Simple cases (assuming uncorrelated errors)
| Operation | Propagated error |
|---|---|
| Sum / difference z = x ± y | σ_z = √(σ_x² + σ_y²) |
| Product / quotient z = x·y or x/y | (σ_z/z)² = (σ_x/x)² + (σ_y/y)² |
| Power z = x^n | σ_z/z = |n| · σ_x/x |
| Log z = ln(x) | σ_z = σ_x / x |
| Exp z = e^x | σ_z / z = σ_x |
| Constant multiple z = c·x | σ_z = |c| · σ_x |
General formula
| Function of many vars | σ_z² ≈ Σ (∂f/∂xᵢ · σ_xᵢ)² + 2 Σ ∂f/∂xᵢ · ∂f/∂xⱼ · cov(xᵢ, xⱼ) |
|---|---|
| Uncorrelated | Covariance term = 0 → σ_z² = Σ (∂f/∂xᵢ · σ_xᵢ)² |
Correlation matters
- The formulas above assume uncorrelated errors — if variables are correlated, include the covariance term.
- Calibration drift, temperature, and voltage supply often introduce correlation.
- Monte Carlo with the joint distribution is the safest approach when correlations are messy.
Best practices
- Report uncertainty and unit with every measurement: "12.3 ± 0.2 V" not just "12.3 V".
- Use standard uncertainty (1σ) unless stated otherwise.
- Expanded uncertainty uses a coverage factor (k = 2 is common, ~95% confidence for normal).
- Always distinguish systematic (bias) from random (noise) errors.
Was this article helpful?