Monte Carlo uncertainty (JCGM 101)

Run the same rows as the GUM budget through the Monte Carlo method of JCGM 101 (GUM Supplement 1): Type A rows as t-distributions, rectangular, triangular and U-shaped limits sampled as they are. Get u(y), the probabilistically symmetric and shortest coverage intervals, and whether the GUM interval is validated for your budget. Seeded, so a result can be reproduced.

Monte Carlo uncertainty · JCGM 101

v0.1.0 · Y = Σcᵢ·Xᵢ, seeded

Row 1 · Type A · Repeatability

Source of uncertaintyTypeUnitSensitivity cResult unit per input unitReadingsSeparate with spaces or new lines; use a point for decimals. u = s/√n, ν = n − 1.

Row 2 · Type B · Reference certificate

Source of uncertaintyTypeUnitSensitivity cResult unit per input unitDistributionStated expanded uncertainty UIts coverage factor k1 if you entered a standard uncertaintyDegrees of freedom νBlank for infinite

Row 3 · Type B · Display resolution

Source of uncertaintyTypeUnitSensitivity cResult unit per input unitDistributionHalf-width a (±)Degrees of freedom νBlank for infinite

Result unitCoverage probabilityTrialsMeaningful digits in u(y)Sets the numerical tolerance δInterval compared with the GUMSeedSame seed, same result

Press Run to draw the distribution. A million trials takes about a second.

y = mean of M model values · u(y) = their standard deviation · interval from the sorted values (JCGM 101 7.7)Type A rows are t-distributions (ν = n − 1); certificates with finite ν are scaled t. Inputs are treated as independent.

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The inputs are in the link itself; nothing is stored or sent.

How this is calculated

Equation: Y = Σcᵢ·Xᵢ; draw M values y_r; y = mean, u(y) = s(y_r); q = ⌊pM + ½⌋; symmetric interval [y_(r), y_(r+q)], r = ⌊(M − q + 1)/2⌋; shortest: the r minimising y_(r+q) − y_(r); GUM validated if |y − U − y_low| ≤ δ and |y + U − y_high| ≤ δ
Source: JCGM 101:2008 (GUM Supplement 1): distributions 6.4.2, 6.4.5, 6.4.6, 6.4.7, 6.4.9; discrete representation, estimate and coverage intervals 7.5–7.7; adaptive procedure and numerical tolerance 7.9; validation of the GUM uncertainty framework 8.1 and 8.2; enhanced Wichmann–Hill generator C.3.3, Box–Muller C.4, t generator C.6. Checked against the JCGM 101 clause 9.2 examples and analytic quantiles (SciPy 1.18.1).
Valid range: Linear model, uncorrelated inputs; coverage probability 0.5 to 0.9999; 10 000 to 5 000 000 trials; Type A needs at least 2 readings
Rounding: Shown to 4 significant figures by default, rounded half up; full precision is always available. The numerical tolerance δ says how many of those digits the run supports.
Version: v0.1.0 · changelog

Calculated value, not a certified result. Accredited laboratories must validate any software they use (ISO/IEC 17025 clause 7.11).

Test vectors for this tool

The reference cases this calculator is tested against on every change. Use them to check your own software. All test vectors · Download JSON

ID	Inputs	Expected	Tolerance	Source
MC-01	rows = B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1, coverageProbability = 0.95, trials = 1000000, seed = 1, output = u	2	± 0.01	JCGM 101:2008 9.2.2: four standard Gaussian inputs, Y ~ N(0, 2²) analytically (table 2)
MC-02	rows = B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricHigh	3.919927969080108	± 0.03	JCGM 101:2008 9.2.2: four standard Gaussian inputs, Y ~ N(0, 2²) analytically (table 2)
MC-03	rows = B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricLow	-3.919927969080108	± 0.03	JCGM 101:2008 9.2.2: four standard Gaussian inputs, Y ~ N(0, 2²) analytically (table 2)
MC-04	rows = B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1, coverageProbability = 0.95, trials = adaptive, seed = 1, output = gumValidated	true	exact	JCGM 101:2008 9.2.2: four standard Gaussian inputs, Y ~ N(0, 2²) analytically (table 2); GUF validated for δ = 0.05
MC-05	rows = B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricHigh	3.879406741347821	± 0.03	JCGM 101:2008 9.2.3: four rectangular inputs, u = 1 each; analytic endpoints ±2√3[2 − (3/5)^¼] (table 3), confirmed by the Irwin–Hall distribution (SciPy 1.18.1)
MC-06	rows = B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricLow	-3.879406741347821	± 0.03	JCGM 101:2008 9.2.3: four rectangular inputs, u = 1 each; analytic endpoints ±2√3[2 − (3/5)^¼] (table 3), confirmed by the Irwin–Hall distribution (SciPy 1.18.1)
MC-07	rows = B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772, coverageProbability = 0.95, trials = 1000000, seed = 1, output = u	2	± 0.01	JCGM 101:2008 9.2.3: four rectangular inputs, u = 1 each; analytic endpoints ±2√3[2 − (3/5)^¼] (table 3), confirmed by the Irwin–Hall distribution (SciPy 1.18.1)
MC-08	rows = B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 17.32050807568877, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricHigh	17.015814099859707	± 0.06	JCGM 101:2008 9.2.4 table 4 (MCM [−17.0, 17.0], u(y) = 10.1, GUF not validated); exact endpoint by numerical convolution (SciPy 1.18.1)
MC-09	rows = B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 17.32050807568877, coverageProbability = 0.95, trials = 1000000, seed = 1, output = u	10.14889156509222	± 0.05	JCGM 101:2008 9.2.4 table 4 (MCM [−17.0, 17.0], u(y) = 10.1, GUF not validated); exact endpoint by numerical convolution (SciPy 1.18.1); u(y) = √103 analytically
MC-10	rows = B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 17.32050807568877, coverageProbability = 0.95, trials = adaptive, seed = 1, output = gumValidated	false	exact	JCGM 101:2008 9.2.4 table 4 (MCM [−17.0, 17.0], u(y) = 10.1, GUF not validated); exact endpoint by numerical convolution (SciPy 1.18.1)
MC-11	rows = B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 1.7320508075688772; B rectangular a = 17.32050807568877, coverageProbability = 0.95, trials = 1000000, seed = 1, output = dLow	2.875647850723386	± 0.06	JCGM 101:2008 9.2.4 table 4 (MCM [−17.0, 17.0], u(y) = 10.1, GUF not validated); exact endpoint by numerical convolution (SciPy 1.18.1); d_low = U_GUM − y_high (8.1.3)
MC-12	rows = B rectangular a = 1, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricHigh	0.95	± 0.002	JCGM 101:2008 6.4.2: R(−a, a) has the 95 % symmetric interval ±0.95a exactly
MC-13	rows = B triangular a = 1, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricHigh	0.7763932022500211	± 0.004	JCGM 101:2008 6.4.5: T(−a, a) has the 95 % symmetric endpoint a(1 − √0.05)
MC-14	rows = B u-shaped a = 1, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricHigh	0.996917333733128	± 0.0005	JCGM 101:2008 6.4.6: arc sine U(−a, a), distribution function arcsin(x/a)/π + ½, endpoint a·sin(0.95π/2)
MC-15	rows = A[10.01 10.03 10.02 10.04 10], coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricHigh	0.01963243161477515	± 0.0003	JCGM 101:2008 6.4.9.2: t₄(x̄, s²/5); endpoint t_0.975,4 · s/√5 (SciPy 1.18.1)
MC-16	rows = A[10.012 10.031 10.018 10.044 10.002 10.027 10.009 10.036 10.021 10.015], coverageProbability = 0.95, trials = 1000000, seed = 1, output = u	0.004657559140027802	± 0.0002	JCGM 101:2008 6.4.9.4 formula (13): u(x) = √((n − 1)/(n − 3)) · s/√n
MC-17	rows = B normal U = 0.02 k = 2, ν = 8, coverageProbability = 0.95, trials = 1000000, seed = 1, output = symmetricHigh	0.02306004135204166	± 0.0003	JCGM 101:2008 6.4.9.7: certificate U, k, ν_eff → t₈(0, (U/k)²); endpoint t_0.975,8 · U/k (SciPy 1.18.1)
MC-18	rows = B normal U = 0.02 k = 2, ν = 8, coverageProbability = 0.95, trials = 1000000, seed = 1, output = u	0.011547005383792514	± 0.0002	JCGM 101:2008 6.4.9.4: variance of t_ν scaled by U/k is (U/k)²·ν/(ν − 2)
MC-19	rows = B normal U = 2 k = 2, coverageProbability = 0.9545, trials = 1000000, seed = 1, output = symmetricHigh	2.0000024438996027	± 0.015	Gaussian N(0, 1) at p = 0.9545: endpoint Φ⁻¹(0.977 25) (SciPy 1.18.1)
MC-20	rows = B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1, coverageProbability = 0.95, trials = 5000, seed = 1, output = u	RangeError	exact	Metrology Bench limit: 10 000 to 5 000 000 trials (JCGM 101 7.2.2 asks for M ≫ 1/(1 − p))
MC-21	rows = B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1, coverageProbability = 0.99999, trials = 1000000, seed = 1, output = u	OutOfRangeError	exact	Coverage probability limited to 0.5–0.9999, as in the budget
MC-22	rows = B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1; B normal U = 1 k = 1, coverageProbability = 0.95, trials = 1000000, seed = 0, output = u	RangeError	exact	Seed must be a whole number from 1 to 2 147 483 646

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