Error Propagation Calculator UK

Error Propagation Formula:

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

Partial Derivative ∂z/∂x:

Uncertainty Δx:

units of x

Partial Derivative ∂z/∂y:

Uncertainty Δy:

units of y

Propagated Uncertainty Δz:

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1. What Is Error Propagation?

Error propagation is a statistical method used to estimate the uncertainty in a derived quantity based on the uncertainties in the measured variables from which it is calculated. It follows the formula for combined standard uncertainty.

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

Explanation: This formula calculates the propagated uncertainty in z when z is a function of two independent variables x and y, each with their own uncertainties.

3. Importance Of Error Propagation

Details: Error propagation is essential in experimental sciences and engineering to determine the reliability of calculated results and to understand how measurement errors affect the final outcome.

4. Using The Calculator

Tips: Enter the partial derivatives and corresponding uncertainties. All values must be valid (uncertainties ≥ 0). The calculator will compute the propagated uncertainty in the same units as the input uncertainties.

5. Frequently Asked Questions (FAQ)

Q1: What types of measurements require error propagation?
A: Error propagation is used whenever calculated results depend on multiple measured variables with known uncertainties, common in physics, chemistry, and engineering experiments.

Q2: How do I determine the partial derivatives?
A: Partial derivatives are obtained by differentiating the function z(x,y) with respect to each variable while treating other variables as constants.

Q3: What if my function has more than two variables?
A: The formula extends to additional variables: \( \Delta z = \sqrt{ \sum \left( \frac{\partial z}{\partial x_i} \Delta x_i \right)^2 } \) for independent variables.

Q4: Are there limitations to this formula?
A: This formula assumes variables are independent and uncertainties are random and normally distributed. Correlated errors require additional covariance terms.

Q5: How should I report the final result with uncertainty?
A: Report as z ± Δz with appropriate units, typically rounding the uncertainty to one or two significant figures.