Garrison TurnerLab 1 - Distributions and Error Analysis5/22/2020TA: Gandalf DumbledoreIntroduction and Objectives

No measurement is perfect and is subject to one or more sources of uncertainty. Somemeasurements are probabilistic in nature such as rolling dice, where each outcome has a strictprobability associated with it. Likewise, the error associated with some measurements tend to berandom. In this laboratory exercise the concepts of distributions, measurements with randomerror, and error analysis were explored. Specifically,the results of rolling two dice in multipletrials were collected and the results collated. For each set, averages and standard deviationscalculated, and histograms of the data were generated. We note that the average of several rollsof two dice should tend toward the number 7, as this is the value with the highest probability, andthe numbers 2 - 6 and 8 - 12 have symmetric probabilities on either side of the value 7. We alsocollected samples of timing a stopwatch from which the effects of random error onmeasurements was investigated.Theory predicts that after a sufficiently large number of probabilistic measurements aremade, if the probabilities are symmetric, the results should tend toward a normal distribution. Itis a major goal of this exercise to attempt to see if this pattern emerges over the number of trialsinvolved. Learning how to use Excel to calculate averages, standard deviations, and generatinghistograms are also major objectives of this exercise.Data and AnalysisThis laboratory exercise consisted of two broad parts. In the first, we rolled two dice forthree trials. In the first trial, the dice were rolled 10 times. In the second, 30, and in the third, 60times. From these data the average and standard deviation were calculated, from which we couldmake predictions about the evolution of these quantities as more and more trials are used andcompare these predictions with our results. Histograms of each trial were also made.

Figure 1. Tables of the data for both experiments.In the second part, we were to start and stop a stopwatch after two seconds, attempting toobtain a measurement as close to 2.00 seconds as possible. This was performed 15 times, andfrom the results the average and standard deviation were calculated, along with a histogram.Finally, the results for the entire student population were collected in a collaborative GoogleSheet, from which we could watch in real time how, as data were entered, the evolution of thedistribution. The random error associated with reaction time should be the dominant source ofuncertainty in this part of the experiment.With regard to the first experiment, we make a prediction about how the average andstandard deviation should change as moreBecause the outcome of rolling a pair of fair dice israndom, it is expected that a small number of rolls would reflect this randomness. As such, wepredict that the average for 10 rolls could vary widely, but we expect the larger sample sizes to

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