29:30 Physics IV Experiment 1

Experiment 1 : Statistics and Probability

Purpose:

To become familiar with the concept of probability and the nature and form of various probability distributions.
To investigate the statistical fluctuations that occur in the counting of random events.
To obtain frequency histograms of data which are distributed according to various probability distributions and to examine the parameters associated with these distributions.

Discussion:

Concepts of statistics and probability find extremely wide application in modern physics, both in theoretical developments (statistical mechanics, quantum mechanics, plasma physics, etc.) and in treatments of experimental data (error analysis, least squares fits, chi-squared analysis, etc.). This experiment is intended to be an introduction to the subject.

An introduction to the concept of probability and discussions of the probability distributions to be encountered in this experiment will be found in the appendix. This material should enhance your appreciation of this experiment and should be read before coming to the laboratory.

Procedure:

In this experiment a Geiger tube will be used to detect ionizing radiation of various intensities. The Geiger tube will be studied in Experiment 2. For the present, accept it as a device which produces a count on its associated scalar whenever a suitable particle with sufficient energy is incident upon it.

The ionizing radiation incident on the Geiger counter in this experiment occurs at random times but at a constant rate and thus the condition that the resulting distribution is Poissonian is satisfied.

After the instructor has set the high voltage and demonstrated how to obtain different count rates, obtain about 100 measurements of the number of counts in some preset time interval (6 seconds is suggested) for four significantly different rates. Suggested rates are such that average total numbers of counts in the time interval are about 1,10, 100 and 1000. It is further suggested that the data taking be divided between the various groups working simultaneously, different groups doing different rates, and the results combined for the final reports, which, of course, should be done individually.
Use sets of 10 presumably unbiased coins. Shake and toss the coins about 100 times and record for each toss the number of heads. Again this can be a combined effort using several sets of coins.

Report:

For each of the four sets of counting data construct frequency histograms showing the number of times each result, or range of results for the higher count rate data, occurs versus the result. For each set compute the mean and the standard deviation from the data. Compare the standard deviations with those expected for a Poisson distribution. Using the computed means, plot the appropriate Poisson distributions on each graph. Also plot normal distributions on each graph using the com- puted means and standard deviations. Discuss points of agreement and disagreement.

For the coin toss data plot a histogram of the type discussed above. Plot on the same graph the binomial distribution expected for the conditions of the experiment. Again discuss the agreement or lack thereof. Also plot on the same graph a normal distribution with the mean being N/2 and the standard deviation being sqrt(N)/2, N being the number of coins tossed. Make the plots carefully enough that the rather small differences between the normal and binomial distributions may be discerned.

APPENDIX

REFERENCES

The subject of extracting physical information from data that is subject to statistical fluctuations is a highly developed one, as is the general theory of probability. There is an enormous bibliography from which we cite only a single source, which itself contains a fairly good list of outside reading material. Data Reduction and Error Analysis for the Physical Sciences. Philip R. Bevington, McGraw-Hill Paperbacks.