Propagation of Uncertainty Calculator

Formula

The formula to calculate the propagation of uncertainty is:

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

Where:

\( \Delta Q \) is the uncertainty in the calculated quantity \( Q \)
\( \frac{\partial Q}{\partial x}, \frac{\partial Q}{\partial y}, \frac{\partial Q}{\partial z}, \ldots \) are the partial derivatives of \( Q \) with respect to each variable \( x, y, z, \ldots \)
\( \Delta x, \Delta y, \Delta z, \ldots \) are the uncertainties in the measured quantities \( x, y, z, \ldots \)

What is Propagation of Uncertainty?

Propagation of uncertainty, also known as error propagation, refers to the effect on a function by a possible variation in the measurement of its variables. It is a method used in statistics and data analysis to determine how the uncertainty in the output can be quantified through the uncertainties of the inputs. This concept is crucial in fields like physics, engineering, and economics where precise measurements and predictions are necessary.

Example Calculation

Let's assume the following values:

Partial Derivative ∂Q/∂x: 2
Uncertainty in x (Δx): 0.5
Partial Derivative ∂Q/∂y: 3
Uncertainty in y (Δy): 0.2
Partial Derivative ∂Q/∂z: 1
Uncertainty in z (Δz): 0.1

Step 1: Calculate each term:

\[ \left(2 \times 0.5\right)^2 = 1, \quad \left(3 \times 0.2\right)^2 = 0.36, \quad \left(1 \times 0.1\right)^2 = 0.01 \]

Step 2: Sum the squares:

\[ 1 + 0.36 + 0.01 = 1.37 \]

Step 3: Take the square root of the sum:

\[ \Delta Q = \sqrt{1.37} \approx 1.17 \]

The uncertainty (ΔQ) is approximately 1.17.