Counting Statistics

Applied Health Physics Applied Health Physics

Oak Ridge Associated Universities
Objectives

z To review the need for statistics in health phz To review measures of central tendency and 

dispersion.

z To review common statistical models (distrib

To review common statistical models (distrib

z To calculate the uncertainty for a count and 

rate.

z To review the statistics for a single count (ra

versus those for a series of counts of the samsample.
Introduction

z It is impossible to directly measure rad

in every situation at every point in spactime.

z It is impossible to know with complete 

certainty the level of radiation when it wdirectly measured.
Introduction

z One of the goals of statistics is to mak

accurate inferences based on incompledata.

z Using statistics, it is possible to extrapo

from a set of measurements to all measurements in a scientifically valid w
Introduction

Statistical tests allow us to answer quesuch as:

z How much radioactivity is present?

y p

z How accurate is the measured result?z Is the detector working properly? (Are the 

results reproducible (precise))?

z Is there bias error in addition to random er

measurement?

z Is this radioactive?
Introduction

z Without statistics, it is difficult to answe

these questions.St ti ti ll t ti 

z Statistics allow us to answer questions 

degree of confidence that we are drawright conclusion.
PART I:

DEFINE THE TERMS
Uncertainty (Error)

Uncertainty (or error) in health physics difference between the true average ratransformation (decay) and the measurate. 

z Accuracy refers to how close the measure

to the true value. 

z Precision refers to how close (reproducible

repeated set of measurements are.
Uncertainty

z Random errors fluctuate from one measur

to the next and yield results distributed absome mean value.

z Systematic (bias) errors tend to shift all 

measurements in a systematic way so the value is displaced.
Uncertainty

z An example of random error results fro

measurement of radioactive decay.Th f di ti d i 

z The process of radioactive decay is ra

in time.

z An estimate of how many atoms will de

can be made but not the ones that will 
Uncertainty

Some examples of bias error are:

z Dose-rate dependence of instrumentsz Throwing out high and/or low data points (z Uncertainty of a radioactivity standard
Uncertainty

Uncertainty results from:

z Measurement errors

S l ll ti

z

Sample collection errors

z Sample preparation errorsz Analysis errorsz Survey design errorsz Modeling errors
Assumptions

z Nuclear transformations are random, 

independent, and occur at a constant rThe physical half life is long compared 

z The physical half-life is long compared 

counting time.
Science, vol. 234
Descriptive Statistics

z Descriptive statistics characterize a se

data.

z Descriptive statistics include:

Descriptive statistics include:

z Measures of central tendency.z Measures of dispersion.

z Descriptive statistics should always be 

performed on a set of data.
Measures Of Central Tendenc

z The mean is the arithmetic midpoint or 

commonly, the average. 

z If the true mean is known it is designa

If the true mean is known, it is designa

μ. 

z For example, the true mean age of the 

people in this classroom could be determined.
Central Tendency

z However, the true mean of radioactive 

samples is not usually known.A ti t f th i d f

z

An estima

te o

f the mean is ma

de 

for 

radioactive samples.

z The estimated mean is designated . X
Central Tendency

The estimated mean :

X

X

∑

i

=

X

•

Xi = an obse

count (result).

n

X

= count (result).

•n = the numobservations (counts) made.

as

n

⎯⎯→

∞

, then 

X

⎯⎯→

Central Tendency

as

n

⎯⎯→

∞, then X

⎯⎯→

This is the basis for pooling data, or comore than once, etc. 
Central Tendency

z The median is the point at which half th

measurements are greater and half are Th di i b tt f 

z The median is a 

b

etter measure o

f cen

tendency for skewed distributions like dosimetry data or environmental data.
Central Tendency

z For example, assume we wanted to ca

the mean salary of everyone in the classroom.

z The mean would probably be a good 

measure of central tendency.
Central Tendency

z But suppose Bill Gates walked into the 

classroom.

z The mean salary would not reflect my 

The mean salary would not reflect my 

z The median would be a better measure 

central tendency for that severely skewdistribution.
Measures of Dispersion

z The range is the difference between th

maximum and minimum values.Th d i ti i th diff b t

z The 

deviation is the difference 

b

e

twee

given value and the true mean, and capositive or negative. 
Dispersion

z The mean deviation is of little use sinc

positive and negative values will canceother out.

z The variance is the square of the mean 

deviation, and assures positive values
Dispersion

z The true variance is:

∑ X i −

=

2

2 ( μ ) σ

z The true standard deviation is:

n

( )

n

∑ X i −

=

2

μ

σ
Dispersion

In health physics, the sample (group) variance and sample (group) standard deviation is used, since the true mean radioactive samples is usually not knowradioactive samples is usually not know
Dispersion

z The sample (group) variance is:

( )

1

2

2

−

−

=

∑

n

X X

S i

z The sample (group) standard deviation 

( )

1

2

−

−

=

∑

n

X X

S i
Distributions

z Distributions are statistical models for dz They are used as theoretical means to 

l l t t ti ti lik th l ( 

calculate statistics, like the sample (gro

standard deviation, for example.
Distributions

z The binomial distribution is a fundame

frequency distribution governing randodichotomous (2-sided) events.

z This fits radioactivity well, since 

transformation is random and dichotom
Distributions

z The Poisson distribution is a special ca

the binomial distribution in which the probability of an event is small and the sample is large.

z The Poisson distribution also fits radioa

very well, since the probability (p) of anatom transforming is small, and a samusually consists of a large number of a(>100).
Distributions

The Poisson distribution is simple to usmodel, because it is characterized by oindependent parameter, the mean.

μ = np
Distributions

z The standard deviation is derived from 

mean:

σ = μ

z For example, μ is the count, and σ is th

square root of the count.
PART II:

SINGLE COUNT
Counting Statistics for a 

Single Count

z The Poisson distribution is important 

counting radioactive samples.

z We assume that a single count is the 

mean of a Poisson distribution, and thsquare root of the count is the standadeviation.
Single Count

z For example, a single count of a samp

X = 209counts

z The standard deviation of that count is

σ = 209 = 14.5 counts
Single Count

This means that 68% of any repeated of this sample would be between:

X 209 ± 14 5 tX = 209 ± 14 .5 counts

194.5 ≤ X ≤ 223.5
Single Count Rate

Many times in health physics, counting results are expressed as a count rate (rather than a total count (X) by dividing count time (t)count time (t).

t

X

R

=
Single Count Rate

Continuing with the previous example, count time t were 10 minutes, the coun(R) would be:

cpm

cts

R 20 .9

10 min

209

=

=
Single Count Rate

z The standard deviation of the count rat

can be calculated two ways:

X

σ

R

=

R

o

r

σ

R

=

z The next slide shows how these two 

equations are equivalent.

t

σ

R

t

R
Single Count Rate

t

Rt

t

X

R

=

R

=

=

=

σ then 

σ

t

X Substitute Rt for 

X :

then Rt

X

R

( )

2

2 Squaring both sides : ⎟

⎟⎠⎞

⎜⎜⎝⎛

=

t

Rt

σ

R

( )

t

R

t

Rt

R

=

=

2

2 Reducing the equation : σ

t

R Taking the square root of both sides :σ

R

=
Single Count Rate

t

X

σ

R

=

t

R

σ

R

=

cpm

counts

RR

1.4

10 min

209

==

σ

σ

cpm

cpm

RR

1.4

10 min

20 .9

==

σ

σ
Single Count Rate

z When calculating the standard deviatio

count rate, many people feel like they amaking a mistake because they are “dby t twice

”

by

t twice .

z But this is correct.

tt

X

t

R

R

(

)

σ

=

=
Single Count Rate

z So, for the previous example:

cpm cpm

R

1.4

10 min

20 .9

σ

=

=

z The result would be expressed:

20.9cpm

±

1.4cpm

or 19.5cpm

≤

R

≤ 22.3cpm
Coefficient of Variation

z Sometimes it is convenient to express 

uncertainty as a percent of the count.Thi i ll d th ffi i t f i ti

z This is call

e

d the coeffi

cien

t o

f variatio

(COV) or the percent variation (%error
Coefficient of Variation

For a given count:

X

X

X

σ

σ

100

%

=

(

) ( )

X ( 

X

2

2 100 100 Squaring both sides: %σ

=

=

(

)

X

X

%

X

2 Squaring both sides: σ

X

Taking the square root of both sides : %

σ

=
Coefficient of Variation

For a given count rate:

R

R

σ

σ

100

%

=

t

X

X 100 Since

σ

= then

%

σ

=

( )

( ) ( ) 

X

t

X

t

X

R

2

2

2

2

2

2

2 100 

100 Squaring both sides : %

σ

=

=

R

Taking the square root of both sides : %

σ

=

t

t

X

R

%

R Since

σ

= , then

σ

=
Coefficient of Variation

Continuing with the previous example:

6.9

%

209

100

%

σ

R

=

=
Coefficient of Variation

z Note that the COV decreases with incr

values of the count.F l if th t 809 i t 

z

For example, if the coun

t was 809 ins

te

209:

3.5

%

809

100

%

σ

R

=

=
Coefficient of Variation

The COV can be reduced to an arbitravalue by increasing the counting time:

2

%

1 100 ⎟

⎠⎞ ⎜

⎝⎛

=

R

σ

t
Coefficient of Variation

z For example, suppose a 1% uncertai

(error) was desired for the previous example:

z Some survey instruments now displa

%error, so you can count until you rethe desired % error.

478 min or almost 

8 

1

100

20 .9

1

2

⎟ =⎠⎞ ⎜

⎝⎛

⎟⎟⎠⎞

⎜⎜⎝⎛

=

cpm

t
Uncertainty of Net Count Rat

z Often in health physics, measuremen

expressed as net count rates.

R

R

R

z There is uncertainty associated with b

R

g and R

b.

Rnet

=

R

g

−

R

b
Net Count Rate

z Uncertainties cannot simply be added 

subtracted when two numbers are comTh t l l t th bi 

z The proper way 

to calcula

te the combi

uncertainty with more than one term is the squares of the uncertainties, then tthe square root.
Net Count Rate

For example, the uncertainty of a net crate is:

2

2

Rnet

Rg

Rb

σ

=

σ

+

σ
Net Count Rate

Rnet

R

R

t

R

g

b

⎛

⎞

⎛

⎞

σ

=

σ

+

σ

σ

=

2

2

R

2

2 and

b

b

g

g

Rnet

b

b

g

g

Rnet

t

R

t

R

t

R

t

R

=

+

⎟⎟⎠

⎞

⎜⎜⎝⎛

+ ⎟

⎟⎠

⎞

⎜⎜⎝⎛

=

σ

σ

2
Net Count Rate

z For example, suppose the gross count 

sample was 2,000 counts for a 10 mincount.

z The background was 200 counts for a 

minute count.

z What is the net count rate and associa

uncertainty?
Net Count Rate

m

counts

m

counts

Rnet 16

5

200

10

2

,000 ⎟ =

⎠⎞ ⎜

⎝⎛ ⎟ −

⎠⎞ ⎜

⎝⎛

=

⎝ 10

m

⎠

⎝

5

m

⎠

cp

m

cpm

m

cpm

net

5.3

5

40

10

200

σ

=

+

=
Net Count Rate

The result would be expressed:

160

cp

m

±

5.3

cp

m

p

p

154.7 cpm

≤

Rnet

≤165.3 cpm
Net Count Rate

Special case: Rate-meters

The uncertainty associated with a reada count rate meter is:

2 x time constant

count rate

σ crm

=

Time constant = response time
Net Count Rate

The optimum distribution of a given totcounting time between the gross count background count can be determined bdetermining the ratio:determining the ratio:

b

g

b

g

R

R

t

t

=
Net Count Rate

z Using the previous example:

=

R

R

t

t

b

g

b

g

z Then the time allotted for the gross cou

should be 2.23 times that allotted for thbackground.

2.23

40

200

=

=

cpm

cpm

t

t

b

g

PART III:

MULTIPLE COUNTS
Counting Statistics for Sever

Countsz Normally a radioactive sample would b

counted only once, and a Poisson distassumed.

z In other words, the standard deviation 

calculated from the mean.
Several Counts

However, to determine if uncertainty is only to the random nature of radioactivnot other sources, the standard deviatibe determined experimentally from a sbe determined experimentally from

a s

of counts.
Several Counts

z This is the approach taken in quality co

testing of equipment.Th di t ib ti f t f 

z The di

s

t

rib

ution o

f coun

ts 

from a samp

repeatedly counted is best described bGaussian (or normal) distribution.
Several Counts

z The Gaussian distribution is familiar to 

people as the “bell-shaped” curve used grading.

z The percentage of counts that will fall w

given range around the mean are:

z

±

1

σ 68.3%

z

±

2

σ 95.5%

z

±

3

σ 99.7%
Several Counts

The Gaussian distribution can be descby the:

M

X

X

∑ i =

z

Mean.

z Standard deviation.

n

X

=

( ) 1

2

−

−

=

∑

n

X

X

S i
Several Counts

z For a sufficiently large sample (mean > 

the Poisson distribution approximates Gaussian distribution.

z Therefore, if the only uncertainty is from 

radioactivity, the standard deviation of series of counts of the same sample determined assuming a Gaussian distrshould nearly equal the standard deviadetermined assuming a Poisson distrib

Several Counts

If the two standard deviations are not nequal, then other factors (equipment problems) are contributing to uncertainshould be eliminatedshould be eliminated.
Several Counts

z There are several ways to determine “

equal”.Th Chi S T t i th d

z The Chi

-

Square 

Tes

t is one meth

o

d.

z The other method is the Reliability Fac

and we’ll use it in the statistics lab exe
Several Counts

The Reliability Factor is defined as the standard deviation determined by Gaustatistics divided by the Poisson standadeviation

.

X

S

R .F . =
Several Counts

z The RF versus the number of repetitio

then plotted on a graph such as the onyour Radiological Health Handbook.

z One of three conclusions can be drawn 

the position of the RF value on the gra

Several Counts

z If the point lies within the inner set of li

the uncertainty is due only to the randonature of radioactivity.

z If the point lies outside the outer set of 

If the point lies outside the outer set of 

there is a strong indication that other uncertainty is involved (instrument calietc.).

z If the point lies between the lines, the r

inconclusive and needs to be repeated
Repetition Counts

1 209 4492 217 1743 248 3174 235 235 224 38

( )

2

Xi −

X

6 223 527 233 88 250 3929 228 5

10 235 23

Total 2302 1481

Mean 230.2
Several Counts

z Gaussian: 230.2

10

2302

X

=

=

12.8

9

1481

S

=

=

z Poisson:

9

σ

=

X

= 230.2

=15.2

230.2

10

2302

X

=

=
Several Counts

z The RF is:

0.84

15.2

12.8 RF

=

=

z From the graph, this is an acceptable R

10 repetitive counts.

References

Cember H. Introduction to Health Physics(1996).Gollnick, DA. Basic Radiation Protection Technology, 4th edition (20

Knoll, GF. Radiation Detection and Measurement, 2nd edition (1989

National Council on Radiation Protection and Measurements. NCR

#112 Calibration of Survey Instruments Used in Radiation ProteAssessment of Ionizing Radiation Fields and Radioactive SurfacContamination (1991).

Shleien, BS; Slaback Jr., LA; Birky, BK. Handbook of Health Physic

Radiological Health, 3rd edition (1998).