Measurement uncertainty
Complete a compliant GUM / Monte-Carlo uncertainty evaluation in minutes — no need to read JCGM 100/101 first. Pick a scenario, follow the guided steps, watch the budget update live, and export a professional report.
An uncertainty evaluation = a report of “what was measured + how unsure we are”: list every error source (component) with its uᵢ; the tool combines them, derives k, and writes the report.
Not sure which method? Answer two questions
Enter each component’s standard uncertainty uᵢ, sensitivity cᵢ and degrees of freedom νᵢ. The tool combines as u_c = √Σ(cᵢ·uᵢ)², computes effective df ν_eff (Welch–Satterthwaite), then derives k from the t-distribution at ν_eff (or you set k manually).
Where does uncertainty come from? Common sources: repeatability, resolution, calibration certificate, temperature, bias. Click above to add a preset component with sensible defaults.
④ Analysis report
GUM · genericMeasurement Uncertainty Analysis Report
Generic GUM report · GUM (JCGM 100:2008) · 2026-10-22
1 · Executive summary
| Version | Generic (context not applied) | ||
|---|---|---|---|
| Measurand | Length (mm) | ||
| Method | GUM (JCGM 100:2008) | ||
| u_c | 0.025827 | k | 2 |
| U = k·u_c | 0.051655 | U (≤2 s.f.) | 0.052 |
2 · Measurement model
Y = Σ cᵢ·xᵢ (default linear model)
The model relates the measurand Y to input quantities xᵢ; cᵢ = ∂Y/∂xᵢ is the sensitivity coefficient, taken as 1 when dimensionless.
3 · Sources & uncertainty budget
| # | Source of uncertainty | Type | Distribution | Basis of evaluation | uᵢ | cᵢ | |cᵢ|·uᵢ | νᵢ |
|---|---|---|---|---|---|---|---|---|
| 1 | Repeatability (Type A) | A | normal | Type A · 10 repeats (Bessel formula) · df ν=n−1=9 | 0.020000 | 1 | 0.020000 | 9 |
| 2 | Reference standard (Type B) | B | normal | Type B · normal (certificate gives U; ÷ k: 95%→2, 99%→2.58) | 0.015000 | 1 | 0.015000 | 100 |
| 3 | Fixed point / calibration (Type B, uniform) | B | uniform | Type B · rectangular (half-width a, equiprobable) → u=a/√3 | 0.005800 | 1 | 0.005800 | 50 |
| 4 | Resolution (Type B, uniform) | B | uniform | Type B · rectangular (half-width a, equiprobable) → u=a/√3 | 0.002900 | 1 | 0.002900 | 50 |
“Basis of evaluation” states how each uᵢ is obtained (Type A = statistical, Type B = non-statistical); “|cᵢ|·uᵢ” feeds the combination and contribution analysis.
Reference · Type B distribution divisors (GUM / NABL-141)
- Rectangular (half-width a, equiprobable): u = a/√3
- Triangular (half-width a, mid more likely): u = a/√6
- Digital resolution ±d/2: treat as rectangular, u = (d/2)/√3
- Certificate gives expanded U: u = U/k (k is its coverage factor; 95%→k=2, 99%→k=2.58)
- Avoid “double-counting”: an effect already captured by the Type A statistical spread must not be re-entered as a Type B component.
4 · Combined standard uncertainty
u_c = √Σ(cᵢ·uᵢ)² = √((1×0.020000)² + (1×0.015000)² + (1×0.005800)² + (1×0.002900)²) = 0.025827
(rounded to 3 s.f.: u_c ≈ 0.0258)
Combination assumptions
① Input estimates are treated as uncorrelated (rᵢⱼ = 0) and combined by RSS; if any two components are strongly correlated (|r| > 0.5, e.g. one reference standard used for both calibration and check), the covariance form (GUM §5.2) must be used and this report re-checked. ② Non-linearity of the model is negligible for small input uncertainties (PTB/EURAMET practice); otherwise validate by MCM (JCGM 101).
5 · Effective degrees of freedom
ν_eff = u_c⁴ / Σ[(cᵢ·uᵢ)⁴ / νᵢ] = 24
Welch–Satterthwaite formula; an unknown component df is treated by its type default (Type A uses actual n−1, Type B treated as very large → approx. infinite).
6 · Coverage factor
k = 2 · Derived from the t-distribution at ν_eff (two-sided 95% confidence); a manually set k overrides this.
7 · Expanded uncertainty
U = k·u_c = 2 × 0.025827 = 0.051655
Rounded to at most two significant figures: U ≈ 0.052 mm.
8 · Report statement
The measured result of the measurand Length is reported as Y = …; the expanded uncertainty is U = 0.052 mm (k = 2, approximately 95% confidence). The result and its uncertainty are evaluated and expressed per GUM.
9 · Component contribution
| Component | Share | |cᵢ|·uᵢ |
|---|---|---|
| Repeatability (Type A) | 60.0% | 0.020000 |
| Reference standard (Type B) | 33.7% | 0.015000 |
| Fixed point / calibration (Type B, uniform) | 5.0% | 0.005800 |
| Resolution (Type B, uniform) | 1.3% | 0.002900 |
The following components were evaluated and found negligible for this u_c (variance share < 5%): Resolution (Type B, uniform) (1.3%). They are retained in the budget for traceability per GUM 4.1.2, not removed from the model.
10 · Dominant component & recommendations
The expanded uncertainty is dominated by Repeatability (Type A), contributing about 60%.
- Cutting this component by about half (e.g. more repeats to reduce a Type A component, or a higher-grade reference standard to reduce a Type B component) could bring U from 0.0517 down to about 0.0383 mm.
- Check each “Basis of evaluation” above: Type B components should cite a valid certificate/manual and use the correct distribution divisor (rectangular √3, triangular √6, normal per certificate); Type A degrees of freedom ν=n−1 must match the number of repeats.
- Prioritise the top one or two contributors — usually the cheapest way to materially shrink U.
11 · Conformity & reporting requirements (ISO/IEC 17025:2017 · 7.6)
- Uncertainty is evaluated per GUM (JCGM 100:2008); in calibration, CMC states measured quantity / method / range / U in the scope.
- Expanded uncertainty is given at approximately 95% confidence with the coverage factor k stated; relative expressions such as ppm/ppb are avoided.
- Results are reported as “y ± U” with a statement of how the uncertainty was obtained.
- U is rounded to at most two significant figures; u_c keeps one extra figure.
- CMC intervals are closed (no open intervals such as 0<U<x); reported uncertainty must not be smaller than the accredited CMC.
- Testing: uncertainty is evaluated and reported whenever it affects validity of results, client requirements, or conformity decisions.
Linked documents: FR-13 Uncertainty Evaluation Report · FR-38 Uncertainty Evaluation List.
Evaluation & Approval
| Role | Name / Signature | Date |
|---|---|---|
| Evaluated by | ||
| Reviewed by | ||
| Approved by |
Disclaimer: this tool is a preparation aid, not a metrological/calibration certificate. Verify against your national accreditation body’s guidance and the latest standard editions, and take responsibility for the model and its validation.
This result is an internal technical calculation draft, not a calibration or measurement certificate.
• Generic sample — none of your personalization inputs
• Not tailored to country / body / industry
• Watermarked on-screen preview
• Your country / body / industry, etc. woven in — expanded per your personalization inputs
• Tailored to your selected context
• Editable, watermark-free Word (.docx)
- Editable Word (.docx)
- Watermark-free
- Aligned to your shared context
❓ Beginner FAQ
What is uncertainty, really?
It is “how unsure the measurement is”. If you measure 100 N with U = 2 N, the true value is within 98–102 N with about 95% confidence.
Type A vs Type B?
Type A = computed from repeated readings (data speaks); Type B = taken from certificates/handbooks/experience (e.g. U÷k on a calibration certificate). Both feed the same combination formula — neither is “better”.
How do I read U in the report?
Results are written “y ± U”; k = 2 means about 95% confidence. U keeps at most 2 significant figures — rounding is handled automatically per GUM.
Free vs paid?
Free: full calculation + online preview (watermarked). Paid: your accreditation body / industry / stage context is injected, watermark-free Word export, and clause-level conformity notes.
This tool performs the GUM combination algebraically and an approximate Monte Carlo simulation in the browser, and re-computes existing reports against GUM rules. It is a preparation aid, not a metrological certificate. Verify against your national body’s guidance and the latest standard editions.