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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.

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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.

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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?

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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.

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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.

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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.

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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.

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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

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Throwing out high and/or low data points (outliers)

Uncertainty of a radioactivity standard


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Uncertainty

Uncertainty results from:

Measurement errors

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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.

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-counting time.


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Science, vol. 23414


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Descriptive Statistics

Descriptive statistics characterize a set of

data.

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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.

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,

.

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.

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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

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n

n = the number ofobservations

(counts) made.

Xn then,as


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Central Tendency

Xn then,as

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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.

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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.

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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.

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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.

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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.

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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)(

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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

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.


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Dispersion

The sample (group) variance is:

( )1

2

2

=

n

XXS

i

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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

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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.

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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.

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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:

=

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For example, is the count, and is the

square root of the count.


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PART II:

SINGLE COUNT

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Counting Statistics for a

Single Count

The Poisson distribution is important forcounting radioactive samples.

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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

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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:

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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

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.

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:

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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 =

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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

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( )

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==

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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.

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s s ca e t e coe c ent o var at on

(COV) or the percent variation (%error).


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For a given count:

X

XX

100% =

( ) ( )X 222 100100==

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2

XX

100%:sidesbothofrootsquaretheTaking =


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For a given count rate:

R

RR

100% =

t

X

X100

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( )( )

( ) ( )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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Continuing with the previous example:

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%9.6209

100% ==R


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Note that the COV decreases with increasing

values of the count.

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or examp e, t e count was nstea o

209:

%5.3809

100

% ==R


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The COV can be reduced to an arbitrary

value by increasing the counting time:

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2

%

1001

=

Rt


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For example, suppose a 1% uncertainty(error) was desired for the previousexample:

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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.

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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.

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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:

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bg RRRnet +=


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Net Count Rate

RRRnet

t

Rbg

=+=

22

R

22and

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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.

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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

=

=

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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

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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:

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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

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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

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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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PART III:

MULTIPLE COUNTS

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Counting Statistics for Several


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g

Counts

Normally a radioactive sample would be

counted only once, and a Poisson distributionassumed.

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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

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of counts.


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Several Counts

The Gaussian distribution is familiar to many

people as the bell-shaped curve used ingrading.

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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

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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

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.


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Several Counts

There are several ways to determine nearly

equal.

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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

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.

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.

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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.

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,

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

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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

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Poisson:

2.152.230 === X

2.23010

2302==X


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Several Counts

The RF is:

84.02.158.12 ==RF

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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).

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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).

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