lab 3 counting statistics

NERS 315 David SirajuddinLab 3 – Counting Statistics Partners: Nick Krupansky, Yong Ping Qiu

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Lab 3 – Counting Statistics

Introduction

This lab introduced various methods of statistical analysis to be used in interpreting

data obtained in lab. It is our aim to measure radiation that results from nuclear decay of a

radionuclide source. Since radioactive decay is a nonconstant, and thusly random process,

statistical fluctuations are expected and such statistical analysis can help to better analyze the

data in a more lucid, and systematic way. The instruments used to measure are also not ideal.

Measuring instruments such as a detector are prone to errors such as dead time, pileup, and

in some sense: competing forms of decay. Dead time refers to a separation time necessary

between two events to be recorded as two separate pulses by a detector (i.e. processing time).

This technical consequence, therefore, inhibits knowledge of the true rate of radiation. In

paralyzable and non-paralyzable models, this dead time results in either of two possibilities: a)

the exclusion of detection of multiple pulses received by the detector during the dead time,

or b) an ‘addition’ of pulse heights, resulting in a larger than expected pulse amplitude. In

fact, any detector that does not instantaneously detect is prone to these errors.

Often it is of interest to measure a specific type of decay (e.g. β+ decay); however,

unless the source decays with the decay type of interest 100 percent of the time, competing

decay reactions will give a false impression of the measured quantities desired of the decay of

interest. As a general example, x-ray photons emitted from a transitioning electron could

avoid direct detection by being absorbed by another electron, causing it to be ejected (i.e.

auger electrons). These intrinsic fluctuations are built into the framework of nuclear decay

and the instruments used to detect it. Some sources of error in measurement are bound to

be derivatives of these inherent problems. Despite an unavoidable uncertainty, these

fluctuations are useful in the respect that they are characterizable. Statistical models can

model the internal variation of data and be used to determine two important qualities: 1)

whether nuclear counting equipment is operating normally and 2) to provide insight to the

uncertainty of a data set given only a single measurement. It is of prime interest to acclimate

oneself with this statistical analysis first hand as it is integral in measuring radiation.


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The objectives of this lab are to practice diagnosing data as a particular statistical

model, and to use that model to determine quantities that will reflect expected fluctuation,

arithmetic mean, etc. By first beginning with a hypothetical datum, a statistical model was

fitted, and quantities of interest computed. Then, to verify the validity of this choice of

model, more data was given and quantities such as those relating to the internal variation

were measured and compared with those found for the single measurement. The result was

a sound agreement between the two. This lab was a proof of counting statistics through

example, rather than theory. Later in the lab, data was taken in experiment and analyzed in a

similar fashion using both the Poisson and Gaussian models. First a Cs-137 source was used

in tandem with a Geiger-Mueller Tube (GM tube), and counter in aim to measure 30 counts

consistently. Twenty-five measurements were taken, and from this data the experimental

mean, sample variance were measured. A Poisson model was then fitted to the data, and

through comparing this expected standard deviation with that of the experiment, it was

shown to be a good fit. Tallk labout chi square, and the uncertainty comparing with

expected, etc. Finally, it was aimed to measure 5 counts on the counter, one thousand trials

were conducted, and to explicitly show which model (Gaussian, or Poisson) is a better fit,

both distributions were plotted on a graph including the data collected. The theory was

shown to match the data in all accounts.

Procedure

The lab procedure involved two main parts in experiment, and one hypothetical

component with data given, these parts are labeled by number to be consistent with the lab

handout numbering scheme (attached in the appendix). Hypothetical data listed under part

2 in the lab handout is described despite it not being performed in lab as it will be discussed

in the following section, Results and Analysis. Equipment used for all experiments are listed

below:


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Equipment/Materials

Tennelec TC 952 High Voltage Supply (set to 801V)Ortec 572 AmplifierHewlett Packard 54610B Oscilloscope, 500 MhzOrtec 994 Dual Counter/TimerG-M Tube Lead Shield, Model No. AL144, Serial No. 453Ortec Timing SCA SS1Preamp (HV SIG)Cs-137 5 5 μCu, half life = 38 years, May 1989, Nucleus Inc.

2) A source was put in a G-M Tube with a two-minute timing interval. One trial was

conducted and a single count value was yielded on the counter.

2e) the experiment in part two was performed nine more times yielding nine more pieces of

data.

4) The following setup was used:

Figure 3.1

A Cs-137 source was placed inside the GM tube, with the source being placed on the

second shelf from the top, and a time interval on the timer was set to 0.1 s so as to record

approximately 30 counts per 0.1 seconds. A set of twenty-five of these counts were taken.


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8) The same setup as in figure 3.1 was used in aim to record 1000 trials of approximately 5

counts per timing interval. The timing interval used was 0.02 s.

Results and Analysis

Numbering follows consistently with the handout attached in the appendix.

2) Statistical quantities were computed for a single measurement of 10,982 counts per a 2-

minute time interval. The results are displayed in a single table shown below, where σ is the

standard deviation, R is the rate, where R = xi / t. = [min-1] The quantity xi is the single

measurement 10,982 per timing interval, and t is the timing interval, 2 minutes. In regards to

units, min is minutes. Error propagation techniques used are discussed in appendix.

Statistical Quantities Associated with xi = 10,982

xi ± σ xi ± σ / xi (%) R ± σ (min-1) R ± σ / R (%)

10982 + 104.79 10982 + 0.9542 5491 + 52.39 5491 + 0.954

Table 3.1

Taking the original measurement to be approximately Gaussian, xi is defined such that xi =

<xe>, experimental mean = <x> the mean value for the suspected distribution. It follows

by properties of the Gaussian distribution that <x> = σ2, the variance, implying the standard

deviation s = √σ2 = √xi. This is allowable so long as the mean is relatively large. Under

these presumptions, the uncertainty for a single measurement is small (~0.95% < 1%). It is

expected if further data is taken from this same experimental setup that nearly 68% of

further data points will lie within the mean (Knoll, 84).

2e) Nine additional data points were obtained from the same experiment giving a total of ten

data points (data listed in handout, in appendix). The sum ∑, 108714, of these ten

measurements is taken to be a single measurement. A Gaussian distribution is asserted

rendering ∑ = <∑e> = <∑>. Accordingly, <∑> is allowed to be equated with a supposed

variance σ∑2, yielding the result σ∑ = √∑ = √108714 ≈ 329.718 counts per timing interval.

After computing these quantities, the same statistical quantities found in table 3.1 can be


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tabulated, where t in the rate, R∑, equation is now taken to be t = 20 minutes (shown on the

following page):

Statistical Quantities Associated with ∑ = 108714

∑ ± σ∑ ∑ ± σ∑ / ∑ (%) R∑ ± σ∑ (min-1) R∑ ± σ∑ / R∑ (%)

108714 + 329.718 108714 + 0.30329 5435.7 + 16.486 5325.7 + 0.30329

Table 3.2

By measuring counts for ten times the two-minute interval used for the initial calculaltions, precision

was increased. In terms of percent, an increase of 0.65% incurred; however, this results in a 214%

different in regards to percent difference.

To verify if ~68% of the data collected lied within one standard deviation of the two minute

time interval counting exercise, it was noted that a lower bound of [10982 – 104.79 = 10877] counts

per timing interval was found, along with an upper bound of [10982 + 104.79 = 11086] counts per

timing interval. Glancing over the data, it was found that six of the ten measurements lied within

one standard deviation implying that 6 / 10 = 60% lied within one standard deviation which is

approximately 68%. However, This value does differs by 8%, and this statistical error may or may not

be rectified if more measurements are taken. This problem seems to stem from the initial

measurement being an untypical value with regards to the further data collected in part (2e).

Computing the mean of the total set of ten measurements yields: ∑ / 10 = 10,871 counts per

timing interval. This value is differs from the initial count given for xi = 10,982 counts per

timing interval. Using <∑> as the mean, assuming a Gaussian distribution again to avoid

excess calculation, a standard deviation of 104.264 counts per timing interval was found.

Factoring in uncertainty, it was found that 7 / 10 = 70% of the measurements lied within

one standard deviation of this mean. It can then be concluded that using a single

measurement as the mean for a fitted distribution can be done with certain accuracy, but it

may be incur error depending on how typical the measurement may be in an experiment.

3) If the single measurement value xi2 = 10,915 counts per timing interval is taken to be the

only element of a data set. Then, assuming a Gaussian distribution, further analysis yielded

how many elements of the data set lie within various standard deviations found from this

single datum. Identical to the procedure described above, the mean is taken to be the single

measurement xi2 implying a standard deviation σxi2 of approximately 104.47 counts per


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timing interval. The number of measurements out of a total of ten that deviate from various

standard deviations according to this model is tabulated below.

Standard Deviation Number of trials outside of standard

deviation

0.67σxi2 6

1.0σxi2 4

1.6σxi2 2

2.0σxi2 0

Table 3.3

The Gaussian model would predict that 5.028 or about 5 would lie within 0.67 standard deviations,

3.174 (about 3) would lie outside one deviation, 1.096 or about 1 would lie outside of 1.6 deviations,

and 0.456 (nearly none) would lie outside of two standard deviations. Data for the Gaussian

distribution’s expected valued was found via a normal curve table

( http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/normaltable.html ), however, the

actual technique to finding these values involves integrating the Gaussian distribution function from

any ± Z*σ, where Z is a real number. Where the Gaussian Distribution is defined as:

where μ denotes the mean, σ2 is the variance, and σ is the standard deviation. It is evident that the

data differs slightly from the predicted Gaussian number of deviations, but these differences are

small and shows the Gaussian in a good fit.

5) From the data collected in experiment consisting of 25 trials aiming to record 30 counts, the

experimental mean <x>, and sample variance s2 were computed to be:

Experimental Mean 31.84

Sample Variance 27.47

Table 3.4


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Where all numbers are quoted in terms of counts per time interval. Asserting a Poisson model to the

data, it can be presumed that the expected standard deviation σi has the following relationship

with the experimental mean xe:

σi = √xi ≈ √xe

Taking the square root yields σi ≈ 5.64 counts per timing interval, whereas the standard

deviation s from experiment gives a value of s ≈ 5.24 counts per timing interval implying

that s ≈ σi. The significance of σi in this application it reflects predictions given by a

Poisson distribution, with only one value. The experimental equivalent standard deviation s

is approximately equal to σi showing the utility and accuracy of using statistical models. If s

is taken to be the true standard deviation, then the predicted deviation of the Poisson

distribution differs by only 7.63%. This small experiment shows the validity of using

Poisson statistics when only one or few measurements are known.

6) the Chi-squared test was performed using the Chi square formula (postulated by Karl

Pearson in 1900), and was computed to be 20.708. This test quantitatively discerns a

measure of fluctuation in the data. Particularly, it measures how much the ratio of the

sample variance over the modeled variance differs from unity. Using interpolation, along

with a chi square table ( http://www.statsoft.com/textbook/sttable.html#chi) , it is found

that the probability that a random sample from a poisson distribution with the same mean

will show a larger fluctuation is 0.633. This value is high, indicating that the data set

collected though can be modeled by a poisson distribution, would yield substantial

conflictions with the models predictions more than half of the time.

7) using elementary error propagation techniques, it was found that the uncertainty to be

5.89. Comparing this to the standard deviation σi, it is nearly the same. Despite the chi

square test, in some ways the poisson distribution proves to fit the data.

8) Using a 0.02 second timing interval, 1000 trials were taken in aim to record 5 counts on

the counter. A frequency plot Pi of Probability of occurrence of exactly i counts vs i value is

displayed below.


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Figure 3.1

Predictably, the counts of highest frequency resides around the ‘center’ of this data,

around the mean, 4 counts. The data resembles a bell type curve. Which model will fit

this data best is discussed in the following part.

9) Finding the mean, the Poisson and Gaussian distribution functions are graphed on the

same axes as the data distribution to visually see which one fits the best.


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Figure 3.2

From figure 3.2, it is evident that the poisson distribution fits the data best. This is

expected, as one of the stipulations of fitting a data set to poisson is that n should not be

large. In this case n is ~ 5, making poisson a better choice than the Gaussian

distribution. The theory matches the data with accuracy.

Conclusions

It was shown though both hypothetical and experimental data that fitting statistical

models to data, or just a single datum is valid. In all cases, no matter how many

measurements were known, it was proved valid to use either a Poisson distribution when

n is large, or in the opposite case a Gaussian model. It was the approach to measure

nuclear decay through direct measurement, and then to correctly interpret the results

using counting statistics.

In part two, it was found that increasing the measuring time decreased the

error incurred. By having ten times the measuring time, the uncertainty decreased from

0.954% to 0.303%. Experimental quantities were computed and compared to those of

the statistical predictions according to a model, as in the Gaussian in part 3. The number


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of trials differing from the various multiples of the standard deviation agreed more or

less with theory.

In part 4, actual measurement was done and the standard deviation found

from the sample variance of the experiment was found to be different from the

prediction standard deviation of the poisson model by 0.4. The chi square test helped to

quantitatively assess the agreement between the sample variance and the predicted

variance, and the uncertainty was shown to be low, but more importantly in close

agreement with that of theory (a change of 5.64 for the standard deviation of experiment

to 5.89 for that of theory).

Finally, 1000 trials were taken and it was visually shown that the poisson distribution

mapped almost identically onto the data’s frequency functions points.

Appendix

Pages are attached.

lab 3 counting statistics

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