Propagate measurement uncertainties through addition, subtraction, multiplication, or division operations.
Last updated: March 2026
Error propagation (also called uncertainty propagation) is a technique used in science and engineering to determine how measurement uncertainties affect the results of calculations. When you measure physical quantities, each measurement has some degree of uncertainty due to instrument limitations, environmental factors, and human error.
When you combine measured values through mathematical operations (addition, multiplication, etc.), the uncertainties in the original measurements combine to produce an uncertainty in the final result. Error propagation allows you to calculate this final uncertainty rigorously, ensuring that reported results honestly reflect the precision of the underlying measurements.
This is critical in experimental sciences, engineering design, medical diagnostics, and any field where accurate quantification of measurement quality is essential for making informed decisions.
When adding or subtracting measurements, uncertainties combine in quadrature (root sum of squares):
When multiplying or dividing, relative (percentage) uncertainties combine in quadrature:
You measure resistance using Ohm's law: R = V / I
Random errors from independent measurements tend to cancel out partially. Root sum of squares (quadrature addition) reflects this statistical expectation, giving more realistic uncertainty estimates than assuming all errors are perfectly correlated.
For addition, absolute uncertainties combine. For multiplication, relative (fractional) uncertainties combine. This is because multiplying scales both the value and its uncertainty.
Round the uncertainty to 1-2 significant figures, then round the central value to the same decimal place. Example: 5.0 ± 0.1 (not 5.001 ± 0.13). Use the format: (value ± uncertainty) unit.
Standard error propagation assumes independent random uncertainties. If you have correlated uncertainties or complex distributions, you may need more advanced techniques like Monte Carlo error propagation.
Focus on the largest uncertainty contributors first. Identify which measurements dominate the final uncertainty through the propagation formula, then improve precision in those measurements.
This calculator handles linear combinations and products/quotients. For transcendental functions, use the calculus-based partial derivatives method or consult specialized error propagation guides.
Related Tools