Error Propagation Calculator
Calculate uncertainty propagation through mathematical operations with error analysis
Calculate Error Propagation
Operation Preview
Enter values above to calculate error propagation
Example Calculations
Addition Example
Problem: Two rod lengths: (2.00 ± 0.03) m + (0.88 ± 0.04) m
Result: Z = 2.00 + 0.88 = 2.88 m
Error: ΔZ = √(0.03² + 0.04²) = √(0.0009 + 0.0016) = 0.05 m
Final: 2.88 ± 0.05 m
Division Example
Problem: Bird velocity: (120 ± 3) m ÷ (20 ± 1.2) s
Result: Z = 120 ÷ 20 = 6.0 m/s
Relative errors: ΔX/X = 3/120 = 0.025, ΔY/Y = 1.2/20 = 0.06
Error: ΔZ = 6.0 × √(0.025² + 0.06²) = 6.0 × 0.065 = 0.39 m/s
Final: 6.0 ± 0.4 m/s
Propagation Rules
Addition/Subtraction
ΔZ = √((ΔX)² + (ΔY)²)
Absolute errors combine
Multiplication/Division
ΔZ/Z = √((ΔX/X)² + (ΔY/Y)²)
Relative errors combine
Key Concepts
Uncertainties always propagate to increase total uncertainty
Errors combine in quadrature (square root of sum of squares)
Relative error is percentage uncertainty
Independent measurements assumed
Understanding Error Propagation
What is Error Propagation?
Error propagation occurs when you measure quantities with uncertainties and then calculate derived quantities. The uncertainties in the original measurements "propagate" through mathematical operations to affect the uncertainty in the final result.
Why is it Important?
- •Quantifies uncertainty in calculated values
- •Essential for scientific measurements
- •Helps assess reliability of results
- •Required for error analysis reporting
Mathematical Foundation
General Formula
ΔZ = √((∂f/∂X · ΔX)² + (∂f/∂Y · ΔY)²)
Where f(X,Y) is the function and ∂f/∂X are partial derivatives
Simplified Cases
Z = X ± Y: ΔZ = √((ΔX)² + (ΔY)²)
Z = X × Y: ΔZ/Z = √((ΔX/X)² + (ΔY/Y)²)
Z = X / Y: ΔZ/Z = √((ΔX/X)² + (ΔY/Y)²)