Uncertainty Propagation
Propagate measurement error through a formula, and see which quantity dominates the budget.
Expression
Named variables, the operators + − * / ^, brackets, the constants pi and e, and sqrt ln log exp sin cos tan asin acos atan abs. Write every multiplication out.
Examples
Quantities
one row per variable
This expression has no variables, so there is nothing to propagate.
Result
—
- Value
- —
- Standard uncertainty
- — ±
- Relative uncertainty
- — %
Uncertainty budget
share of the variance
Give at least one quantity an uncertainty to see where the error comes from.
What this assumes
The combined uncertainty is the first-order expression σf² = Σ (∂f/∂xᵢ)² σᵢ², with the partial derivatives taken numerically at the values you entered. It assumes the inputs are independent: there is no covariance term. Quantities read off the same instrument, calibrated against the same standard, or derived from one another are correlated, and their covariances would change the answer in either direction. Correlated inputs are out of scope here.
Being first order, it also assumes the formula is near enough linear across a few standard uncertainties. That holds for small errors on a smooth formula and breaks down near a stationary point or a singularity — where the tool says so rather than propagating a derivative it cannot trust.