The basic gist of how Monte Carlo simulations work is that you randomly select inputs, perform calculations on the randomly-selected inputs, and collect the outputs. This process is repeated several times (perhaps thousands, tens of thousands, or even more! As with any statistical sample, the more, the better), and in the end, all the outputs are gathered together and analyzed. To randomly select inputs, you’ll need to specify boundaries for which inputs can be selected from. A statistical model can help with this, such as a Gaussian distribution, which is a fancy term for the familiar “bell curve.” As for the aggregated outputs, statistical analysis would make sense in order to make sense of thousands of data sets. Basically, statistics is a useful tool that compliments the Monte Carlo technique. Also, generally computers are used to perform a Monte Carlo simulation due to the large number of repetitive calculations required.

This is what a bell curve looks like.

Monte Carlo simulations can be used in space sciences. For example, if one wants to analyze the risk of failure of a spacecraft in orbit, one can perform a Monte Carlo simulation with random inputs for how the spacecraft begins its orbit (speed, physical orientation, etc.), since that state cannot be predetermined accurately and instead can be modeled statistically. Then, the laws of orbital mechanics can be applied to the inputs to produce outputs that can be analyzed later. A more simple example of where the Monte Carlo method is used is the classic game of Battleship. Initially, a player would randomly guess locations for where a battleship is located. After the player scores a hit, the player would follow an algorithm (guess points that are in line with the hit) to sink the battleship (the outcome).

(Image from Wikipedia)

There are many different ways to perform quality control, but one basic method that quality engineers can use are quality control charts. There are different kinds of quality control charts for controlling different quantities, such as the average or the variability of a set of data. The basic idea behind constructing quality control charts is to collect a random sample of the product in question during production, and collect a set of data from the sample. For example, a quality engineer can collect a random sample of Sun Chips from the production line and extract the salt content from each chip in the sample pool.

Much work goes into ensuring products such as Sun Chips are of utmost quality to the consumer (such as you and me).

Next, the mean and variance of the set of data are calculated, and the quantity under study is plotted on a control chart. If the average salt content of Sun Chips are under question, then the individial salt contents of each chip are plotted on the control chart. Then, three lines are drawn on the control chart:

• Center line: in this example, the center line is drawn at the mean salt content of the Sun Chips sample under analysis
• Upper control limit: located at a certain distance above the center line. The distance from the center line depends on the variability of the sample, as well as the sample size (sample size is the number of Sun Chips that were colleced from the manufacturing line for the quality control study).
• Lower control limit: located at the same distance below the center line as the upper control limit was located from the center line.

After the three lines and each sample’s data point are plotted on the control chart, one can analyze the chart for variation that may indicate a flaw in the production process. In reality, some variation in a random sample of data is inevitable, but the presence of the upper and lower control limits can help raise alarm. If any of the plotted data points lie outside either of the control limits, then that is an indication that the production process produced an item that was statistically significant, which means it was not likely to be due solely to chance. Perhaps one of the machines in the manufacturing plant is inadvertently putting too much salt on the Sun Chips during production, and someone should look into solving that problem.

Example control chart showing important characteristics

In the example control chart shown, you can see that all the plotted data points lie within the control limits (the topmost and bottommost dashed lines), so you can safely conclude that the process under analysis is “under statistical control,” or that all the variation observed in the product is due to inherent chance and should not be alarming.

Thanks for reading!

(Sun Chips image from Frito Lay, example control chart image from Wikipedia.)