Uncertainty Propagation Calculator

Propagate ± Through a Formula

Enter a formula, give each variable a value and its uncertainty, and read off the result with a propagated 1σ error. Expand the steps to see the symbolic partial derivatives and which variable dominates.

Formula (use variable names, e.g. m / V)

Variables

NameValueUncertainty (±)Type

Options

Coverage factor (k)

Monte Carlo check

Cross-check by sampling

MC distribution

MC samples

Result

Show the calculation steps

How To Use This

Write the formula using variable names and standard operators.
Set each variable's value and its uncertainty. Pick absolute, percent, or exact (a constant with no error).
Read the result as value ± 1σ, then raise the coverage factor k for a wider confidence interval.
Expand the steps to see the partial derivatives and which variable drives the error.

Formula Syntax

Operators: + - * / ^ and ( ).

Constants: pi, e.

Functions:

explnlog10log2sqrtcbrtabs sincostanasinacosatan sinhcoshtanh

Variable names may include letters, digits, and underscores, e.g. m_out, T2.

The Math

For independent variables, first-order (Taylor) propagation gives

σ_f = √( Σ (∂f/∂xᵢ · σᵢ)² )

Each term ∂f/∂xᵢ · σᵢ is that variable's share of the total error, added in quadrature.

Scope & Limits

Estimates only - verify before you rely on them

This uses first-order propagation and assumes variables are independent (uncorrelated). It is accurate when uncertainties are small relative to the values, so the formula is roughly linear across ±σ.

For strongly nonlinear formulas or large uncertainties, enable the Monte Carlo check. When it disagrees with the Taylor result, trust the sampled spread. Correlated inputs are not modeled.

Provided as-is with no warranty of accuracy or fitness. Verify any result you depend on against an independent method and qualified engineering judgment before acting on it.