What error propagation means
In engineering measurements, a result often depends on several measured quantities. If the measured
quantities have uncertainty, then the calculated result also has uncertainty.
\[
y=f(x_1,x_2,\ldots,x_n)
\]
Error propagation estimates how uncertainty in \(x_1,x_2,\ldots,x_n\) affects the uncertainty in \(y\).
Central value
First, evaluate the formula using the central measured values.
\[
y_0=f(x_{1,0},x_{2,0},\ldots,x_{n,0})
\]
This gives the best estimate of the output before uncertainty is added.
Partial derivatives
A partial derivative measures how sensitive the result is to one input while the other inputs are held fixed.
\[
\frac{\partial f}{\partial x_i}
\]
If this derivative is large, a small uncertainty in \(x_i\) can create a large uncertainty in the final result.
Statistical uncertainty propagation
If the input uncertainties are independent, the standard uncertainty of the result is commonly estimated by
root-sum-square combination:
\[
u_y
=
\sqrt{
\left(\frac{\partial f}{\partial x_1}u_1\right)^2
+
\left(\frac{\partial f}{\partial x_2}u_2\right)^2
+
\cdots
+
\left(\frac{\partial f}{\partial x_n}u_n\right)^2
}.
\]
Here \(u_i\) is the standard uncertainty of input \(x_i\).
Expanded uncertainty
Many engineering reports give an expanded uncertainty:
\[
\Delta y = k u_y.
\]
The factor \(k\) is called the coverage factor. A common engineering choice is \(k=2\), which is often used
as an approximate 95% coverage level when the distribution is close to normal.
Worst-case uncertainty
A conservative estimate adds absolute contributions instead of combining them statistically:
\[
\Delta y_{\max}
\approx
\sum_i
\left|
\frac{\partial f}{\partial x_i}
\right|
\Delta x_i.
\]
This method assumes the input errors all act in the direction that makes the result as large as possible.
It is more conservative than RSS propagation.
Direct max/min corner testing
Another way to estimate maximum and minimum possible outputs is to test combinations of input bounds:
\[
x_i = x_{i,0}\pm \Delta x_i.
\]
The calculator evaluates the formula at the corners of the uncertainty box and reports the smallest and
largest finite output found.
Monte Carlo simulation
Monte Carlo propagation samples each input many times from its uncertainty distribution, then evaluates the
formula for every sample.
\[
x_i \sim \text{selected distribution},
\qquad
y=f(x_1,x_2,\ldots,x_n).
\]
The resulting collection of \(y\)-values estimates the output distribution. This is especially useful for
nonlinear formulas.
Example: resistance from voltage and current
If resistance is calculated from
\[
R=\frac{V}{I},
\]
then the partial derivatives are
\[
\frac{\partial R}{\partial V}
=
\frac{1}{I},
\qquad
\frac{\partial R}{\partial I}
=
-\frac{V}{I^2}.
\]
Therefore,
\[
u_R
=
\sqrt{
\left(\frac{u_V}{I}\right)^2
+
\left(\frac{V u_I}{I^2}\right)^2
}.
\]
Sensitivity and variance contribution
Each input contributes a term to the output variance:
\[
c_i
=
\left(
\frac{\partial f}{\partial x_i}u_i
\right)^2.
\]
A large contribution means that improving that measurement would reduce the output uncertainty the most.
Common mistakes
- Adding uncertainties directly when a statistical RSS method is required.
- Using percent uncertainty as if it were absolute uncertainty.
- Forgetting units in the final uncertainty.
- Ignoring the largest sensitivity contributor.
- Using linear propagation for a strongly nonlinear formula without checking Monte Carlo results.
- Confusing standard uncertainty \(u\) with expanded uncertainty \(ku\).
- Assuming all input errors are independent when they are actually correlated.
