Table of Contents
What Is Error Propagation?
Error propagation (uncertainty propagation) determines how measurement uncertainties in input quantities affect the uncertainty of a calculated result. When you measure length with ±0.1 cm and width with ±0.2 cm, error propagation tells you the uncertainty in the calculated area.
This is essential in physics experiments, engineering design, analytical chemistry, and any field where measured quantities are combined through mathematical operations. The standard approach uses partial derivatives and assumes independent, normally distributed errors.
Propagation Rules
Summary Table
| Operation | Error Rule | Add Absolute/Relative |
|---|---|---|
| A + B | Quadrature sum | Absolute |
| A - B | Quadrature sum | Absolute |
| A × B | Relative quadrature | Relative |
| A / B | Relative quadrature | Relative |
| A^n | n × relative error | Relative |
Examples
If length = 10.0 ± 0.5 cm and width = 5.0 ± 0.3 cm: Area = 50.0 cm², relative error = √((0.5/10)² + (0.3/5)²) = 7.81%, so Area = 50.0 ± 3.9 cm².
Frequently Asked Questions
Why use quadrature (square root of sum of squares)?
Quadrature addition assumes errors are independent and random. Simply adding errors would overestimate uncertainty because it assumes all errors are in the same direction simultaneously, which is unlikely.
What if errors are correlated?
For correlated errors, include covariance terms: δf² = (df/dA)²δA² + (df/dB)²δB² + 2(df/dA)(df/dB)Cov(A,B). This calculator assumes independent errors.
Absolute vs relative error?
Absolute error has the same units as the measurement. Relative error is dimensionless (usually expressed as %). Use relative error for multiplication/division; absolute for addition/subtraction.