Uncertainty Calculator Wolfram Alpha

Uncertainty 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 in x (Δx):

units of x

Partial Derivative (∂z/∂y):

Uncertainty in y (Δy):

units of y

Uncertainty in z (Δz):

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1. What Is Uncertainty Propagation?

Uncertainty propagation is a method used to estimate the uncertainty in a calculated result based on the uncertainties in the input measurements. It follows the formula for error propagation in measurements.

2. How Does The Calculator Work?

The calculator uses the uncertainty 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 the measurement of x
\( \frac{\partial z}{\partial y} \) — Partial derivative of z with respect to y
\( \Delta y \) — Uncertainty in the measurement of y

Explanation: This formula calculates how uncertainties in input variables (x and y) propagate through a function to create uncertainty in the output (z).

3. Importance Of Uncertainty Calculation

Details: Calculating uncertainty is crucial in scientific measurements and engineering to understand the reliability of results and to make informed decisions based on experimental data.

4. Using The Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: When should I use uncertainty propagation?
A: Use it when you have calculated values derived from measurements with known uncertainties, particularly in physics, chemistry, and engineering experiments.

Q2: What if I have more than two variables?
A: The formula can be extended to include additional terms for each variable: \( \Delta z = \sqrt{\sum\left(\frac{\partial z}{\partial x_i} \Delta x_i\right)^2} \)

Q3: Are there limitations to this formula?
A: This formula assumes uncertainties are independent and random, and works best for small uncertainties relative to the measured values.

Q4: What if my function is not differentiable?
A: Numerical methods or alternative approaches may be needed for functions that aren't easily differentiable.

Q5: How precise are the results from this calculator?
A: The calculator provides results with 6 decimal places, but the actual precision depends on the precision of your input measurements.