Lowenthal G., Airey P. Practical Applications of Radioactivity and Nuclear Radiations

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In this chapter calculations will be concerned only with m and s, which

should always be calculated from as many repeat measurements as practical,

all taken in identical conditions. There are many applications when it is

impractical to make more than a few repeat measurements in identical

conditions, so adding to uncertainties which can be allowed for using the

Student t factor (Table 6.4(b)).

6.6.3 Gaussian statistics

The Gaussian distribution deals with differences between results of individual

counts n

,..., n

(all taken in identical conditions and corrected as

required), and their arithmetic mean m. Being symetrical, it can be applied,

e.g. to the scatter of results of repeated weighings of a solution about their

mean. These and similar results follow Gauss statistics but not Poisson

statistics.

Writing m7n

= d

, the normalised Gauss function is generated from the

expression:

Pn

 w expÿd

I=2s

; (6.10)

where w is a constant depending on the Gaussian standard deviation s

and

so on the width of the distribution which, when processing decay rates, has to

conform to the width calculated in accordance with the Poisson distribution.

As noted earlier, both m and s

should be calculated from as many repeat

measurements as possible, all taken in identical conditions, so obtaining n

results (degrees of freedom). For the large majority of applications i would

6.6 Poisson and Gaussian statistics 171

Figure 6.10. The Poisson distribution (Eq. 6.7) calculated for a mean (m) of 2.5 cps

when the asymmetry is pronounced (Debertin and Helmer, 1988, Figure 1.22).

rarely be as large as 20. Uncertainties due to a smaller number of repeat

measurements could be allowed for using the Student t factor (Table 20,

NCRP, 1985). Bearing this in mind, the arithmetic mean is given by:

m 

i1

=i: 6:11

Having to calculate m reduces the number of independently obtained

results (degrees of freedom) from n

to (n

71).

One can now calculate s

from the sum of the squares of the differences

between the mean m and individual n

values i.e.

i1

m ÿ n



when:

Measurements and results172

Table 6.4. Con®dence limits and Student t factors.

(a) Con®dence limits applying to Poisson and Norm al distributions (Section 6.6.4).

Con®dence limits about the

mean as multiples of the 0.674 1.00 1.64 2.00 3.00 3.30

standard deviation s

Probability (%)

(a)

50 68.3 90 95.4 99.7 99.9

(b) Student t factors.

(b)

Number of counts (n) t factors for use in the stated intervals, see (a)

68.3 95.4 99.7

3 1.32 4.3 19.2

4 1.20 3.2 9.2

5 1.15 2.8 6.9

6 1.11 2.6 5.5

8 1.08 2.4 4.5

10 1.06 2.3 4.1

20 1.03 2.1 3.4

50 1.01 2.0 3.2

100 1.00 2.0 3.1

(a)

These are the probabilities that countrates will, on average, fall within the stated

range.

(b)

If ®ve successive measurements of a countrate yield m = 25.0 cps the standard

deviation is not simp ly 25.0

1/2

but 1.1625.0

1/2

, i.e. 6 to the nearest integer.

25  5 could be assumed to apply for >20 repeat measurements. t factors are often

ignored; whether or not they should be used depends on the required accuracy.



i1

m ÿ n



n

ÿ 1

1=2

: 6:12

With the mean m, s

and s

de®ned and assuming that m is large enough

for Gaussian statistics to apply, it can be expected (Table 6.4(a)), that 68.3%

of the individual results are, on average, within (ms

) and 31.7% are

outside these limits. It should be rare (on average no more than one case in

twenty) that the difference exceeds 2s

, and in only about three cases in

1000 should the difference exceed 3s

(Figure 6.11). If differences between

the mean and individual counts are signi®cantly smaller or larger than

expected from the rules just stated, this should be investigated in case

randomness has been interfered with.

The series of measurements used to calculate m in Eq. (6.11) could be

repeated, say n

times in otherwise identical conditions to obtain a series of

means, m

, m

...,m

. Making decay corrections, if necessary, and assuming

randomness, these mean values will agree more closely among themselves

6.6 Poisson and Gaussian statistics 173

Figure 6.11. A Gaussian distribution calculated from Eq. (6.10), showing con®dence

intervals in terms of the standard deviation (Eq. (6.12)).

than did the counts n

, n

,...,n

and their mean m. The Gaussian standard

deviation for these means, s

, can be shown to be given by

 s

=n



1=2

; (6.13)

which is closely equal to:



i1

m ÿ n



n

ÿ 1

1=2

: 6:14

If calculations are made using the Poisson standard deviation (Eq. (6.8)),

one obtains s

= m

1/2

and for i consecutive counts in identical conditions and

for the same counting period one has:

i  m



1=2

 100i  m



ÿ1=2

%: (6.15)

6.6.4 Con®dence limits

Con®dence limits provide information on the reliability of uncertainty

statements and should always be quoted together with the uncertainty to

which they refer. When countrates conform to Poisson and Gaussian

statistics, the con®dence limit is equal to the standard deviation or a multiple

thereof (Table 6.4(a)). However, occasional deviations from a result expected

on statistical grounds could always occur by chance.

If the number of repeat counts has to be kept small (say <10), the

veri®cation of randomness should be effected using the Student t factor

(Table 6.4(b)). For instance, to be con®dent of a 68.3% con®dence level for a

distribution due to only six repeat counts, the calculated value of s

(Eq.

(6.9)) should be multiplied by 1.11 (Table 6.4(b)). To realise a 99.7%

con®dence level, the multiplier is not 3 (Table 6.4(a)) but 5.5 (Table 6.4(b)).

This increase in the standard deviation is introduced to compensate for the

added uncertainty due to the small number of repeat measurements.

The situation is different when estimating con®dence levels to allow for

systematic uncertainties (Section 6.5.2). To obtain reliable estimates it may be

advisable for the measurements to be repeated often enough to permit this to

be done realistically since con®dence levels, to be helpful, should be neither

too large nor too small.

Measurements and results174

6.7 Other characteristics of results and statistical tests

6.7.1 Countrates and their combination

Whenever practical, results of counts of radioactive decays should be stated

as countrates, r = n/t where n is the number of counts in time t seconds so that

n/t is expressed in counts per second (cps).

If the mean of a series of countrates is r

, Poisson statistics predict:

 s

 n=t  n

1=2

=t  r

r

=t

1=2

; (6.16)

while the fractional standard deviation (Eq. (6.9)), expressed as a countrate

becomes:

r

=t

1=2

 r

 1=r

t

1=2

 r

 100=r

t

1=2

 r

 100=n

1=2

6:17

since r  t  n (see above).

There are experimental situations when several results, A, B, . . . , each

with its standard deviation, s

, s

, . . . , have to be combined by either

addition or subtraction. One obtains the standard deviation s

of the sum or

difference by quadrature summing (Section 6.5.2), i.e.

s

 s



1=2

: (6.18)

Similarly, when two or more results are combined as a product, say

 R

or a quotient R

with standard deviations s

and s

, the

fractional standard deviation of the product s

/P is given by:

=P s



s

1=2





1=2

: (6.19)

Results of statistical distributions are the more accurately de®ned the

larger their mean m and therefore the smaller the fractional standard

deviation s/m. However, all countrates must be equally probable or they

should be weighted to allow for any differences.

If countrates are low, a large value of m can be realised only over a long

counting time. This could create dif®culties due to instabilities in the counting

equipment especially so if the count plus background rate, r

is similar to the

background rate, r

. It is then helpful to divide the available counting time

between t

and t

so that r

, the countrate due to the source calculated from

is measured with the smallest realisable uncertainty. It can be shown

that this applies if the counting times t

and t

per source are in the ratio:

6.7 Tests for randomness 175

r



1=2

: (6.20)

The standard deviation applicable to the source is then:

r

 r



1=2

: (6.21)

Clearly set out examples are offered by Shapiro (1972, Pt. IV, Section 6).

6.7.2 Tests for accuracy and consistency

Accuracy

Figures 6.12(a) and (b) show two superimposed histograms of counting

results. Each was obtained with 100 repeat counts taken in identical

conditions at a countrate close to 1.1 cps, but the counting times were

respectively 10 and 100 seconds. The standard deviation, s

, was calculated

from Eq. (6.12). For the 10s counts, s

/m was 3.22/11.25 i.e. s

is about

29% of m; for the 100s counts s

/m equals 9.10/111.2 i.e. s

is about 8.2% of

m. The result is a signi®cantly more tidy histogram. For highly accurate

results one aims at a near-smooth distribution when s/m values should be

0.1% or less. In that case a plot of the observed data (assuming random

conditions), can be expected to closely approximate a Gaussian curve. The

ratio of the two s

/m values, 29%/8.2%, is 3.5, which is close to the ratio of

Measurements and results176

Figure 6.12. Two histograms each obtained for 100 repeat counts at a count rate

close to 1.1 cps. (a) The histogram for 10 s counts. Average = 11.25, standard

deviation = 3.22. (b) The histogram for 100 s counts. Average = 111.2, standard

deviation = 9.10 (Shapiro, 1972, Fig. 4.19).

the square root of the mean counts, 111.2

1/2

/11.25

1/2

, or 3.1 as predicted by

Eq. (6.9).

Figure 6.11 shows a Gaussian distribution obtained from a very large

number of calculated results (Eq. (6.10)) so explaining the smooth curve in

contrast to the histograms in Figure 6.12. The area between the curve and the

base line (the Gaussian function extends from 7? to +?) represents all

possible results while the areas between m1s, m2 s and m3s represent

respectively 68.3, 95.4 and 99.7% of the total to a high degree of accuracy

(Table 6.4(a)).

Consistency

It is often important to verify that conditions in a system that is regularly

employed for countrate measurements either remain unchanged from one

day to the next (or longer), or change by a speci®ed amount in line with

changes in experimental parameters. A procedure developed for such tests,

which causes minimum interference with normal requirements, is the so-

called t-distribution. The data used for the example below can be found in

NCRP (1985, Table 20).

Following two sets of measurements taken at different times but otherwise

in identical conditions, one calculates the t factor from the expression:

t m

ÿ m

=s

 s



1=2

=dn

1=2

; (6.22)

where the symbols have the same meaning as in Section 6.6.3 and s could be

obtained either from Eq. (6.8) or Eq. (6.12). For a comparison between two

sets of readings one has dn =2.

Let sets of ten readings each be taken with a long-lived source (decay is

negligible) one week apart, to verify the stability of a process. Suppose the

recorded results were: m

= 3830, m

= 3604, s

= 87.8, s

= 195.1, with dn =2

when Equation (6.22) then yields t = 2.99.

From Table 20 in NCRP (1985) the critical value for 18 degrees of freedom

(10 + 10 7 2) and a probability level of 0.01 is close to 2.90, less than the

calculated 2.99, so suggesting that the probability for these two sets to

indicate consistent conditions is less than 1% and certainly less than 5%,

suggesting that the process could have been disturbed.

6.7.3 Tests for randomness

A simple test to verify randomness is known as the double check test. It can

be applied when the mean, m , exceeds at least 15 and so is large enough for

the standard deviations to be calculated either by Poisson or Gaussian

6.7 Tests for randomness 177

statistics. If the results obtained with the two statistics agree within less than

about 10% this can be accepted as evidence for randomness. A failure to pass

this test, especially when values of m are known to be accurate, say 1% or

better, should put experimenters on the alert when looking for errors.

A more powerful test is the x

(or chi squared) test. The data needed for

this test when the number n of repeat counts is between 6 and 25 are shown in

Table 6.5. Using the same notation as in Equation (6.12), the x

function can

be shown to be de®ned by:



i1

m ÿ n





m: 6:23

For a distribution which obeys Poisson and Gaussian statistics, one has:

 m

Measurements and results178

Table 6.5. A selection of chi squared values.

Degrees of

There is a probability of

freedom 0.99 0.95 0.90 0.50 0.10 0.05 0.01

(n71)

(a)

that the calculated value of chi squared will be equal to or greater than:

5 0.554 1.145 1.610 4.351 9.236 11.070 15.086

6 0.872 1.635 2.204 5.348 10.645 12.592 16.812

7 1.239 2.167 2.833 6.346 12.017 14.067 18.475

8 1.646 2.733 3.490 7.344 13.362 15.507 20.090

9 2.088 3.325 4.168 8.343 14.684 16.919 21.666

10 2.558 3.940 4.865 9.342 15.987 18.307 23.209

11 3.053 4.575 5.578 10.341 17.275 19.675 24.725

12 2.571 5.226 6.304 11.340 18.549 21.026 26.217

13 4.107 5.892 7.042 12.340 19.812 22.362 27.688

14 4.660 6.571 7.790 13.339 21.064 23.685 29.141

15 5.229 7.261 8.547 14.339 22.307 24.996 30.578

16 5.812 7.962 9.312 15.338 23.542 26.296 32.000

17 6.408 8.672 10.085 16.338 24.769 2 7.587 33.409

18 7.015 9.390 10.865 17.338 25.989 2 8.869 34.805

19 7.633 10.117 11.651 18.338 27.204 30.144 36.191

20 8.260 10.851 12.443 19.337 28.412 31.410 37.566

21 8.897 11.591 13.240 20.337 29.615 32.671 38.932

22 9.542 12.338 14.041 21.337 30.813 33.924 40.289

23 10.196 13.091 14.848 22.337 32.007 35.172 41.638

24 10.856 13.848 15.659 23.337 33.196 36.415 42.980

(a)

The number of degrees of freedom is usually one less than n, the number of repeat

measurements.

(see Eq. (6.8)). Also squaring both sides of Eq. (6.12) yields:



i1

m ÿ n



=n

 1:

Substituting these relations into Eq. (6.23) yields:

n

ÿ 1: (6.24)

Here n

again represents the number of equally probable observations (see

Table 6.5). One degree of freedom was used to calculate the mean, making it

necessary to replace n

by n

71 as was done in Eq. (6.12).

A high probability for randomness applies when x

does not differ greatly

from 0.50 (50%) though many experimenters accept x

in the range 10 to 90%

as admissable evidence for randomness. However, results smaller than 10%

or larger than 90% strongly suggest non-randomness.

Suppose an experimenter measured the countrate from a radioactive

source, accumulating 20 counts to obtain m = 312 cps, s

= 5.3 cps, s

= 3.9

cps and x

= 33. These results suggest a signi®cant non-random effect. The

double check test shows a difference between s

and s

of over 30% instead

of under 10% as expected for randomness. Also, with (n71) = 19 (Table 6.5),

a x

value of 33 suggests a less than 5% probability for randomness. Given

these very different results from what is expected for randomness, the

experimenter has to discover what caused them and how the results could be

improved.

6.8 Moving on to applications

This chapter ends the introductory part of this book. It was written to

familiarise readers, if only to some extent, with the characteristics of the

nuclear radiations commonly employed for applications in industry, tech-

nology and related ®elds, and with the basic properties of the radionuclides

that emit these radiations. This was done at somewhat greater length than is

normal for books dealing with applications because the basic nuclear sciences

are no longer as widely taught in schools and universities as they used to be.

Nevertheless, experimenters will frequently ®nd it necessary to look for

additional information among the quoted references, in other textbooks or

via the Internet, as listed in Table A3.1 in Appendix 3.

The next three succeeding chapters are the core of this book. The nuclear

radiation applications discussed in these chapters will often deal with large-

scale undertakings: malfunction in large industrial plants, extensive searches

6.8 Applications 179

for water or oil, or safe procedures for the dispersal of high-activity nuclear

waste to name just a few. Descriptions of large projects make it necessary to

arrange the material in Chapters 7, 8 and 9 in a somewhat different way from

what went before. Whenever practical, use will be made of the foregoing

discussions of the characteristics of radionuclides and nuclear radiations.

More often it will be necessary to refer experimenters to the scienti®c

literature. On the whole readers will be encouraged to develop their own

procedures and apply them to suit their interests.

Measurements and results180