Error Propagation Calculator

What Is Error Propagation?

Error propagation estimates how measurement uncertainties combine when you add, subtract, multiply, or divide quantities. If you measure two lengths with known uncertainties, this tool computes the uncertainty of their sum, difference, product, or quotient. For converting between absolute and percent uncertainty on a single value, see the Absolute Uncertainty Calculator.

Propagation Formulas

For addition and subtraction, uncertainties combine in quadrature:

$$\Delta Z = \sqrt{(\Delta X)^2 + (\Delta Y)^2}$$

For multiplication and division, relative errors combine the same way:

$$\Delta Z = |Z| \cdot \sqrt{\left(\frac{\Delta X}{X}\right)^2 + \left(\frac{\Delta Y}{Y}\right)^2}$$

Example: Adding Two Rod Lengths

Rod A: $2.00 \pm 0.03$ m. Rod B: $0.88 \pm 0.04$ m. Sum $Z = 2.88$ m. Propagated uncertainty: $\Delta Z = \sqrt{0.03^2 + 0.04^2} = 0.05$ m. Result: $2.88 \pm 0.05$ m.

Related Tools

• Percent Error Calculator
• Accuracy Calculator