# Monte Carlo Error Propagation

Learning Objectives

How to apply the concepts of Monte Carlo to propagate errors.

# Review of the assumptions of our data

Review of assumptions of the data that we are working under

1. The set of the infinite number of possible measurements of a continuous variable like thickness will be a normal distribution.
• Mean μ
• Standard deviation σ
2. The mean of our sample of size N is a good estimate of the mean of the population μ.

3. The standard deviation of our sample is a good estimate of the standard deviation of our population.

Below you can see our example data that we’ve been using throughout this lab: 10 measurements of radius and 10 measurements of the height or thickness. The mean and standard deviations previously calculated are also shown. We are assuming that these measurements are independent: that the thickness of the of the nickel and its radius are not correlated with each other in any way. For example, in observation number six, the radius is above the mean while the height is actually below the mean. This is what we mean when we say that they’re independent: just because the radius is high doesn’t necessarily mean that the thickness is also high.

 Observation Radius [cm] Height [cm] 1 1.060 0.190 2 1.055 0.200 3 1.050 0.180 4 1.060 0.190 5 1.055 0.200 6 1.050 0.150 7 1.025 0.150 8 1.050 0.150 9 1.050 0.150 10 1.025 0.170 Mean 1.048 0.176 Std. Dev. 0.013 0.020
Now, let’s go back to our assumptions. The first assumption is that all the possible true values of these continuous variables of radii and height are from normal distribution. For the radii, it will have a mean of 1.048cm and a standard deviation of 0.013cm (the figure on the left below), while the heights will fill out a normal distribution of mean 0.176cm and thickness 0.020cm (figure on the right).

# Principles of Monte Carlo Error propagation

Now, let’s talk about the principles of Monte Carlo error propagation. The basic idea is you choose randomly from the known distributions, in our case these Normal distributions for height and thickness, and then do your calculation. After you’ve calculated you add your result to a table and begin to build up a sample of results of your calculation: one entry for each set of random values that you’ve chosen. Do that a whole mess of times, as many times as you basically have time for, and that leaves you with a sample of results of your calculation from which you can measure the mean and standard deviation of this sample of answers. The mean of the sample of answers is your central value and the standard deviation is your uncertainty.

In the case of our data

In our example, we are going to choose a random value for each variable: we’re going to choose a random height and a random radius from our normal distribution, then we’re going to go and calculate volume For each pair of height and radius, we’re going to get a volume and build up a sample of volumes. We’re going to repeat this a bunch of times and then we can measure the mean and standard deviation of this sample of volumes and that will give us our result.

# Doing Monte Carlo error propagation “by hand”

So how are we going to practice this technique? The rest of this section will focus on how to do this “by hand” in a very tactile and easy to understand way using the data that you’ve collected. We will only do 10 Monte Carlo iterations, 10 times through this loop, just to give you a sense of how this works. Then, in a latter section of the lab you will learn how to do a more thorough and accurate job by using a spreadsheet to do a full and complete Monte Carlo of your results.
1. Take your measurements and write them on little scraps of paper: you should have 10 radii and 10 heights.
2. Put them in a boxes (ideally with lids): one for radii and one for heights.
3. Shake and pull out one radius and one thickness.
4. Calculate volume.
5. Put the radii and height back in their respective boxes.
6. Repeat steps 1 – 5 ten times to get a sample of 10 volumes.
7. Determine the mean and standard deviation of those results. 