Uncertainty Propagation Calculator

General Uncertainty Propagation Formula:

\[ \text{Absolute Uncertainty} = \sqrt{\sum\left(\frac{\partial f}{\partial x_i} \times \Delta x_i\right)^2} \]

Function Value:

Absolute Uncertainty:

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1. What is Uncertainty Propagation?

Uncertainty propagation is the process of determining the uncertainty in a function's output based on the uncertainties in its input variables. It's essential in experimental sciences and engineering to understand the reliability of calculated results.

2. How Does the Calculator Work?

The calculator uses the general uncertainty propagation formula:

\[ \Delta f = \sqrt{\sum\left(\frac{\partial f}{\partial x_i} \times \Delta x_i\right)^2} \]

Where:

\( \Delta f \) — Uncertainty in the function result
\( \frac{\partial f}{\partial x_i} \) — Partial derivative of f with respect to variable x_i
\( \Delta x_i \) — Uncertainty in variable x_i

Explanation: The formula combines the effects of all input uncertainties through their partial derivatives, accounting for how each variable affects the final result.

3. Importance of Uncertainty Calculation

Details: Proper uncertainty analysis is crucial for understanding the reliability of experimental results, comparing measurements, and making informed decisions based on calculated values.

4. Using the Calculator

Tips:

Enter the mathematical function using standard operators (+, -, *, /, ^)
List all variables separated by commas
Provide values and their uncertainties in the same order as the variables
The calculator will compute both the function value and its uncertainty

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can this calculator handle?
A: The calculator can handle most differentiable functions of multiple variables, including polynomial, exponential, and trigonometric functions.

Q2: How are partial derivatives calculated?
A: The calculator uses symbolic differentiation to compute the partial derivatives automatically from your input function.

Q3: What if my variables are correlated?
A: This calculator assumes independent variables. For correlated variables, additional covariance terms would be needed in the uncertainty calculation.

Q4: Can I use this for non-differentiable functions?
A: The propagation formula requires differentiable functions. For non-differentiable cases, consider Monte Carlo methods.

Q5: What units should I use for uncertainties?
A: Use the same units as your input values. The output uncertainty will be in the units of your function result.