Propagation Of Error Calculator

Error Propagation Formula:

\[ \delta z = \sqrt{ \left( \frac{\partial z}{\partial x} \delta x \right)^2 + \left( \frac{\partial z}{\partial y} \delta y \right)^2 } \]

Propagated Error δz:

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1. What Is Error Propagation?

Error propagation is a statistical method used to estimate the uncertainty of a derived quantity based on the uncertainties of the input variables. It's essential in experimental sciences where measurements have inherent errors.

2. How Does The Calculator Work?

The calculator uses the error propagation formula:

\[ \delta z = \sqrt{ \left( \frac{\partial z}{\partial x} \delta x \right)^2 + \left( \frac{\partial z}{\partial y} \delta y \right)^2 } \]

Where:

\( \frac{\partial z}{\partial x} \) — Partial derivative of z with respect to x
\( \frac{\partial z}{\partial y} \) — Partial derivative of z with respect to y
\( \delta x \) — Uncertainty in x measurement
\( \delta y \) — Uncertainty in y measurement

Explanation: This formula calculates the combined uncertainty in z when z is a function of two independent variables x and y with known uncertainties.

3. Importance Of Error Propagation

Details: Understanding error propagation is crucial for determining the reliability of experimental results, comparing theoretical predictions with measurements, and making informed decisions based on uncertain data.

4. Using The Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: When should I use error propagation?
A: Use error propagation when you need to calculate the uncertainty of a quantity that depends on other measured values with known uncertainties.

Q2: What if I have more than two variables?
A: The formula extends to multiple variables: \( \delta z = \sqrt{ \sum \left( \frac{\partial z}{\partial x_i} \delta x_i \right)^2 } \) for independent variables.

Q3: Are there limitations to this method?
A: This method assumes uncertainties are independent and normally distributed. It works best for small uncertainties and linear approximations.

Q4: How do I determine the partial derivatives?
A: Partial derivatives are calculated from the mathematical relationship between z and the input variables. They represent how z changes with each variable.

Q5: Can this handle correlated errors?
A: This calculator uses the formula for independent errors. For correlated errors, additional covariance terms are needed in the formula.