Probably the most general error-propagation technique is called Monte-Carlo analysis. You can use this technique to solve many difficult statistical problems. Calculating how SEs propagate through a formula for y as a function of x works like this:
Generate a random number from a normal distribution whose mean equals the value of x and whose standard deviation is the SE of x.
Plug the x value into the formula and save the resulting y value.
Repeat this step a large number of times.
The resulting set of y values will be your simulated sampling distribution for y.
Calculate the SD of the y values.
The SD of the simulated y values is your estimate of the SE of y. (Remember, the SE of a number is the SD of the sampling distribution for that number.)
You can perform these calculations very easily using the free program Statistics 101. With very little extra effort, this software can give you the confidence interval and even a histogram of the simulated areas. And simulation can easily and accurately handle non-normally distributed measurement errors.
Consider the example of estimating the SE of the area of a circle whose diameter is 2.3 cm, with a SE of 0.2 cm. The formula for the area of a circle, in terms of its diameter (d) is A = (π/4)r2
This problem can easily be solved by simulation, using the Statistics 101 program. The program (only four lines long) generates the output shown. The SE of the coin area from this simulation is about 0.72, in good agreement with the value obtained by the other methods.
![[Credit: Screenshot courtesy of John C. Pezzullo, PhD]](https://www.dummies.com/wp-content/uploads/439888.image0.jpg)