EVERYDAY MATH
Measurement uncertainty calculator
Combine independent standard uncertainty components with sensitivity coefficients and inspect each component's variance contribution.
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Enter standard uncertainty components and their sensitivity coefficients. Unused rows may remain zero.
Combined measurement uncertainty
- Combined variance
- 0.08
- Combined standard uncertainty uc
- 0.28284271
- Dominant contributor
- Component 1
| Component | Signed cᵢuᵢ | Variance contribution | Share |
|---|---|---|---|
| 1 | 0.2 | 0.04 | 50% |
| 2 | 0.2 | 0.04 | 50% |
| 3 | 0 | 0 | 0% |
| 4 | 0 | 0 | 0% |
This first-order root-sum-square model assumes independent, uncorrelated inputs and supplied sensitivity coefficients. It does not include covariance, coverage factors or a measurement model.
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Verification and calculation details
- Purpose
- Measurement uncertainty calculator
- Variables and units
- Use the labelled fields and units shown in the calculator. Results use those units unless a conversion is stated.
- Formula and method
- The exact formula and worked example are shown below. Deterministic formula executed locally in the browser.
- Assumptions
- Only entered values and stated assumptions are included. Local fees, rules and professional requirements are excluded unless explicitly shown.
- Engine version
- 1.0.0
- Production verification baseline
- Release 63 · source
998803f - Verification run
release-v63-2026-09-20- Last verified
- — Automated release checks passing
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How this calculator works
For independent, uncorrelated inputs with standard uncertainties u(xᵢ) and sensitivity coefficients cᵢ=∂f/∂xᵢ, the variance contribution is [cᵢu(xᵢ)]² and the combined standard uncertainty is u_c(y)=√Σ[cᵢu(xᵢ)]².
Variables and inputs
- Standard uncertainty components
- Required numeric input. 1 to 20 components; finite; non-negative; no more than 1 trillion each.
- Sensitivity coefficients
- Required numeric input. one per component; finite; signed; absolute value no more than 1 trillion.
How to interpret the result
The calculator returns combined variance, combined standard uncertainty, signed component contributions, variance contribution percentages, dominant contributor. Read each output against the assumptions you entered, compare alternatives one variable at a time, and use the result as an estimate rather than a promise.
Common mistakes to avoid
- Mixing units or entering a rate as a decimal when the field requests a percentage.
- Rounding intermediate values instead of the final displayed result.
- Leaving out a cost, timing rule or local requirement listed in the assumptions below.
Practical uses
Use the measurement uncertainty calculator to check a real scenario, compare a changed assumption and explain the arithmetic before relying on a quote, specification or decision.
Worked example
For u₁=0.2 with c₁=1 and u₂=0.1 with c₂=2, both signed contributions are 0.2. Each variance contribution is 0.04, combined variance is 0.08 and combined standard uncertainty is √0.08≈0.28284271.
Assumptions and limitations
- all entered uncertainty components are standard uncertainties
- inputs are independent and uncorrelated
- sensitivity coefficients are correct for the user's measurement model
- the first-order linear approximation is adequate near the estimate
- all contributions resolve to one consistent output unit
- correlated inputs require covariance terms and are outside this calculator's scope
- does not derive sensitivity coefficients or construct a measurement model
- does not calculate expanded uncertainty or choose a coverage factor
- strongly nonlinear models may require a different propagation method
- results do not replace a laboratory's documented uncertainty budget or accreditation requirements
Important notes
Results are estimates based on the values entered. Confirm rates, prices, units and professional requirements that apply in your location. Review the Answer Atlas Calculation Standard and corrections policy for formula, privacy and correction rules.
Frequently asked questions
What is a sensitivity coefficient?
It describes how much the output changes for a small change in one input. Enter the coefficient from your measurement model in units consistent with the desired output.
Can I enter a negative sensitivity coefficient?
Yes. The signed contribution may be negative, but its variance contribution is squared and therefore non-negative.
Can this calculator combine correlated inputs?
No. Correlated inputs require covariance terms. This calculator deliberately covers only independent, uncorrelated components.