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Counting Statistics
Oak Ridge Associated Universities
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Introduction
It is impossible to directly measure radiation
in every situation at every point in space and
time.
3
It is impossible to know with complete
certainty the level of radiation when it was not
directly measured.
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Introduction
One of the goals of statistics is to make
accurate inferences based on incomplete
data.
4
Using statistics, it is possible to extrapolate
from a set of measurements to all
measurements in a scientifically valid way.
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Introduction
Statistical tests allow us to answer questionssuch as:
How much radioactivit is resent?
5
How accurate is the measured result?
Is the detector working properly? (Are the detectorresults reproducible (precise))?
Is there bias error in addition to random error in ameasurement?
Is this radioactive?
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Introduction
Without statistics, it is difficult to answer
these questions.
6
tat st cs a ow us to answer quest ons w t adegree of confidence that we are drawing the
right conclusion.
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PART I:
DEFINE THE TERMS
7
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Uncertainty (Error)
Uncertainty (or error) in health physics is the
difference between the true average rate oftransformation (decay) and the measuredrate.
8
Accuracy refers to how close the measurement isto the true value.
Precision refers to how close (reproducible) arepeated set of measurements are.
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Uncertainty
Random errors fluctuate from one measurement
to the next and yield results distributed aboutsome mean value.
9
Systematic (bias) errors tend to shift allmeasurements in a systematic way so the mean
value is displaced.
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Uncertainty
An example of random error results from the
measurement of radioactive decay.
10
e process o ra oact ve ecay s ran omin time.
An estimate of how many atoms will decaycan be made but not the ones that will decay.
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Uncertainty
Some examples of bias error are:
Dose-rate dependence of instruments
11
Throwing out high and/or low data points (outliers)
Uncertainty of a radioactivity standard
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Uncertainty
Uncertainty results from:
Measurement errors
12
amp e co ec on errors Sample preparation errors
Analysis errors
Survey design errors Modeling errors
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Assumptions
Nuclear transformations are random,
independent, and occur at a constant rate.
13
-counting time.
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Science, vol. 23414
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Descriptive Statistics
Descriptive statistics characterize a set of
data.
15
Measures of central tendency.
Measures of dispersion.
Descriptive statistics should always beperformed on a set of data.
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Measures Of Central Tendency
The mean is the arithmetic midpoint or more
commonly, the average.
16
,
.
For example, the true mean age of thepeople in this classroom could bedetermined.
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Central Tendency
However, the true mean of radioactive
samples is not usually known.
17
n est mate o t e mean s ma e orradioactive samples.
The estimated mean is designated .
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Central Tendency
The estimated mean :
Xi Xi = an observed
18
n
n = the number ofobservations
(counts) made.
Xn then,as
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Central Tendency
Xn then,as
20
This is the basis for pooling data, or counting
more than once, etc.
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Central Tendency
The median is the point at which half the
measurements are greater and half are less.
21
e me an s a etter measure o centratendency for skewed distributions like
dosimetry data or environmental data.
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Central Tendency
For example, assume we wanted to calculate
the mean salary of everyone in theclassroom.
22
The mean would probably be a good
measure of central tendency.
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Central Tendency
But suppose Bill Gates walked into the
classroom.
23
The median would be a better measure ofcentral tendency for that severely skeweddistribution.
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Measures of Dispersion
The range is the difference between the
maximum and minimum values.
24
e ev at on s t e erence etween agiven value and the true mean, and can be
positive or negative.
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Dispersion
The mean deviation is of little use since
positive and negative values will cancel eachother out.
25
The variance is the square of the mean
deviation, and assures positive values.
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Dispersion
The true variance is:
X i =
2
2)(
26
The true standard deviation is:n
( )n
Xi =2
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Dispersion
In health physics, the sample (group)
variance and sample (group) standarddeviation is used, since the true mean of
27
.
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Dispersion
The sample (group) variance is:
( )1
2
2
=
n
XXS
i
28
The sample (group) standard deviation is:
( )1
2
=
n
XXS
i
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Distributions
Distributions are statistical models for data.
They are used as theoretical means to
29
ca cu ate stat st cs, e t e samp e groupstandard deviation, for example.
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Distributions
The binomial distribution is a fundamental
frequency distribution governing random anddichotomous (2-sided) events.
30
This fits radioactivity well, since
transformation is random and dichotomous.
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Distributions
The Poisson distribution is a special case of
the binomial distribution in which theprobability of an event is small and thesample is large.
31
The Poisson distribution also fits radioactivityvery well, since the probability (p) of any one
atom transforming is small, and a sampleusually consists of a large number of atoms(>100).
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Distributions
The standard deviation is derived from the
mean:
=
33
For example, is the count, and is the
square root of the count.
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PART II:
SINGLE COUNT
34
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Counting Statistics for a
Single Count
The Poisson distribution is important forcounting radioactive samples.
35
We assume that a single count is themean of a Poisson distribution, and the
square root of the count is the standard
deviation.
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Single Count
For example, a single count of a sample was:
counts209=X
36
The standard deviation of that count is:
counts5.14209 ==
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Single Count
This means that 68% of any repeated counts
of this sample would be between:
37
counts.=
5.2235.194 X
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Single Count Rate
Many times in health physics, counting
results are expressed as a count rate (R)rather than a total count (X) by dividing by the
38
.
t
XR =
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Single Count Rate
Continuing with the previous example, if the
count time t were 10 minutes, the count rate(R) would be:
39
cpmcts
R 9.20
min10
209==
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Single Count Rate
The standard deviation of the count rate (R)
can be calculated two ways:
X=
Ror =
40
The next slide shows how these two
equations are equivalent.
t t
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Single Count Rate
t
Rtt
XR
R ==
==
thent
X:XforRtSubstitute
XRtthen
R
41
( )
2
2:sidesbothSquaring
=
tRt
R
( )t
R
t
RtR == 2
2:equationtheReducing
t
RR =:sidesbothofrootsquaretheTaking
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Single Count Rate
t
XR =
t
RR =
42
cpm
counts
R
R
4.1
min10209
=
=
cpm
cpm
R
R
4.1
min109.20
=
=
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Single Count Rate
When calculating the standard deviation for a
count rate, many people feel like they aremaking a mistake because they are dividing
43
.
But this is correct.
tt
X
t
RR
)(
==
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Single Count Rate
So, for the previous example:
cpmcpm
R 4.1min10
9.20==
44
The result would be expressed:
cpmcpm 4.19.20
cpmRcpm 3.225.19or
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Coefficient of Variation
Sometimes it is convenient to express the
uncertainty as a percent of the count.
45
s s ca e t e coe c ent o var at on
(COV) or the percent variation (%error).
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Coefficient of Variation
For a given count:
X
XX
100% =
( ) ( )X 222 100100==
46
2
XX
100%:sidesbothofrootsquaretheTaking =
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Coefficient of Variation
For a given count rate:
R
RR
100% =
t
X
X100
47
( )( )
( ) ( )XX
X
t
X
t
X
R
2
2
2
2
2
2
2
2 100100100
%:sidesbothSquaring ===
XR
100%:sidesbothofrootsquaretheTaking =
t
Xt RR ,
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Coefficient of Variation
Continuing with the previous example:
48
%9.6209
100% ==R
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Coefficient of Variation
Note that the COV decreases with increasing
values of the count.
49
or examp e, t e count was nstea o
209:
%5.3809
100
% ==R
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Coefficient of Variation
The COV can be reduced to an arbitrary
value by increasing the counting time:
50
2
%
1001
=
Rt
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Coefficient of Variation
For example, suppose a 1% uncertainty(error) was desired for the previousexample:
51
Some survey instruments now display%error, so you can count until you reachthe desired % error.
hours8almostormin4781
1009.201
2
=
=
cpmt
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Uncertainty of Net Count Rate
Often in health physics, measurements areexpressed as net count rates.
52
There is uncertainty associated with both
Rg and Rb.
bgnet =
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Net Count Rate
Uncertainties cannot simply be added or
subtracted when two numbers are combined.
53
e proper way to ca cu ate t e com ne
uncertainty with more than one term is to sum
the squares of the uncertainties, then take
the square root.
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Net Count Rate
For example, the uncertainty of a net count
rate is:
54
bg RRRnet +=
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Net Count Rate
RRRnet
t
Rbg
=+=
22
R
22and
55
b
b
g
g
Rnet
b
b
g
g
Rnet
t
R
t
R
t
R
t
R
+=
+
=
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Net Count Rate
For example, suppose the gross count of a
sample was 2,000 counts for a 10 minutecount.
56
The background was 200 counts for a 5
minute count.
What is the net count rate and associated
uncertainty?
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Net Count Rate
cpmcountscounts
Rnet 160200000,2
=
=
57
cpmm
cpm
m
cpmnet 3.5
5
40
10
200=+=
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Net Count Rate
The result would be expressed:
cpmcpm 3.5160
58
cpmRcpm net 3.1657.154
Net Count Rate
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et Cou t ate
Special case: Rate-meters
The uncertainty associated with a reading on
a count rate meter is:
59
constantx time2
ratecount=crm
Time constant = response time
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Net Count Rate
The optimum distribution of a given total
counting time between the gross count andbackground count can be determined by
60
b
g
b
g
R
R
t
t=
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Net Count Rate
Using the previous example:
=R
R
t
t
b
g
b
g
61
Then the time allotted for the gross countshould be 2.23 times that allotted for thebackground.
23.240200 ==cpmcpm
ttb
g
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62
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PART III:
MULTIPLE COUNTS
63
Counting Statistics for Several
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g
Counts
Normally a radioactive sample would be
counted only once, and a Poisson distributionassumed.
64
In other words, the standard deviation is
calculated from the mean.
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Several Counts
However, to determine if uncertainty is due
only to the random nature of radioactivity andnot other sources, the standard deviation can
65
of counts.
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Several Counts
The Gaussian distribution is familiar to many
people as the bell-shaped curve used ingrading.
67
The percentage of counts that will fall within agiven range around the mean are:
1 68.3%
2 95.5% 3 99.7%
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Several Counts
The Gaussian distribution can be described
by the:
X i
68
ean.
Standard deviation.
n
( )
1
2
=
n
XXS
i
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Several Counts
If the two standard deviations are not nearly
equal, then other factors (equipmentproblems) are contributing to uncertainty, and
71
.
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Several Counts
There are several ways to determine nearly
equal.
72
e - quare est s one met o .
The other method is the Reliability Factor,
and well use it in the statistics lab exercise.
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Several Counts
The Reliability Factor is defined as the
standard deviation determined by Gaussianstatistics divided by the Poisson standard
73
.
X
SFR =..
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Several Counts
The RF versus the number of repetitions is
then plotted on a graph such as the one inyour Radiological Health Handbook.
74
One of three conclusions can be drawn fromthe position of the RF value on the graph.
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Several Counts
If the point lies within the inner set of lines,
the uncertainty is due only to the randomnature of radioactivity.
76
,
there is a strong indication that otheruncertainty is involved (instrument calibration,
etc.).
If the point lies between the lines, the result isinconclusive and needs to be repeated.
Repetition Counts ( )2XXi
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1 209 449
2 217 174
3 248 317
4 235 23
5 224 38
77
6 223 52
7 233 8
8 250 392
9 228 5
10 235 23
Total 2302 1481
Mean 230.2
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Several Counts
Gaussian: 2.230
10
2302==X
8.121481
==S
78
Poisson:
2.152.230 === X
2.23010
2302==X
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Several Counts
The RF is:
84.02.158.12 ==RF
79
From the graph, this is an acceptable RF for
10 repetitive counts.
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ReferencesCember H. Introduction to Health Physics(1996).
Gollnick, DA. Basic Radiation Protection Technology, 4th edition (2000).
Knoll, GF. Radiation Detection and Measurement, 2nd edition (1989).
81
National Council on Radiation Protection and Measurements. NCRP Report#112 Calibration of Survey Instruments Used in Radiation Protection for the
Assessment of Ionizing Radiation Fields and Radioactive Surface
Contamination (1991).
Shleien, BS; Slaback Jr., LA; Birky, BK. Handbook of Health Physics and
Radiological Health, 3rd edition (1998).