Enter the partial derivatives of the calculated quantity with respect to each variable and the uncertainties in the measured quantities into the calculator to determine the propagation of uncertainty.

Propagation Of Uncertainty Formula

The following formula is used to calculate the propagation of uncertainty. 

ΔQ = √[(∂Q/∂x * Δx)² + (∂Q/∂y * Δy)² + (∂Q/∂z * Δz)² + ...]

Variables:

  • ΔQ is the uncertainty in the calculated quantity
  • Q∂Q/∂x, ∂Q/∂y, ∂Q/∂z, ... are the partial derivatives of Q with respect to each variable x, y, z,
  • ...Δx, Δy, Δz, ... are the uncertainties in the measured quantities x, y, z, ...

To calculate the propagation of uncertainty, first calculate the partial derivative of the calculated quantity with respect to each variable. Then, multiply each partial derivative by the uncertainty in the corresponding variable. Square each of these products and add them together. Finally, take the square root of the sum to get the uncertainty in the calculated quantity.

What is a 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.

How to Calculate Propagation Of Uncertainty?

The following steps outline how to calculate the Propagation of Uncertainty using the given formula:

  1. First, identify the partial derivatives (∂Q/∂x, ∂Q/∂y, ∂Q/∂z, ...) of the calculated quantity Q with respect to each variable (x, y, z, ...).
  2. Next, determine the uncertainties (Δx, Δy, Δz, ...) in the measured quantities (x, y, z, ...).
  3. Using the formula ΔQ = √[(∂Q/∂x * Δx)² + (∂Q/∂y * Δy)² + (∂Q/∂z * Δz)² + ...], calculate the uncertainty in the calculated quantity Q.
  4. Finally, evaluate the result to obtain the uncertainty in the calculated quantity Q.
  5. After inserting the values and calculating the result, check your answer for accuracy.

Example Problem:

Use the following variables as an example problem to test your knowledge:

∂Q/∂x = 2, ∂Q/∂y = 3, ∂Q/∂z = 4

Δx = 0.5, Δy = 0.2, Δz = 0.3