1. Physics 114: Lecture 11-a Error Analysis, Part III
John Federici
NJIT Physics Department

2. Physics Cartoons

3. Experimental Design – Error Analysis
In this section of material, we will demonstrate how to use error analysis in the design of experiments. Please note that the DETAILS of the experiment are not as important for this class as to the METHODOLOGY of determining which errors are DOMINATING the noise.
Lets start with a simple example from an exam
Problem 4. You are given the following measurements,
a=[ ];
which represent counts of an event (say cosmic rays) recorded from a detector. The counts are taken in 1 second intervals. Find:
a) The mean of the sample of five measurements. Write down the equation you used.
b) The standard deviation of the sample of five measurements. Write down the equation you used.
c) Now, assume that you are told that there is a systematic noise in the detector which adds a background count of 11 counts/s to your data. By correcting your measurement for the systematic error, what is the CORRECTED mean number of cosmic rays detected in a 1 second interval?
(d) Using the results of (b) and (c), what is the Signal-to-noise (SNR) ratio for the data?
e) Assuming that the experiment obeys Poission statistics, estimate how long the time interval for detection should be so that the SNR=100 [after correction for the systematic noise].

4. Experimental Design – Error Analysis
Problem 4. You are given the following measurements, a=[ ];
which represent counts of an event (say cosmic rays) recorded from a detector. The counts are taken in 1 second intervals. Find:
a) The mean of the sample of five measurements. Write down the equation you used.
b) The standard deviation of the sample of five measurements. Write down the equation you used.
c) Now, assume that you are told that there is a systematic noise in the detector which adds a background count of 11 counts/s to your data. By correcting your measurement for the systematic error, what is the CORRECTED mean number of cosmic rays detected in a 1 second interval?
d) Using the results of (b) and (c), what is the Signal-to-noise (SNR) ratio for the data?
e) Assuming that the experiment obeys Poisson statistics, estimate how long the time interval for detection should be so that the SNR=100 [after correction for the systematic noise].

5. Experimental Design – Error Analysis
Problem 4. You are given the following measurements, a=[ ];
which represent counts of an event (say cosmic rays) recorded from a detector. The counts are taken in 1 second intervals. Find:
a) The mean of the sample of five measurements. Write down the equation you used.
b) The standard deviation of the sample of five measurements. Write down the equation you used.

6. Experimental Design – Error Analysis
Problem 4. You are given the following measurements, a=[ ];
which represent counts of an event (say cosmic rays) recorded from a detector. The counts are taken in 1 second intervals. Find:
a) The mean of the sample of five measurements
b) The standard deviation of the sample of five measurements
c) Now, assume that you are told that there is a systematic noise in the detector which adds a background count of 11 counts/s to your data. By correcting your measurement for the systematic error, what is the CORRECTED mean number of cosmic rays detected in a 1 second interval?
d) Using the results of (b) and (c), what is the Signal-to-noise (SNR) ratio for the data?

7. Experimental Design – Error Analysis
OK, now we have a signal-to-noise ratio for the measurement. How do we use our knowledge of statistics to IMPROVE the measurement?
e) Assuming that the experiment obeys Poisson statistics, estimate how long the time interval for detection should be so that the SNR=100 [after correction for the systematic noise].
For Poisson statistics, we know that
So we want to INCREASE our measurement time interval so that the MEAN of our counts is 104. But we KNOW that there are roughly 95.2 counts per second. SO the time interval required to count 104 counts is

8. Can not I just take more data?
The previous example suggests that the way to improve the SNR ratio is to INCREASE the time interval over which one counts the cosmic rays.
Rather than taking 5 measurements each 105 seconds in duration, can I take instead 5×105=525 measurements of 1 second intervals?
If you are INCREASING the number of measurements (but keeping the time interval the same), you are changing from a sample population of 5 samples to a sample population of 525 samples, but you HAVE NOT CHANGED the PARENT population which depends on the MEAN of the measurement.
Since you have not changed the mean, you have not changed the standard deviation, nor the SNR ratio!

9. NEW IN 2018
Start in class HW analysis of measuring deviation of laser beam through block of glass to infer index of refraction.