Lab 1 - Monte Carlo Methods - Volume of the Library (1)

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Physics

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Apr 3, 2024

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Lab 1 - Monte Carlo Methods - Volume of the Library By the end of the lab prep, you should be able to… ● Know the definition of mean and standard deviation and be able to calculate these quantities for any data set ● Recognize measurements as continuous, rather than discrete, values. ● Use a spreadsheet to calculate the result of a function ● Recognize characteristics of a normal distribution. ● Know that the number of digits in data representation should be based upon uncertainties and not upon fixed significant figures rules ● Use the idea of standard deviation to express the number of appropriate digits for any calculation ● Compare representations of data using significant figures and using uncertainties ● Describe the difference between statistical and systematic uncertainties By the end of this lab, you should be able to… ● Categorize an uncertainty as either statistical or systematic ● Characterize data with a probability distribution based upon mean and standard deviation ● Critique the limitations of the Gaussian assumption ● Use a spreadsheet to generate random numbers with both a flat distribution and a normal distribution with a given mean and standard deviation ● Describe the fundamental principle of Monte Carlo-based uncertainty propagation as being based upon pulling values from a probability distribution and putting them in a formula. ● Use a probability distribution to get possible values ● Create a spreadsheet to generate the possible values ● Compute a set of possible results of a formula using a spreadsheet ● Synthesize the results of those formula results into a single representation of your final measurement ● Set-up a spreadsheet to propagate uncertainties using Monte Carlo methods for a new set of data and a through a new mathematical formula University of Massachusetts - Amherst Spring 2024 Physics Department 1/6

Overview In this lab, you will be designing a method to measure the height of the library. Once you collect the data, you will be analyzing it using probability distributions and the Monte Carlo Method. In this lab, you will be working in your team of four. However, you will be turning in lab reports as pairs. Part 1 - Measure the Volume of the Library First, you and your lab team will design a method to measure the length, width, and height of the W.E.B. du Bois Library here on campus. The only measuring instrument you will have is a meter stick. You will only measure only the tower of the library. By Ktr101 - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=17831764 NOTE: You are not being graded on your accuracy. The variations within a team and among teams are the critical focus of this lab! University of Massachusetts - Amherst Spring 2024 Physics Department 2/6

1. Describe your methods of measuring the dimensions of the library. Once your team has decided on a method of measuring the library, each team member needs to execute the plan individually . Everyone should follow the agreed upon plan as close as possible. Again, you are not being graded on your accuracy. The variation in your results will be critical to the statistical analysis portion of this lab.Try to avoid sharing data with each other until the second part of this lab. WARNING: DO NOT ENTER THE FENCED IN SPACE AROUND THE LIBRARY! THIS IS A RESTRICTED SPACE! 2. Record your team’s measurements and calculate the volume of the library using the dimensions you measured, and record them for the second part of this lab. Record your team’s measurements in the table below: Teammate Length l (m) Width w (m) Height h (m) Volume V (m 3 ) 1 2 3 4 Part 2 - Statistical Analysis You may have learned in previous classes ways of determining uncertainties. A more rigorous method that always works is the Monte Carlo method with which you should be familiar with the prep. This method is the method used in most research today. Representing your data 3. Calculate the mean and standard deviation of each of the three dimensions (length/width/height) using the four measurements taken by your team using the same method. University of Massachusetts - Amherst Spring 2024 Physics Department 3/6

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4. Calculate the volume of the library using the means for each dimension. Just making an educated guess, what do you think is the uncertainty in this volume? What was the reasoning behind your guess? Physical simulation of the Monte Carlo Method Each team member should write their three measurements on three separate slips of paper, and place these slips into three separate cups: one for the height, one for the length, and one for the width. ● Take turns with your teammates randomly drawing a slip from each cup, record the value of length, width, and height drawn in the Monte Carlo Method spreadsheet. Make sure to select the tab Physical Simulation at the bottom of the spreadsheet. ● Put the slips of paper back into their respective cups. And pass the cups to your teammate. ● The next teammate should mix the slips of paper in their respective cups and randomly draw a new set of length, height, and width. ● Repeat this until you have 24 values of length, width, and height. ● Calculate the volume of the library for each of the 24 randomly selected values of length, width, and height. 5. What is the average and standard deviation for each dimension (length, width, and height), and for the volumes you have calculated using this method? Using the spreadsheet functions =AVERAGE() and =STDEV() to calculate the averages and standard deviations. 6. Calculate the percent uncertainty on each of the three dimensions. The percent uncertainty is the ratio of standard deviation to the average value. For example, if the average value of the height of the library is and the uncertainty in the height is , then the percent uncertainty is: , NOT 7. Using the percent uncertainty, which of the three dimensions will have the greatest impact on the uncertainty of the volume? Justify your answer. Computer Simulation of the Monte Carlo Method The more times you pull numbers from the cups, the more you can fill out your probability distribution, which will give you a better sense of your data. At ten draws, it doesn’t take too long to do these calculations by hand, but what if you wanted a hundred, or even a thousand draws? At this point, we can turn to computer simulations to “make random draws’’ for us. University of Massachusetts - Amherst Spring 2024 Physics Department 4/6

Before diving into the simulations, let’s play around with some of the spreadsheet functions first. In particular, we’ll be looking at two functions RAND() and NORMINV(). This time, we will be providing you a template you can use for your spreadsheet, available on Moodle. As we proceed through the course, we will be expecting you to begin to develop spreadsheets on your own, so it’s important to understand what exactly the spreadsheet is trying to do. Click on the tab Example Distributions. There is a list of numbers created using the built in functions RAND() and NORMINV(). The charts provided are histograms of the values in each list. If you press CTRL+R for Windows or Command+R for Macs, you can refresh the data, and have the sheet give you new values. 8. What does RAND() do? Feel free to use the internet to look up this function. 9. What does NORMINV() do? To help you understand how this function works, it may be helpful to adjust the values of the average and standard deviation. Feel free to use the internet to look this up as well. Using a Computer to do the Monte Carlo Method Now let’s move onto the tab Monte Carlo Simulation. Essentially, we are getting the spreadsheet to pick out our random values instead of drawing slips of paper from a cup. we are drawing numbers from a normal distribution of each dimension, using the NORMINV() function: At the top of the sheet enter your team’s average and standard deviation for each of the dimensions of the library. The spreadsheet will now calculate a random value for each dimension 3000 times! You will have to scroll down the sheet to see all of the random values. University of Massachusetts - Amherst Spring 2024 Physics Department 5/6

10. Why do we use a normal distribution to represent our measurement here? Are there situations where you would want to use a different distribution (a flat distribution, for example)? 11. What is the mean and standard deviation of the volume of the library calculated using the spreadsheet? Upload a copy of your spreadsheet to your lab report. You can upload your spreadsheet to Moodle assignment for Lab 1 when you upload your report. 12. Compare your uncertainty measured with this technique to the educated guess you made in question 4. Was your educated guess of the uncertainty too high? Too low? Just right? 13. Calculate the percent uncertainty on the volume, and compare this value to the percent uncertainty of the dimensions. Is the percent uncertainty equal to the sum of the percent uncertainty or the percent uncertainties of each dimension multiplied together? 14. Why do you need to look at percent uncertainties when comparing the volume with its other dimensions? 15. Make an educated guess on how your measured volume compares to the actual volume of the library. Do you think it will be higher or lower? Explain your reasoning. 16. Evaluate your method of measuring the volume of the library. What are the strengths and weaknesses of your method? 17. Ask your TA for the actual volume of the library, and give them your measured value of the volume. How many standard deviations is your measurement off by? 18. Now that you know how your measurement compares with the actual volume, what are some of the factors that could have affected your measurements? University of Massachusetts - Amherst Spring 2024 Physics Department 6/6

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