Propagated error

Definition

Propagated error is the measure of how uncertainties in variables affect the uncertainty in a function derived from those variables. It is often calculated using partial derivatives to approximate the total differential of the function.

5 Must Know Facts For Your Next Test

1. Propagated error can be approximated using differentials, specifically by summing up the absolute values of partial derivatives multiplied by their respective variable errors.
2. When dealing with multiple variables, the propagated error is influenced by the sensitivity of each variable as indicated by its partial derivative.
3. The propagated error formula for a function $f(x, y)$ is given by $df \approx \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$, where $dx$ and $dy$ are small changes in $x$ and $y$, respectively.
4. In practical scenarios, reducing measurement errors in highly sensitive variables (those with larger partial derivatives) can significantly reduce overall propagated error.
5. Understanding propagated error is essential for accurate predictions and reliable results in real-world applications involving measurements and approximations.

Review Questions

Related terms

Differentials: Small changes in a variable used to approximate changes in a function through linear approximation.

Partial Derivatives: Derivatives of functions with respect to one variable while keeping other variables constant.

$\Delta x$: $\Delta x$ represents a small change or increment in the variable $x$.