First-order propagation of measurement uncertainty
For a result z = f(x,y), the first-order law of propagation combines sensitivity coefficients with input standard uncertainties. When inputs are independent, covariance terms are zero and contributions combine by root sum of squares.
The method is a local linear approximation. Correlation, nonlinearity, asymmetric distributions, systematic effects, degrees of freedom, and coverage factors require a complete measurement model or Monte Carlo method.
How to use the uncertainty propagation calculator
- Select arithmetic operation: Choose addition, subtraction, multiplication, or division.
- Enter values and uncertainties: Use nonnegative standard uncertainties in the same units as their corresponding values.
- Calculate: Review the result and combined standard uncertainty.
- Check the model: Add covariance and other effects if inputs are correlated or the function is materially nonlinear.
Formula and variables
Addition and subtraction use √(uₓ²+uᵧ²); multiplication and division use the corresponding absolute derivative sensitivities.
u²(z) ≈ (∂f/∂x)²u²(x) + (∂f/∂y)²u²(y)- x, y — Input estimates
- Measured or estimated input quantities
- u(x), u(y) — Standard uncertainties
- One-standard-deviation-equivalent input uncertainties
- u(z) — Combined standard uncertainty
- First-order uncertainty of the calculated result
Product uncertainty
Multiply x = 10 ± 0.1 by y = 5 ± 0.2 with independent standard uncertainties.
- x
- 10 ± 0.1
- y
- 5 ± 0.2
- z = 10 × 5 = 50
- u(z) = √[(5 × 0.1)² + (10 × 0.2)²]
Result: z = 50 with standard uncertainty approximately 2.06.
Coverage or confidence language requires an appropriate distribution and coverage factor beyond this result.
Understanding your results
Report uncertainty with context
The output is a combined standard uncertainty under the stated independence and linearization assumptions.
- Round uncertainty appropriately and align the result’s decimal place.
- Addition and subtraction combine absolute uncertainties.
- Multiplication and division are often described with relative uncertainties, but the derivative form also handles a zero numerator.
Assumptions
- Inputs are independent and their standard uncertainties are correctly evaluated.
- First-order linearization is adequate near the estimates.
- The mathematical operation is the complete measurement model.
Limitations
- Does not include covariance, systematic correction models, effective degrees of freedom, coverage factors, confidence intervals, or Monte Carlo propagation.
- Does not support powers or arbitrary functions.
- Does not determine input uncertainties from raw replicate data.
Common mistakes
- Entering expanded uncertainty as though it were standard uncertainty.
- Ignoring correlation from shared calibration or environment.
- Adding uncertainties linearly without a reason.
- Reporting too many significant digits.
Practical use cases
Laboratory calculation checks
Combine two independent standard uncertainties through basic arithmetic.
Measurement education
Compare absolute derivative and relative-uncertainty forms.
Frequently asked questions
Why use root sum of squares?
For independent input effects represented by standard uncertainties, variances add under first-order propagation.
What if x and y are correlated?
Include covariance terms; this calculator intentionally assumes zero covariance.
Is ±u a 95% confidence interval?
Not automatically. Standard uncertainty is typically one-standard-deviation-equivalent; expanded coverage requires more assumptions and a coverage factor.
Sources and review
- Guide to the Expression of Uncertainty in Measurement — Introduction — Joint Committee for Guides in Metrology. Accessed 2026-07-13.
Reviewed 2026-07-13.