Error Propagation Calculator

Calculate uncertainty propagation through mathematical operations with error analysis

Calculate Error Propagation

Operation Preview

(0 ± 0) + (0 ± 0) = Z ± ΔZ

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)