Propagated Error Calculator Python

Propagated Error 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:

Δx (Uncertainty in x):

units of x

Partial Derivative ∂z/∂y:

Δy (Uncertainty in y):

units of y

Propagated Error Δz:

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1. What Is Propagated Error?

Propagated error refers to the uncertainty in a calculated result that arises from uncertainties in the input measurements. It quantifies how errors in measured variables affect the final computed value.

2. How Does The Calculator Work?

The calculator uses the standard 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 combined uncertainty in z when it depends on multiple variables with known uncertainties.

3. Importance Of Error Propagation

Details: Understanding error propagation is essential in scientific measurements, engineering calculations, and data analysis to determine the reliability and precision of computed results.

4. Using The Calculator

Tips: Enter the partial derivatives and corresponding uncertainties. All values must be valid numbers with uncertainties ≥ 0.

5. Frequently Asked Questions (FAQ)

Q1: What if my function has more than two variables?
A: The formula can be extended to include additional terms: \( \Delta z = \sqrt{\sum (\frac{\partial z}{\partial x_i} \Delta x_i)^2} \)

Q2: When is this formula applicable?
A: This formula works best when measurement errors are independent and normally distributed.

Q3: How do I determine the partial derivatives?
A: Partial derivatives are calculated by differentiating your function with respect to each variable while treating other variables as constants.

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

Q5: Can this be implemented in Python?
A: Yes, this calculation can be easily implemented in Python using the math.sqrt() function for square root operations.