Uncertainty budget (GUM)

Build a bench uncertainty budget row by row: Type A from your readings, Type B from limits or a certificate with a rectangular, triangular, U-shaped or normal distribution. Combines by root sum of squares, finds ν_eff by Welch–Satterthwaite and k from the Student t distribution, and shows every row’s share.

Uncertainty budget · GUM

v0.1.0 · RSS, Welch–Satterthwaite, Student t

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 · Resolution

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

Result unitCoverage probability

Truncate ν_eff to a whole number before finding k (as with a printed t table)

Expanded uncertainty U (p = 95.45 %)0.02597 mVu_c = 0.01258 mV · ν_eff = 40.11 · k = 2.064

Uncertainty budget
Source	Type	Divisor	u (input unit)	c	\|c\|·u (mV)	% of u_c²	ν
Repeatability	A (n = 5)	√n	0.007071 mV	1.000	0.007071	31.6	4.000
Reference certificate	B · normal	k = 2	0.01000 mV	1.000	0.01000	63.2	∞
Resolution	B · rectangular	√3	0.002887 mV	1.000	0.002887	5.3	∞

u_c = √Σ(cᵢuᵢ)² · ν_eff = u_c⁴ / Σ((cᵢuᵢ)⁴/νᵢ) · U = k·u_c, k from Student t at ν_effInputs are treated as uncorrelated. Round U for a certificate separately (at most two significant figures).

ShowFiguresRounding

Show full precision

How this is calculated

Equation: u_A = s/√n, ν = n − 1; u_B = a/√3, a/√6, a/√2 or U/k; u_c = √Σ(cᵢuᵢ)²; ν_eff = u_c⁴ / Σ((cᵢuᵢ)⁴/νᵢ); U = k·u_c with k = t_p(ν_eff)
Source: JCGM 100:2008 (GUM): Type A evaluation 4.2.3 eq. (5) with ν = n − 1 (4.2.6); Type B evaluation 4.3.3–4.3.4 (stated U and k), 4.3.7 eq. (7) (rectangular), 4.3.9 eq. (9b) (triangular); U-shaped (arc sine) variance from JCGM 101:2008 6.4.6; combined standard uncertainty for uncorrelated inputs 5.1.2 eq. (10) and 5.1.3 eq. (11a); Welch–Satterthwaite effective degrees of freedom G.4.1 eq. (G.2b), truncation option G.4.1 Note 1; k = t_p(ν_eff) as in G.3.2 and G.6.4. Student-t coverage factor computed numerically. Cross-checked against GTC 1.5.1 and SciPy 1.18.1.
Valid range: Coverage probability 0.5 to 0.9999; inputs uncorrelated; Type A needs at least 2 readings
Rounding: Shown to 4 significant figures by default, rounded half up; full precision is always available. Rounding U for a certificate (two significant figures) is a separate step.
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
UNC-K-01	p = 0.95, dof = 1	12.706204736174694	± 1e-9	Hand-derived closed form: k = tan(πp/2) for ν = 1 (Cauchy). Cross-checked: SciPy 1.18.1 scipy.stats.t.ppf
UNC-K-02	p = 0.9545, dof = 2	4.526550760081992	± 1e-9	Hand-derived closed form: k = p√2/√(1 − p²) for ν = 2. Cross-checked: SciPy 1.18.1 scipy.stats.t.ppf
UNC-K-03	p = 0.95, dof = 10	2.228138851986274	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, dof); agrees with mpmath 1.4.1 at 40 digits and GTC 1.5.1 reporting.k_factor
UNC-K-04	p = 0.99, dof = 16	2.9207816224251	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, dof); agrees with mpmath 1.4.1 at 40 digits and GTC 1.5.1 reporting.k_factor
UNC-K-05	p = 0.9545	2.0000024438996027	± 1e-9	SciPy 1.18.1 scipy.stats.norm.ppf((1 + p)/2); agrees with mpmath 1.4.1 at 40 digits and GTC 1.5.1 reporting.k_factor
UNC-K-06	p = 0.95	1.959963984540054	± 1e-9	SciPy 1.18.1 scipy.stats.norm.ppf((1 + p)/2); agrees with mpmath 1.4.1 at 40 digits and GTC 1.5.1 reporting.k_factor
UNC-K-07	p = 0.9545, dof = 16.3	2.165590984821532	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, dof); agrees with mpmath 1.4.1 at 40 digits and GTC 1.5.1 reporting.k_factor
UNC-K-08	p = 0.95, dof = 1000000	1.959966356814107	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, dof); agrees with mpmath 1.4.1 at 40 digits
UNC-K-09	p = 0.99999, dof = 10	OutOfRangeError	exact	Supported coverage probability is 0.5 to 0.9999
UNC-K-10	p = 0.4	OutOfRangeError	exact	Supported coverage probability is 0.5 to 0.9999
UNC-B1-01	rows = A[10.1 10.3 10.2 10.4 10]; B rectangular a = 0.3; B triangular a = 0.6; B u-shaped a = 0.2; B normal U = 0.2 k = 2, coverageProbability = 0.9545, output = u_c	0.3535533905932738 mm	± 1e-9	Hand-derived: u² = 0.025/5 + 0.3²/3 + 0.6²/6 + 0.2²/2 + (0.2/2)² = 0.125, u_c = √0.125. Cross-checked with GTC 1.5.1
UNC-B1-02	rows = A[10.1 10.3 10.2 10.4 10]; B rectangular a = 0.3; B triangular a = 0.6; B u-shaped a = 0.2; B normal U = 0.2 k = 2, coverageProbability = 0.9545, output = nu_eff	2500	± 2.5e-7	Hand-derived: ν_eff = 0.125² / (0.005²/4) = 2500. Cross-checked with GTC 1.5.1
UNC-B1-03	rows = A[10.1 10.3 10.2 10.4 10]; B rectangular a = 0.3; B triangular a = 0.6; B u-shaped a = 0.2; B normal U = 0.2 k = 2, coverageProbability = 0.9545, output = k	2.0010029372621485	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, ν) at the GTC df
UNC-B1-04	rows = A[10.1 10.3 10.2 10.4 10]; B rectangular a = 0.3; B triangular a = 0.6; B u-shaped a = 0.2; B normal U = 0.2 k = 2, coverageProbability = 0.9545, output = U	0.7074613730561325 mm	± 1e-9	k·u_c with u_c from GTC 1.5.1 and k from SciPy 1.18.1
UNC-B2-01	rows = B normal U = 0.2 k = 2, c = 3; B triangular a = 0.6, c = -0.5; B rectangular a = 0.3, c = 2, coverageProbability = 0.95, output = u_c	0.4743416490252569 V	± 1e-9	Hand-derived: (3·0.1)² + (0.5·0.6/√6)² + (2·0.3/√3)² = 0.09 + 0.015 + 0.12 = 0.225, u_c = √0.225. Cross-checked with GTC 1.5.1
UNC-B2-02	rows = B normal U = 0.2 k = 2, c = 3; B triangular a = 0.6, c = -0.5; B rectangular a = 0.3, c = 2, coverageProbability = 0.95, output = nu_eff	Infinity	exact	Hand-derived: every ν is infinite, so ν_eff is infinite. Cross-checked with GTC 1.5.1
UNC-B2-03	rows = B normal U = 0.2 k = 2, c = 3; B triangular a = 0.6, c = -0.5; B rectangular a = 0.3, c = 2, coverageProbability = 0.95, output = k	1.959963984540054	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, ν) at the GTC df
UNC-B2-04	rows = B normal U = 0.2 k = 2, c = 3; B triangular a = 0.6, c = -0.5; B rectangular a = 0.3, c = 2, coverageProbability = 0.95, output = U	0.9296925484568423 V	± 1e-9	k·u_c with u_c from GTC 1.5.1 and k from SciPy 1.18.1
UNC-B3-01	rows = B normal U = 1 k = 1, ν = 4; B normal U = 1 k = 1, ν = 4, coverageProbability = 0.95, output = u_c	1.4142135623730951 Ω	± 1e-9	Hand-derived: u_c = √(1² + 1²) = √2. Cross-checked with GTC 1.5.1
UNC-B3-02	rows = B normal U = 1 k = 1, ν = 4; B normal U = 1 k = 1, ν = 4, coverageProbability = 0.95, output = nu_eff	8	± 1e-9	Hand-derived: ν_eff = 2² / (1/4 + 1/4) = 8. Cross-checked with GTC 1.5.1
UNC-B3-03	rows = B normal U = 1 k = 1, ν = 4; B normal U = 1 k = 1, ν = 4, coverageProbability = 0.95, output = k	2.306004135204166	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, ν) at the GTC df
UNC-B3-04	rows = B normal U = 1 k = 1, ν = 4; B normal U = 1 k = 1, ν = 4, coverageProbability = 0.95, output = U	3.2611823228941716 Ω	± 1e-9	k·u_c with u_c from GTC 1.5.1 and k from SciPy 1.18.1
UNC-B4-01	rows = B normal U = 1 k = 1, ν = 3; B normal U = 1 k = 1, ν = 5, coverageProbability = 0.95, output = nu_eff	7.5	± 1e-9	Hand-derived: ν_eff = 2² / (1/3 + 1/5) = 7.5. Cross-checked with GTC 1.5.1
UNC-B4-02	rows = B normal U = 1 k = 1, ν = 3; B normal U = 1 k = 1, ν = 5, coverageProbability = 0.95, output = k	2.333039626864974	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, ν) at the GTC df
UNC-B5-01	rows = B normal U = 1 k = 1, ν = 3; B normal U = 1 k = 1, ν = 5, coverageProbability = 0.95, output = k, truncateDof = true	2.364624251592784	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, ν truncated to an integer) at the GTC df
UNC-B6-01	rows = A[1 2 3 4], coverageProbability = 0.95, output = u_c	0.6454972243679028 g	± 1e-9	Hand-derived: s = √(5/3), u = s/√4. Cross-checked with GTC 1.5.1
UNC-B6-02	rows = A[1 2 3 4], coverageProbability = 0.95, output = nu_eff	3	± 1e-9	Hand-derived: ν = n − 1 = 3. Cross-checked with GTC 1.5.1
UNC-B6-03	rows = A[1 2 3 4], coverageProbability = 0.95, output = k	3.182446305283708	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, ν) at the GTC df
UNC-B6-04	rows = A[1 2 3 4], coverageProbability = 0.95, output = U	2.0542602567605206 g	± 1e-9	k·u_c with u_c from GTC 1.5.1 and k from SciPy 1.18.1
UNC-H1-01	rows = B normal U = 25 k = 1, ν = 18; B normal U = 5.8 k = 1, ν = 24; B normal U = 3.9 k = 1, ν = 5; B normal U = 6.7 k = 1, ν = 8; B rectangular a = 0.000002, c = 0; B rectangular a = 0.000001, ν = 50, c = 5000062.3; B rectangular a = 0.05, ν = 2, c = -575.0071645; B normal U = 0.2 k = 1, c = 0; B u-shaped a = 0.5, c = 0, coverageProbability = 0.99, output = u_c	31.663879111008633 nm	± 3.1663879111008635e-9	GTC 1.5.1: sum of cᵢ·xᵢ as GTC uncertain numbers (type_a.estimate for readings; type_b.uniform, triangular, arcsine for Type B), giving u and df (model: JCGM 100:2008 Annex H.1 end-gauge calibration as coded in GTC examples/GUM_H1.py)
UNC-H1-02	rows = B normal U = 25 k = 1, ν = 18; B normal U = 5.8 k = 1, ν = 24; B normal U = 3.9 k = 1, ν = 5; B normal U = 6.7 k = 1, ν = 8; B rectangular a = 0.000002, c = 0; B rectangular a = 0.000001, ν = 50, c = 5000062.3; B rectangular a = 0.05, ν = 2, c = -575.0071645; B normal U = 0.2 k = 1, c = 0; B u-shaped a = 0.5, c = 0, coverageProbability = 0.99, output = nu_eff	16.751855737627242	± 1.675185573762724e-9	GTC 1.5.1: sum of cᵢ·xᵢ as GTC uncertain numbers (type_a.estimate for readings; type_b.uniform, triangular, arcsine for Type B), giving u and df (model: JCGM 100:2008 Annex H.1 end-gauge calibration as coded in GTC examples/GUM_H1.py)
UNC-H1-03	rows = B normal U = 25 k = 1, ν = 18; B normal U = 5.8 k = 1, ν = 24; B normal U = 3.9 k = 1, ν = 5; B normal U = 6.7 k = 1, ν = 8; B rectangular a = 0.000002, c = 0; B rectangular a = 0.000001, ν = 50, c = 5000062.3; B rectangular a = 0.05, ν = 2, c = -575.0071645; B normal U = 0.2 k = 1, c = 0; B u-shaped a = 0.5, c = 0, coverageProbability = 0.99, output = k	2.9035476304491388	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, ν) at the GTC df (model: JCGM 100:2008 Annex H.1 end-gauge calibration as coded in GTC examples/GUM_H1.py)
UNC-H1-04	rows = B normal U = 25 k = 1, ν = 18; B normal U = 5.8 k = 1, ν = 24; B normal U = 3.9 k = 1, ν = 5; B normal U = 6.7 k = 1, ν = 8; B rectangular a = 0.000002, c = 0; B rectangular a = 0.000001, ν = 50, c = 5000062.3; B rectangular a = 0.05, ν = 2, c = -575.0071645; B normal U = 0.2 k = 1, c = 0; B u-shaped a = 0.5, c = 0, coverageProbability = 0.99, output = U	91.9375811635971 nm	± 9.19375811635971e-9	k·u_c with u_c from GTC 1.5.1 and k from SciPy 1.18.1 (model: JCGM 100:2008 Annex H.1 end-gauge calibration as coded in GTC examples/GUM_H1.py)
UNC-H2-01	rows = B normal U = 25 k = 1, ν = 18; B normal U = 5.8 k = 1, ν = 24; B normal U = 3.9 k = 1, ν = 5; B normal U = 6.7 k = 1, ν = 8; B rectangular a = 0.000002, c = 0; B rectangular a = 0.000001, ν = 50, c = 5000062.3; B rectangular a = 0.05, ν = 2, c = -575.0071645; B normal U = 0.2 k = 1, c = 0; B u-shaped a = 0.5, c = 0, coverageProbability = 0.99, output = k, truncateDof = true	2.9207816224251	± 1e-9	SciPy 1.18.1 scipy.stats.t.ppf((1 + p)/2, ν truncated to an integer) at the GTC df (model: JCGM 100:2008 Annex H.1 end-gauge calibration as coded in GTC examples/GUM_H1.py)
UNC-H2-02	rows = B normal U = 25 k = 1, ν = 18; B normal U = 5.8 k = 1, ν = 24; B normal U = 3.9 k = 1, ν = 5; B normal U = 6.7 k = 1, ν = 8; B rectangular a = 0.000002, c = 0; B rectangular a = 0.000001, ν = 50, c = 5000062.3; B rectangular a = 0.05, ν = 2, c = -575.0071645; B normal U = 0.2 k = 1, c = 0; B u-shaped a = 0.5, c = 0, coverageProbability = 0.99, output = U, truncateDof = true	92.48327620212405 nm	± 9.248327620212404e-9	k·u_c with u_c from GTC 1.5.1 and k from SciPy 1.18.1 (model: JCGM 100:2008 Annex H.1 end-gauge calibration as coded in GTC examples/GUM_H1.py)
UNC-B7-01	rows = B normal U = 1 k = 1, ν = 4; B normal U = 1 k = 1, ν = 4, coverageProbability = 1, output = U	OutOfRangeError	exact	Supported coverage probability is 0.5 to 0.9999

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