest (minimum length); they coincide only for symmetric outputs.
Assumptions and limits
Interval endpoints are sample quantiles. u(y) converges as 1/√M — MC noise 6.93e-5 here — while the 95% endpoints move more between runs: Adaptive M reports scatter 2.5e-4 and 2.1e-4 for the endpoints against 3.7e-5 and 2.7e-5 for y and u(y).
Sensitivity shares rank inputs; they are not a variance decomposition. The Monte Carlo shares vary one input at a time, so interactions are missing; with correlated inputs the covariance term is excluded from the shares and printed separately (4.500e-1 on the two-resistor preset, where both rows still read 50%). At M = 100 000 that preset's §8 check cannot resolve the comparison; the verdict says so. What it does not do: it does not turn raw repeated measurements into an estimate and u, convert units, or assemble a calibration certificate. For a sample mean and standard deviation to type in, use the statistics calculator ; for first-order propagation without Monte Carlo, the Science Lab error panel is lighter. Why the coverage factor is not always 2 The 95% GUM interval above uses k = 2.1434 , not 1.96: T is declared with ν = 9, Welch–Satterthwaite returns ν_eff = 14.096 , and k is the t quantile t₀.₉₇₅(ν_eff). The Monte Carlo row shows k = 1.9622 for the same 95% — the empirical half-width divided by the sample standard deviation. Both factors sit in one table; the §8 verdict turns on endpoints, not k: the standard uncertainties agree to 0.126%, yet the t-based interval reaches 9.8575604 where the sampled interval stops at 9.8537341, and that gap exceeds δ. The effect grows as ν falls — the t-repeat preset (ν = 3) puts GUM k at 3.1824 against MC k 1.8558. State which interval you report.
Where it fits Nonlinear models The square preset, y = x² with x ~ N(0, 1), is the case linearisation cannot see: ∂f/∂x = 2x vanishes at 0, so GUM reports u = 0 while Monte Carlo reads u(y) = 1.41508 .
Correlated inputs Two 1 kΩ resistors calibrated against the same standard, u = 0.5 Ω each and r = 0.9: the combined standard uncertainty reads 0.974679 against 0.707107 treated as independent, and the sensitivity footnote carries the covariance contribution 4.500e-1.
Choosing M On the pendulum, Adaptive M converged after 16 sequences × 20 000 trials — 320 000 draws — with y = 9.8107167 , u(y) = 0.021842 and the shortest 95% interval [9.76794, 9.85328].
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References
BIPM JCGM, JCGM 100:2008 — Evaluation of measurement data: Guide to the expression of uncertainty in measurement , bipm.org （访问日期：2026-10-01）— first-order propagation and effective degrees of freedom.
BIPM JCGM, JCGM 101:2008 — Supplement 1: Propagation of distributions using a Monte Carlo method , bipm.org （访问日期：2026-10-01）— Monte Carlo trials, coverage intervals, §8 tolerance.
NIST, Technical Note 1297: Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results , nist.gov （访问日期：2026-10-01）— coverage factors.
Wikipedia, Propagation of uncertainty , en.wikipedia.org （访问日期：2026-10-01）— covariance terms.
Wikipedia, Welch–Satterthwaite equation , en.wikipedia.org （访问日期：2026-10-01）— effective degrees of freedom.
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Updated 2026-10-01