Error Propagation Calculator guide
Error Propagation Calculator: Combine Measurement Uncertainties
Use our error propagation calculator to find the uncertainty in a calculated result from the uncertainties in each measurement, using partial derivatives.
By the AllCalculator editorial teamUpdated 15 October 20262 min read
Every measurement has some uncertainty. When you combine measurements in a formula, their uncertainties combine too. Our Error Propagation Calculator works out the uncertainty in your result for any formula.
How to use the calculator (Error Propagation Calculator)
- Type your formula using single letters, for example x × y / z or m × g × h.
- Enter each value and uncertainty, like x = 2.0 ± 0.1, y = 3.5 ± 0.2.
You'll see the result with its uncertainty, the relative uncertainty and how much each measurement contributes.
The method
For independent uncertainties, the calculator uses the standard formula:
σf² = (∂f/∂x · σx)² + (∂f/∂y · σy)² + …
Each term is a measurement's uncertainty multiplied by how sensitive the result is to that measurement. They're added in quadrature (squared, summed, then square-rooted).
Shortcuts you might know
- Adding or subtracting: add the absolute uncertainties in quadrature.
- Multiplying or dividing: add the relative (percentage) uncertainties in quadrature.
- Powers: multiply the relative uncertainty by the power. For x², double it.
The calculator applies the general method, so it works for any formula, including ones with trig, logs and roots.
A worked example
With the calculator's default settings, f = xy/z with x = 2.0 ± 0.1, y = 3.5 ± 0.2 and z = 1.2 ± 0.05:
- f = 2.0 × 3.5 ÷ 1.2 = 5.83
- The relative uncertainties are 5%, 5.7% and 4.2%, which combine to 8.7%
- So f = 5.83 ± 0.51
The contributions table shows y has the biggest effect, so measuring it more precisely would help most.
For summary statistics of repeated measurements, see our Statistics Calculator.
Frequently asked questions
What if my uncertainties aren't independent?
This method assumes they are. Correlated uncertainties need extra covariance terms.
Should I round the uncertainty?
Usually to one or two significant figures, then round the result to the same decimal place.
What's the difference between absolute and relative uncertainty?
Absolute is in the same units as the value (± 0.1cm). Relative is a fraction or percentage of the value (± 5%).
Use the Error Propagation Calculator for lab reports and coursework.
Sources and further reading
- NPL: Good practice guides on measurementnpl.co.uk
- BIPM: Guide to the expression of uncertainty in measurement (GUM)bipm.org
Links open official and independent websites in a new tab.
This guide is general information to help you understand your options. It isn't financial, tax, legal or building advice and doesn't take your personal circumstances into account. Figures were checked when the guide was last updated and can change, so confirm anything important with GOV.UK, HMRC, a regulated adviser or a qualified tradesperson.