The propagation of uncertainties is a method used to determine how errors or uncertainties in the measured or input quantities of a mathematical expression affect the uncertainty in the final result. It’s a fundamental concept in experimental science, engineering, and other fields where measurements and calculations are involved. The goal is to provide a quantitative estimate of the uncertainty in the final result based on the uncertainties in the input values.
Procedure to Calculate Uncertainty in Result
- Express the mathematical relationship: Start with the mathematical expression that relates the input variables (X, Y, Z, etc.) to the output variable (W), such as W = f(X,\ Y,\ Z).
- Determine the partial derivatives: Calculate the partial derivatives of the output variable (W) with respect to each input variable (X, Y, Z), denoted as \frac{\partial{W}}{\partial{X}}, \frac{\partial{W}}{\partial{Y}}, \frac{\partial{W}}{\partial{Z}},. These represent how changes in each input affect the output.
- Find the uncertainties: Determine the uncertainties (standard deviations or measurement errors) associated with each input variable (\Delta{X}, \Delta{Y}, \Delta{Z},).
- Calculate the uncertainty in the output: Use the following formula to calculate the uncertainty (ΔW) in the output variable (W):
\begin{aligned} \Delta{W} &= \sqrt{(\frac{\partial{W}}{\partial{X}}\times \Delta{X})^2 + (\frac{\partial{W}}{\partial{Y}}\times \Delta{Y})^2 + (\frac{\partial{W}}{\partial{Z}}\times \Delta{Z})^2}\\\end{aligned}
It’s important to note that this method assumes that the uncertainties in the input variables are independent and normally distributed. Also, the calculated uncertainty is typically given as a standard deviation or a percentage of the measured value.
Here’s a simple example:
Suppose you want to calculate the area (A) of a rectangle, given its length (L = 100\ \rm cm) and width (W = 50\ \rm cm).
The formula for the area is:
\begin{aligned}A &= L \times W\\\end{aligned}
If you know that \Delta{L} = 0.1\ \rm cm and \Delta{L} = 0.2\ \rm cm, and you want to find \Delta{A}, you can use the propagation of uncertainties:
- Express the mathematical relationship: A = L \times W.
- Determine the partial derivatives: \frac{\partial{A}}{\partial{L}} = W and \frac{\partial{A}}{\partial{W}} = L.
- Find the uncertainties: \Delta{L} = 0.1\ \rm cm and \Delta{W} = 0.2\ \rm cm.
- Calculate the uncertainty in A:
\begin{aligned}\Delta{A} &= \sqrt{(\frac{\partial{A}}{\partial{L}}\times \Delta{L})^2 + (\frac{\partial{A}}{\partial{W}}\times \Delta{W})^2 }\\&= \sqrt{(W \times \Delta{L})^2 + (L \times \Delta{W})^2 }\\&= \sqrt{(50\ \rm cm \times 0.1\ \rm cm)^2 + (100\ \rm cm \times 0.2\ \rm cm)^2 }\\&= \sqrt{425\ \rm cm^4}\\&= 20.6\ \rm cm^2\\\end{aligned}
This will give you the uncertainty in the calculated area, \Delta{A}.
Propagation of Error for the Expression x = a + b
When you have an expression x = a + b, and you want to calculate the propagation of error in x due to uncertainties or errors in a and b, you can use the following approach:
The uncertainty (\Delta{x}) in x is determined by considering the uncertainties (\Delta{a} and \Delta{b}) in a and b.
\begin{aligned}\Delta{x} &= \sqrt{\Delta{a}^2 + \Delta{b}^2}\\\end{aligned}
Here, \Delta{x} represents the propagated uncertainty or error in the calculated value of x. This approach assumes that the uncertainties in a and b are independent and random. If there’s any correlation between the errors in a and b, you would need to take that into account in the error propagation calculation.
This error propagation method is a simplified approach, and it assumes that the uncertainties in a and b are small compared to their values. For more complex expressions or situations where the relationship between variables is not linear, more advanced techniques like Taylor series expansion may be used to estimate error propagation.
Propagation of Error for the Expression x = a – b
When you have an expression x = a – b, and you want to calculate the propagation of error in x due to uncertainties or errors in a and b, you can use the following approach:
The uncertainty (\Delta{x}) in x is determined by considering the uncertainties (\Delta{a} and \Delta{b}) in a and b.
\begin{aligned}\Delta{x} &= \sqrt{\Delta{a}^2 + \Delta{b}^2}\\\end{aligned}
Here, \Delta{x} represents the propagated uncertainty or error in the calculated value of x. This approach assumes that the uncertainties in a and b are independent and random. If there’s any correlation between the errors in a and b, you would need to take that into account in the error propagation calculation.
This error propagation method is a simplified approach, and it assumes that the uncertainties in a and b are small compared to their values. For more complex expressions or situations where the relationship between variables is not linear, more advanced techniques like Taylor series expansion may be used to estimate error propagation.
Propagation of Error for the Expressions x = ab or x = \dfrac{a}{b}
In this case, you can use the formula for error propagation:
\begin{aligned}\Delta{x} &= |x| \times \sqrt{(\dfrac{\Delta{a}}{a})^2 + (\dfrac{\Delta{b}}{b})^2}\\\end{aligned}
Keep in mind that when dividing or multiplying variables, the fractional error in the result depends on the fractional errors in the input variables. In both cases, the error of the result will be the same.
Propagation of Error for the Expressions x = \dfrac{a^2b^3}{c^5}
In this case, you can use the formula for error propagation:
\begin{aligned}\Delta{x} &= |x| \times \sqrt{2\times (\dfrac{\Delta{a}}{a})^2 + 3\times (\dfrac{\Delta{b}}{b})^2+ 5\times (\dfrac{\Delta{c}}{c})^2}\\\end{aligned}