First-order propagation for independent inputs. Not a measurement uncertainty budget. The headline figure is a 1σ Taylor result that assumes your variables are uncorrelated and your uncertainties are small enough for the formula to be roughly linear across ±σ. The hard part of an uncertainty analysis is the numbers you typed in, and this page cannot check any of them. Estimates only, verify before you rely on them. Open for the full scope limits.
What this tool actually does
Parses the formula, differentiates it symbolically, and combines each variable's contribution in quadrature as first-order propagation for independent variables. It also runs an optional Monte Carlo sample of the same formula as a cross-check, and reports which variable dominates the total.
Propagation is the easy half. Deciding what each input's uncertainty actually is, and whether it is Type A from repeated observation or Type B from a datasheet, a calibration certificate or judgement, is the half that decides whether the answer means anything.
The assumptions, and where they break
- Independence. Correlated inputs are not modelled. If two variables come from the same instrument, the same calibration, the same reference or the same operator, they are correlated, and the quadrature sum is wrong in a direction this page cannot tell you.
- Linearity across ±σ. First order is accurate when uncertainties are small relative to the values. For a strongly nonlinear formula or a large uncertainty, enable the Monte Carlo check; where it disagrees with the Taylor result, trust the sampled spread.
- Coverage factor. The expanded figure is the 1σ result multiplied by the k you selected. Reading k = 2 as 95% assumes the output is approximately normal and the effective degrees of freedom are large. With few observations or one dominant Type B contribution, neither holds, and the honest route is a Welch-Satterthwaite effective degrees of freedom and a Student t coverage factor, which this page does not compute.
- Uncertainty is not error. A systematic bias, a miscalibration, a wrong reference value or a mistake in the formula is not an uncertainty and does not propagate this way. A tight ± on a wrong number is still a wrong number.
- Distribution shape. Entering a half-width from a datasheet as though it were a standard uncertainty overstates it by roughly √3 for a rectangular distribution, which is one of the most common mistakes in an uncertainty budget.
Never use this for
- An accredited measurement uncertainty budget or a calibration certificate under ISO/IEC 17025 or the GUM (JCGM 100:2008), which require a documented model, stated coverage, effective degrees of freedom and traceability this page has none of.
- Conformity decisions against a specification or a regulatory limit, which need an agreed decision rule such as those in ISO/IEC 17025 clause 7.1.3 and ILAC-G8.
- Clinical laboratory result reporting, or any measurement a diagnosis or treatment rests on.
- Safety margins, tolerance stack-ups or acceptance criteria that a design or a purchase depends on.
- Published results, submissions or test reports, without an independent recalculation.
What it is checked against
The parser, the symbolic derivatives, the quadrature and the Monte Carlo path are pinned by the test suite against hand calculations, including a worked density example. That proves the arithmetic reproduces first-order propagation. It proves nothing about your model, your input uncertainties, or whether the variables you combined are independent.
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.
Type a formula above to list its variables.
First order, independent inputs, 1σ. This assumes the variables are uncorrelated and the formula is roughly linear across ±σ. Correlated inputs are not modelled, and a coverage factor of 2 only means 95% if the output is approximately normal with large effective degrees of freedom. For a nonlinear formula or a large uncertainty, run the Monte Carlo check and trust the sampled spread where the two disagree.