Propagation of Uncertainty Calculator

Propagation of Uncertainty Formula:

\[ \sigma_f = \sqrt{\left(\frac{\partial f}{\partial x} \cdot \sigma_x\right)^2 + \left(\frac{\partial f}{\partial y} \cdot \sigma_y\right)^2} \]

Propagated Uncertainty (σf):

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1. What is Propagation of Uncertainty?

Propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties on the uncertainty of a function based on them. When variables are measured with uncertainty, this calculator determines how that uncertainty propagates through calculations.

2. How Does the Calculator Work?

The calculator uses the propagation of uncertainty formula:

\[ \sigma_f = \sqrt{\left(\frac{\partial f}{\partial x} \cdot \sigma_x\right)^2 + \left(\frac{\partial f}{\partial y} \cdot \sigma_y\right)^2} \]

Where:

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

Explanation: The formula combines the individual uncertainties from each variable, weighted by how sensitive the function is to each variable (the partial derivatives).

3. Importance of Uncertainty Calculation

Details: Understanding how uncertainties propagate through calculations is essential in experimental sciences, engineering, and any field where measurements are involved. It helps determine the reliability of final results.

4. Using the Calculator

Tips:

Enter the partial derivatives of your function with respect to each variable
Enter the known uncertainties (standard deviations) for each measurement
All uncertainties must be non-negative values
The calculator handles two variables, but can be extended to more

5. Frequently Asked Questions (FAQ)

Q1: When should I use this formula?
A: Use it when you need to determine how measurement uncertainties affect the uncertainty of a calculated result, especially when variables are independent.

Q2: What if my variables are correlated?
A: For correlated variables, you need to include covariance terms in the uncertainty calculation.

Q3: Can this handle more than two variables?
A: The principle extends to any number of variables - just add more terms under the square root.

Q4: What units does the uncertainty have?
A: The uncertainty has the same units as your function f. Make sure all partial derivatives and uncertainties are in consistent units.

Q5: How accurate is this method?
A: It provides a first-order approximation that works well when uncertainties are relatively small compared to the measured values.