# Measurement Uncertainty Calculator

> GUM first-order uncorrelated uncertainty budget with entered sensitivities and coverage factor. Coverage probability is not inferred.

Page: https://freecalculator.tech/tools/measurement-uncertainty-calculator
Last reviewed: 2026-10-23

## Formula

For uncorrelated inputs: u_c = √Σ(cᵢuᵢ)²; expanded uncertainty U = k u_c.

- Methodology type: scientific equation.
- Methodology: JCGM 100:2008 GUM §§5–6, first-order propagation restricted to uncorrelated input quantities.
- Observations on this page are input standard uncertainties uᵢ. Sensitivity coefficients must convert each contribution into the output's unit.
- Correlated inputs require covariance terms, omitted only under the declared uncorrelated-input assumption. Significant nonlinearity needs a different method.
- k = 2 does not universally guarantee 95% coverage; the coverage basis is external to this calculator.
- For uncorrelated inputs: u_c = √Σ(cᵢuᵢ)²; expanded uncertainty U = k u_c.

## Inputs

- Input standard uncertainties uᵢ (not raw observations)
- Sensitivity coefficients cᵢ in matching order
- Declared coverage factor k (not a confidence percentage)

## Worked example: Combined standard uncertainty

u_c = √(3² + 4²) = 5 output units. U = 2 × 5 = 10 output units; no coverage probability inferred.
1. u_c = √(3² + 4²) = 5 output units.
2. U = 2 × 5 = 10 output units; no coverage probability inferred.

## Common questions

### Are these raw observations?

No. Enter already evaluated standard uncertainties and matching sensitivity coefficients.

### What if inputs are correlated?

This version does not support covariance. Do not use it when correlations contribute materially.

### Does k = 2 mean exactly 95%?

No. Coverage depends on the output distribution, effective degrees of freedom and justified coverage method.

### How do I use the Measurement Uncertainty Calculator?

Enter the required values for Input standard uncertainties uᵢ (not raw observations), Sensitivity coefficients cᵢ in matching order, Declared coverage factor k (not a confidence percentage). The calculator applies the formula on this page and shows the main result with any supporting breakdown so you can check the arithmetic.

### What formula does the Measurement Uncertainty Calculator use?

For uncorrelated inputs: u_c = √Σ(cᵢuᵢ)²; expanded uncertainty U = k u_c. The visible formula section above lists the calculation path and the edge cases the page handles, so the result can be checked without relying on the form alone.

### Can the Measurement Uncertainty Calculator be used for exact decisions?

Use it as a calculation aid, not as a substitute for checking the underlying rule, contract, policy, or professional advice that applies to your situation. When a result depends on local rules, personal details, prices, or dates, enter those values yourself and confirm them before acting.

## Limitations and assumptions

- Uncorrelated first-order uncertainty budget only.
- No covariance, nonlinear Monte Carlo or effective degrees-of-freedom calculation.
- For education and operational analysis only. No diagnosis, treatment recommendations, prescribing, dosing instructions, or reimbursement entitlement. Scientific calculations require appropriate laboratory verification; operational estimates do not replace local clinical, safety or financial policies.
- The result depends on the values you enter for this measurement uncertainty calculator; it does not supply missing rates, rules, prices, dates, or assumptions for you.

## Sources

- JCGM / BIPM — https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf

## Cite this page

- APA: Free Calculator. (2026, October 23). Measurement Uncertainty Calculator. https://freecalculator.tech/tools/measurement-uncertainty-calculator
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