Poisson Statistics Lab Summary
Steve Koch 20:36, 21 December 2010 (EST):Good summary and good job on this lab.
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.
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 Still there are some revisions to this because the manual is out of date.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.
All of my Data is here but I want to cite a few data sets that are interesting.
I looked up what the Poisson Function is:
Where k is the number of events per time interval and where λ is the expected value of the number of events per interval.
I looked up what the Gaussian Function is:
Where μ is the mean, σ is the standard deviation and x 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:
Standard Deviation calculated in Google Docs
The Difference between the square root of the mean and the standard deviation.
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.
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
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.