1. What is Uncertainty Propagation?

Uncertainty propagation is a statistical method used to estimate the uncertainty in a function's output based on the uncertainties in its input variables. It follows the general formula for error propagation in measurements.

2. How Does the Calculator Work?

The calculator uses the uncertainty propagation formula:

Where:

• \( \frac{\partial f}{\partial x} \) — Partial derivative of function f with respect to x
• \( \sigma_x \) — Uncertainty in measurement x
• \( \frac{\partial f}{\partial y} \) — Partial derivative of function f with respect to y
• \( \sigma_y \) — Uncertainty in measurement y
• \( \sigma_f \) — Propagated uncertainty in the final result

Explanation: The formula calculates the combined uncertainty by considering how uncertainties in individual measurements affect the final result through their partial derivatives.

3. Importance of Uncertainty Analysis

Details: Uncertainty propagation is crucial in scientific measurements, engineering calculations, and experimental data analysis to determine the reliability and precision of calculated results.

4. Using the Calculator

Tips: Enter the partial derivatives and corresponding uncertainties. All values must be valid numerical values with uncertainties being non-negative.

5. Frequently Asked Questions (FAQ)

Q1: When should I use uncertainty propagation?
A: Use it when you have a function of multiple measured variables and want to determine the overall uncertainty in the result.

Q2: What if my function has more than two variables?
A: The formula extends to any number of variables: \( \sigma_f = \sqrt{ \sum \left( \frac{\partial f}{\partial x_i} \sigma_{x_i} \right)^2 } \)

Q3: Are there assumptions in this method?
A: Yes, it assumes uncertainties are independent, random, and normally distributed, and that the function is approximately linear near the measurement point.

Q4: What if the variables are correlated?
A: For correlated variables, additional covariance terms must be included in the propagation formula.

Q5: Can this handle non-linear functions?
A: The formula provides a first-order approximation. For highly non-linear functions, higher-order terms or Monte Carlo methods may be needed.