Understanding Error Propagation
Educational Tool
When you measure physical quantities and use them in calculations, each measurement has some uncertainty. Error propagation tells us how these uncertainties combine to affect the final result. This is essential for scientific experiments, engineering analysis, and any quantitative work where measurement precision matters.
Sum/Difference Formula
σ_f = √(σ₁² + σ₂² + σ₃² + ...)
For sums and differences, add the variances (σ²) of each term, then take the square root. The sign (+ or −) in the formula doesn't affect the uncertainty—each term contributes the same to the spread.
Example: Perimeter = L₁ + L₂ + L₃
Product/Quotient Formula
(σ_f/|f|)² = (σ₁/x₁)² + (σ₂/x₂)² + (σ₃/x₃)²
For products and quotients, add the squared relative uncertainties, then take the square root. Multiply by |f| to get absolute uncertainty.
Example: Density = mass / volume
Power Formula
σ_f/|f| = |n| × σ_x/|x|
The exponent multiplies the relative uncertainty. Squaring (n=2) doubles it; square root (n=0.5) halves it. High powers can amplify small errors significantly.
Example: Area of circle = π × r²
Key Concepts
- Absolute uncertainty (σ): Same units as measurement (e.g., ±0.1 cm)
- Relative uncertainty: σ/|x| (dimensionless fraction)
- Percent uncertainty: Relative × 100%
- Variance (σ²): Square of standard deviation
- Independence: Variables must be uncorrelated for these formulas
Assumptions & Limitations
- Independence: All measured quantities must be independent (uncorrelated). If measurements share systematic errors, use the covariance formula.
- Small uncertainties: These formulas are first-order approximations. They work best when σ/|x| is small (typically <10%).
- Linear approximation: For highly nonlinear functions or large uncertainties, Monte Carlo simulation may be more accurate.
- Gaussian assumption: Results assume normally distributed errors. Non-normal distributions may require different treatment.
Frequently Asked Questions
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