Comparison of Nuclear Decay Data against Gaussian and Poisson
Distribution Models
H. Potter, and E. Kager
(Completed 24 October 2005)
The Poisson distribution, with a p-value of 25.55%, was determined to be more
likely to provide a valid theoretical model for random nuclear decay count data
than the Gaussian distribution, which had a p-value of 5.36%.
I. Introduction
In the 1800’s several mathematicians began thoroughly investigating random
processes. The fruit of their labors was the development of a new field of mathematics
known now as statistics. Two such pioneers in statistics were Carl Friedrich Gauss, who
analyzed what is now known as the Gaussian distribution, and Siméon Denis Poisson,
who analyzed what is now known as the Poisson distribution.
II. Experiment
A distribution of random numbers was created by recording the number of counts
in a radioactive detector for 320 4 second intervals. A radioactive source with an
extremely long half-life was used so that it was safe to assume that the counts would be
randomly distributed about a constant mean value for every observation. Through a
statistical analysis of this data a determination of whether the Gaussian or Poisson
distribution was more likely to provide an accurate statistical model for predicting such
radioactive decay counts was to be made.
III. Results
Value
Observed
10
0
11
0
12
0
13
2
14
0
15
4
16
7
17
13
18
12
19
16
20
17
21
22
22
22
23
31
24
35
25
23
26
28
27
6
28
19
29
14
30
11
31
10
32
7
33
5
34
4
35
2
36
5
37
3
38
0
39
1
40
1
41
0
n
Mean
320
24.3313
Normal Poisson
0.424
0.174
0.736
0.384
1.228
0.780
1.969
1.459
3.033
2.536
4.489
4.113
6.385
6.255
8.727
8.953
11.462
12.102
14.464
15.497
17.539
18.853
20.436
21.844
22.880
24.159
24.614
25.557
25.444
25.910
25.273
25.217
24.122
23.598
22.122
21.266
19.495
18.479
16.508
15.504
13.432
12.575
10.501
9.870
7.889
7.504
5.695
5.533
3.950
3.960
2.633
2.753
1.686
1.860
1.038
1.223
0.614
0.783
0.349
0.489
0.190
0.297
0.100
0.176
2
s
s
24.9808
4.9981
Table 1: The observed frequency for each data value observed, as well as several that
were not observed but were needed for later statistical analysis, alongside the expected
frequency of each value according to both the Gaussian and Poisson distributions.
Summary statistics that were calculated from the data set in order to calculate the
expected values for the two theoretical distributions are also provided at the bottom.
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IV. Analysis and Discussion
Bins
10--17
18--25
26--33
34--41
SUMS:
Observed
26
178
100
16
320
P-Values:
2
χ reduced
Normal Poisson
26.991
24.654
162.111 169.138
119.764 114.329
10.559
11.542
319.425 319.662
5.36%
25.55%
2.553
1.352
χ2N
0.036
1.557
3.261
2.804
7.659
χ2P
0.074
0.464
1.796
1.722
4.056
Table 2: Binning of data and the associated observed and expected values, along with pvalues for the Gaussian and Poisson distributions using a χ2 analysis. χ2 and reduced χ2
values are also given to further illustrate the difference in fits between the two
distributions.
In order to compare how well the Gaussian and Poisson distributions fit the data
by using a χ2 goodness of fit analysis, the data had to be binned so that the observed and
expected values were above 5 for each bin. In order to keep all of the bins the same size,
some values in the largest and smallest bins were included even though they were not
actually observed.
At first glance the distribution that would be more likely to provide a better
theoretical fit for future observed data can be determined by comparing the reduced χ2
values for each distribution. The closer a reduced χ2 value is to 1, the better the data fit
that particular distribution; thus, the Poisson distribution is seen to fit the data better than
the Gaussian distribution. A more quantified comparison of how well each distribution
fits the observed data can be gleaned from the p-values for each distribution. The
meaning of a p-value in this context is that if the observed data were actually governed by
the specified theoretical distribution, then the probability that the data that were observed
would actually be observed is the specified p-value; therefore, the larger that a p-value is,
the more likely it is that the specified distribution actually provides an accurate
theoretical model for the observed data. The Poisson distribution, with a p-value of
25.55%, is thus more likely to provide a better theoretical model for the observed data
than the Gaussian distribution, which had a p-value of 5.36%.
A more intuitive grasp regarding which model better fits the data can be had by
glancing at the graph provided in Figure 1 below, which contains a graph of both the
theoretical Gaussian and Poisson distributions, as well as a plot of each observed data
point. Note that the bins used in the specific calculations indicated above are indicated
by the scale on the x-axis.
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Distribution Comparison
40
35
Frequency
30
25
Observed
20
Normal
Poisson
15
10
5
0
10
18
26
34
Values
Figure 1: A graph of both theoretical distributions against the observed data points.
V. Conclusion
All indicators of which distribution is more likely to be a better theoretical model
for the observed data indicate that the Poisson distribution fits the data better than the
Gaussian distribution: the Poisson distribution has a reduced χ2 value closer to 1, a larger
p-value, and a graph that is in slightly better agreement with the observed data. It is thus
concluded that the Poisson distribution is more likely to be a better theoretical model for
predicting radioactive decay counts than the Gaussian distribution.
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