How To Calculate Propagated Error

Propagated Error Formula:

\[ \delta z = \left|\frac{\partial z}{\partial x}\right| \delta x + \left|\frac{\partial z}{\partial y}\right| \delta y \]

Partial Derivative (∂z/∂x):

Error in x (δx):

Partial Derivative (∂z/∂y):

Error in y (δy):

Propagated Error (δz):

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1. What Is Propagated Error?

Propagated error refers to the uncertainty in a calculated result that arises from uncertainties in the measured input variables. It quantifies how errors in measurements affect the final calculated value.

2. How Does The Calculator Work?

The calculator uses the propagated error formula:

\[ \delta z = \left|\frac{\partial z}{\partial x}\right| \delta x + \left|\frac{\partial z}{\partial y}\right| \delta y \]

Where:

\( \delta z \) — Propagated error in the result
\( \frac{\partial z}{\partial x} \) — Partial derivative of z with respect to x
\( \delta x \) — Error in measurement of x
\( \frac{\partial z}{\partial y} \) — Partial derivative of z with respect to y
\( \delta y \) — Error in measurement of y

Explanation: This formula calculates the maximum possible error by summing the absolute contributions from each variable's uncertainty.

3. Importance Of Error Propagation

Details: Understanding error propagation is crucial in scientific measurements, engineering calculations, and experimental physics to determine the reliability and precision of final results.

4. Using The Calculator

Tips: Enter the partial derivatives and corresponding measurement errors. All values must be valid numerical inputs with errors ≥ 0.

5. Frequently Asked Questions (FAQ)

Q1: When should I use propagated error analysis?
A: Use it whenever you perform calculations with measured values that have associated uncertainties, particularly in scientific and engineering applications.

Q2: What's the difference between absolute and relative error?
A: Absolute error is the actual uncertainty in units, while relative error is the uncertainty expressed as a fraction or percentage of the measured value.

Q3: Can this method handle more than two variables?
A: Yes, the formula extends to multiple variables: δz = |∂z/∂x₁|δx₁ + |∂z/∂x₂|δx₂ + ... + |∂z/∂xₙ|δxₙ

Q4: When is the root-sum-square method preferred?
A: For independent random errors, the root-sum-square method (δz = √[(∂z/∂x δx)² + (∂z/∂y δy)²]) often provides a more realistic error estimate.

Q5: How do I determine measurement errors?
A: Measurement errors can come from instrument precision, calibration uncertainty, reading errors, or statistical variations in repeated measurements.