This free measurement uncertainty calculator builds a GUM uncertainty budget without a spreadsheet. Build a complete uncertainty budget in your browser. Add each source of uncertainty, pick its distribution, and the calculator converts it to a standard uncertainty, combines everything by root-sum-of-squares, works out the effective degrees of freedom, and gives the expanded uncertainty at about 95% confidence.
Uncertainty components
| Source | Type / distribution | Value | k or n | Sensitivity c |
|---|
Value meaning: Type A = standard deviation of repeated results (enter n); Normal = expanded uncertainty from a certificate (enter k); Rectangular or Triangular = half-width ±a (tolerance, resolution); Standard = standard uncertainty already known.
How to use the Measurement Uncertainty Calculator
- Choose the budget type, then enter the measured result and its unit.
- Add each component with + Add component: Type A from repeat data, or Type B from certificates, specifications or tolerances.
- Pick the coverage factor: a fixed k or one from the effective degrees of freedom.
- Press Calculate uncertainty to see the combined and expanded uncertainty.
Report the result from the measurement uncertainty calculator as x ± U, with k and the confidence level stated. Worked examples are in the Eurachem/CITAC guide Quantifying Uncertainty in Analytical Measurement.
How the uncertainty budget is calculated
This calculator follows the Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100) and the Eurachem/CITAC guide, which ISO/IEC 17025 laboratories use to estimate measurement uncertainty.
Choosing the right distribution
- Normal: calibration certificates and CRM certificates that state an expanded uncertainty with a coverage factor.
- Rectangular: tolerances with no further information, such as glassware class tolerances, balance and instrument resolution, or manufacturer specifications.
- Triangular: when values near the centre are more likely, for example glassware filled carefully to the mark.
- Type A: anything you measured repeatedly: repeatability, intermediate precision, or the SD from control charts.
Relative or absolute budget?
For chemistry results that come from multiplying and dividing (concentration = mass × purity ÷ volume and so on), working in relative uncertainties (%) is simplest, with sensitivity coefficients of 1. For additive models, or when components have different units, use an absolute budget with the correct sensitivity coefficients (the partial derivatives of your model equation).
Frequently asked questions
What confidence level does the expanded uncertainty give?
With k = 2 (or k from the t-distribution at 95.45%), the interval covers the true value with about 95% confidence, which is what ISO/IEC 17025 reports usually state.
When should I use k from the degrees of freedom instead of k = 2?
If a Type A component with few replicates dominates the budget, the effective degrees of freedom can be low and k should be larger than 2. The t-based option handles this automatically. With ν_eff above about 30, k is close to 2.
Should I include bias or recovery in the budget?
If you do not correct results for a known bias, many guides recommend including the uncertainty of the bias (or recovery) estimate as a component. Significant uncorrected bias should be investigated and corrected where possible.
Can I use my intermediate precision SD as a single component?
Yes. The top-down (Nordtest / Eurachem) approach combines long-term within-lab reproducibility with a bias component, often from CRMs or PT results. It is fully accepted for ISO/IEC 17025.
What is the expanded uncertainty formula?
U = k × uc, where uc = √(u₁² + u₂² + …) for independent components in the same units or relative terms. With k = 2 the interval covers about 95% for a normal distribution.
Can you give an uncertainty budget example?
For a result of 10.0 mg/L with relative components of 1.5% (precision), 1.0% (calibration standard) and 0.8% (volumetric), uc = √(1.5² + 1.0² + 0.8²) = 1.97%. U = 2 × 1.97 = 3.9%, so report 10.0 ± 0.4 mg/L (k = 2).
Related tools
For guidance only. Your uncertainty procedure must follow your accreditation body’s policy (for example ILAC P14 for calibration). Built by an ISO/IEC 17025 laboratory professional.