Poisson Statistics Lab Summary

Steve Koch 20:36, 21 December 2010 (EST):Good summary and good job on this lab.

Purpose

The purpose of this lab is to see that as a Poisson distribution strays away from zero and the numbers become sufficiently large the distribution goes to a Gaussian. We want to learn more about the Poisson distribution and how it relates to physics.

Procedure

Much of this Procedure is online linked here. Still there are some revisions to this because the manual is out of date. Basically one need only ensure that the Photo Multiplier Tube is correctly hooked up to the High Voltage provider and everything is correctly hooked up to the computer. From there collecting data is simple. Open the program that collects the data and make sure you choose the Amp-in before selecting the data collection function. Using the Pre-amp method creates graphs that are not Poisson or Gaussian distributions. I showed this in my Labbook.

Data

All of my Data is here but I want to cite a few data sets that are interesting.

{{#widget:Google Spreadsheet |key=0ArI06ZBK1lTAdF9KclhmZzI2bGhEbjhSUk5vdUlHQUE |width=830 |height=700 }}

{{#widget:Google Spreadsheet |key=0ArI06ZBK1lTAdDFDQ0VsTEdOeGlhUGtXX1h0cUREa2c |width=830 |height=700 }}

{{#widget:Google Spreadsheet |key=0ArI06ZBK1lTAdFZIV09vSXpRdUVIU3EteC1MZWV4MFE |width=830 |height=700 }}

{{#widget:Google Spreadsheet |key=0ArI06ZBK1lTAdHZRbFFkN0FaNVJLSHRwUHNmYS1RRlE |width=830 |height=700 }}

Calculations

I looked up what the Poisson Function is:

[math]\displaystyle{ f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!},\,\! }[/math]

Where [math]\displaystyle{ k }[/math] is the number of events per time interval and where [math]\displaystyle{ \lambda }[/math] is the expected value of the number of events per interval.

I looked up what the Gaussian Function is:

[math]\displaystyle{ f(x)=\frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}},\,\! }[/math]

Where [math]\displaystyle{ \mu }[/math] is the mean, [math]\displaystyle{ \sigma }[/math] is the standard deviation and [math]\displaystyle{ x }[/math] is the number of occurrences at a point.

From the lab manual we find that the standard deviation should equal the square root of the mean:

[math]\displaystyle{ standard deviation=\sqrt{mean},\,\! }[/math]

10ms: stdev=0.54767796013852

20ms: stdev=0.78642612359645

40ms: stdev=1.11239409182849

80ms: stdev=1.58183385316939

100ms: stdev=1.7488917560004

200ms: stdev=2.45148330520545

400ms: stdev=3.46438365068596

800ms: stdev=4.90067440998601

Standard Deviation calculated in Google Docs

10ms: stdev=0.54594837315979

20ms: stdev=0.7775635504387

40ms: stdev=1.14761292926255

80ms: stdev=1.62063612220736

100ms: stdev=1.76461434607371

200ms: stdev=2.41024659670614

400ms: stdev=3.41192722689169

800ms: stdev=4.93394466988019

The Difference between the square root of the mean and the standard deviation.

10ms: Difference=0.001729586978738

20ms: Difference=0.008862573157752

40ms: Difference=0.03521883743406

80ms: Difference=0.038802269037964

100ms: Difference=0.01572259007331

200ms: Difference=0.04123670849931

400ms: Difference=0.052456423794275

800ms: Difference=0.033270259894178

Conclusions

I showed the 10ms, 20ms, 400ms and 800ms data sets here because the 10ms and 20ms data sets show a strong Poisson Overlay and the 400ms and 800ms data sets show a strong correlation between the Poisson and Gaussian Overlays.

As I said before the Pre-amp data sets, 40pre and 100pre, are not used because they were shown to not be Poisson or Gaussian.

Now I have also show that the difference between the square root of the mean and the calculated standard deviation increase as the time interval increases. This means that as the time interval goes to zero the standard deviation will approach the square root of the mean. This is consistent with statistical theory.

So I have shown that the Amp-in version of the count of events per time interval of a PMT follows a Poisson Distribution and as the interval increases it follows and Gaussian.

Citation

1) I got the information for the Q-Q plot here

2) I got the information for the Poisson Distribution here

3) I got the information for the Gaussian Distribution here

Thanks

1)To Steve Koch for assistance in the lab specifically telling me about the COUNTIF function

2)To Katie Richardson for assistance with google docs and the amusing Chicago Piano Tuners Problem.

3)To Nathan for assistance in the lab and being a good lab partner.