Laboratory and research measurements

Measurement Uncertainty Calculator

Estimate how input measurement uncertainties affect a calculated result. Research Mode includes propagation for common arithmetic operations and selected functions, with an option to account for correlation between inputs.

Open Uncertainty Tools

What you can do

  • Combine two measured values while retaining their standard uncertainties.
  • Use correlation coefficients when measurement errors are not independent.
  • Propagate uncertainty through arithmetic and selected mathematical functions.
  • See the result as an estimated value and a standard uncertainty, not a guarantee of accuracy.

Formula and interpretation

For independent measurements a ± u(a) and b ± u(b), the standard uncertainty of their sum is u(a+b)=√[u(a)²+u(b)²]. With correlation coefficient ρ, the variance includes +2ρu(a)u(b). Correlation is a property of the input measurements, not an adjustment to make results look more precise.

Worked uncertainty propagation

Two independent lengths are a = 12.0 ± 0.2 mm and b = 8.0 ± 0.1 mm, with uncertainties interpreted as standard uncertainties. Their sum is 20.0 mm. The combined standard uncertainty is √(0.2²+0.1²)=√0.05≈0.224 mm. A sensible result is (20.0 ± 0.22) mm, before considering calibration bias or expanded coverage.

How to use the workspace

  1. Open Research Mode and locate Measurement Uncertainty.
  2. Select addition and enter values 12.0 and 8.0 with uncertainties 0.2 and 0.1.
  3. Set correlation to 0 only if treating the measurements as independent.
  4. Run the calculation and report the value, units, and uncertainty convention together.

Assumptions and common errors

  • Not the same as tolerance: standard uncertainty and engineering design tolerance have different interpretations.
  • Systematic effects: calibration bias and omitted uncertainty sources are not removed by propagation.
  • Distribution and coverage: reporting a 95% interval requires assumptions beyond simply displaying one standard uncertainty.

Frequently asked questions

Does zero correlation mean independent measurements?
Independence implies zero correlation in this context, but zero correlation alone does not prove independence.
Can uncertainty be negative?
No. Standard uncertainty is nonnegative; the signed error of an observation is a different concept.
Does this calculate an expanded uncertainty automatically?
The core propagation reports standard uncertainty; choose and justify any coverage factor separately.

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