Making the Most of Uncertain
Low-Level Measurements
Presented to the Savannah River Chapter of the Health Physics Society
Aiken, South Carolina, 2011 April 15
Daniel J. Strom, Kevin E. Joyce, Jay A. MacLellan, David J. Watson,
Timothy P. Lynch, Cheryl. L. Antonio, Alan Birchall, Kevin K.
Anderson, Peter A. Zharov
Pacific Northwest National Laboratory
strom@pnl.gov +1 509 375 2626
PNNL-SA-75679
Prologue
• Uncertainty is different for sets of sets
of data than it is for single data points
• If you have more than one uncertain
measurement, you need to learn about
measurement error models
• HPs generally do not speak the
language of statisticians well enough
to be comprehended
– σ is not a synonym for standard deviation
– s is not σ is not ˆ
• We have to get smarter!
– Or some biostatistician will commit
regression calibration on our numbers!
Carroll RJ, D Ruppert, LA Stefanski, and CM Crainiceanu. 2006. Measurement Error in
Nonlinear Models: A Modern Perspective. Chapman & Hall/CRC, Boca Raton.
2
Outline
•
•
•
•
•
•
•
•
Censoring
The lognormal distribution
Measurements and measurands
Requirements and assumptions for this novel method
Population variability and measurement uncertainty
Disaggregating the variance
Distribution of measurands
The “everybody” prior
3
Outline 2
•
•
•
•
•
•
•
Probability distributions for individual measurands
The Bayesian approach
The “everybody else” prior
Applications to real radiobioassay data
The importance of accurate uncertainty
Bohr’s correspondence principle
Conclusions
4
Censoring
• Changing a measurement result
• Common practices
– Set negative values to 0
– Set all results less than some value to
• 0
• ½ the value
• The value
• A non-numeric character like “M”
• Changing measurement results causes great problems in
statistical inference
– DR Helsel. 2005. Nondetects and data analysis. Statistics for censored
environmental data. John Wiley & Sons.
• This method requires uncensored data
5
The Lognormal Distribution
• Frequently observed in Nature
• Multiplication of arbitrary distributions results in
lognormals
Ott WR. 1990. A Physical Explanation of the Lognormality of Pollutant Concentrations. J.Air Waste
Mgt.Assoc. 40 (10):1378-1383
6
Measurand, Measurement, Error, and
Uncertainty (ISO)
• measurand: particular quantity subject to measurement
– also, the “true value of the quantity subject to measurement”
• result of a measurement: value attributed to a
measurand, obtained by measurement
• error: the unknown difference between the measurand
and the measurement
– this is a different meaning from the theoretical concept in
statistics!
• uncertainty: a quantitative estimate of the magnitude of
the error
– statisticians often do not distinguish between error and
uncertainty and may use them synonymously
7
Requirements and Assumptions
• This method requires uncensored data
– small values are reported as they are calculated, with no
rounding, setting negative values to zero, or otherwise
changing
• Assume measurands are lognormally distributed
– Many populations in nature are lognormally distributed
– Lognormal common in radiological and environmental
measurements
– Other functions could be used as long as they have a mean
8
Population Variability and
Measurement Uncertainty
• The sample variance of a set of measurements on a
population arises from two sources:
– population variability
– measurement error
• If measurements have no error, then all observed sample
variance is due to variability in the population
9
Measurement Error Model
• True values (measurands) ti give rise to measured
values mi
• We have good independent estimates of the combined
standard uncertainty ui of each measurement mi
mi = ti + ui
ui ~ N(0, ui2)
• We calculate the sample variance of mi
• We use sample variance and a summary measure of the
ui to estimate the variance due to population variability
of ti
10
Spread of Measurement Results
(Sample Variance) Is Due to 2 Causes
“Average”
Measurement
Uncertainty
Variability within Population
11
Spread of Measurement Results
(Sample Variance) Is Due to 2 Causes
uRMS
Variability within Population
12
Spread of Measurement Results
(Sample Variance) Is Due to 2 Causes
uRMS
θ
ˆ (ti )
2
s 2 (mi )  ˆ 2 (ti )  uRMS
13
Estimating the
Variance of the Distribution of Measurands
ˆ 2  i 
Estimated Variance
of the Measurands
☑ Calculated

s 2  xi 
Sample Variance
of the Measurements
☑ Known
-
2
uRMS
Mean Square
Measurement
Uncertainty
☑ Known
• The “reliability” or “attenuation” or “variability fraction”
2
2
is
ˆ
ˆ

(
t
)

(ti )
2
i
r  2
 2
2
ˆ (ti )  uRMS s (ti )
• Analogous to a correlation coefficient
– r2: fraction of variance explained by model
– r′ 2: fraction of variance due to measurand variability
14
Distribution of Measurands
• The estimated variance of the measurands is ˆ 2 (ti )
• Assume measurands are lognormally distributed
• Assume the expectation of the measurands equals the
mean of the measurements:
E (t )  m
– measurements are unbiased
– this assumption respects the data
• Calculate the parameters of the lognormal
– geometric mean
– geometric standard deviation sG
• This is the distribution of “possibly true values”
15
Analysis of Baseline Radiobioassay Data
•
90Sr:
128 baseline urine bioassays
– Everyone is exposed to global fallout
– gas proportional counter
– 100-minute counts
•
137Cs:
5,337 baseline in vivo bioassays
– Everyone is exposed to global fallout & Chernobyl
– coaxial high-purity germanium (HPGe) scanning system
– 10-minute scans
•
239+240Pu:
3,270 baseline urine bioassays
– All exposure is occupational; essentially no environmental
exposure in North America
– α-spectrometry
– ~2,520 minute counts
16
probability density
The “Everybody”
Probability Density
Function (PDF):
A Distribution of
Possibly True Values
Histogram
of data
90Sr
(mBq/day)
probability density
probability density
• Histogram and PDF have
identical arithmetic means
PDF of
measurands
239Pu
(µBq/sample)
137Cs
(mBq/kg)
Probability Distributions
for Individual Measurands
• Now that we have the lognormal PDF of all measurands,
what can we say about individual measurands?
• Each individual’s measurand is somewhere within the
population of measurands
• We now assume that each mi, ui pair is the mean and
standard deviation of the Normal “likelihood” PDF for
individual i
• Assume the ith measurement was the last one made in the
population
– When the ith measurement was made, the other M1 m and u
values were known
• Use this with Bayes’s theorem
18
The Bayesian Approach to Assigning
Possibly True Results to Individuals
(Likelihoo d PDF)(Prior PDF)
Posterior PDF 
Normalizing Factor
Thomas Bayes 1702 – 1761
19
Bayesian Method for Individuals
• Instead of the “everybody” PDF, the “everybody else”
PDF is used as the prior for each individual
• Each individual’s likelihood is a normal distribution
with mean mi and standard deviation ui
• Using Bayes’s theorem, we developed a method to
derive a posterior probability density function (PDF) for
each individual’s measurand ti
p (ti | mi , ui ) 

p (mi | ti , ui ) p (ti | {mk i , uk i })
 p(m | t , u ) p(t  | {m
i
i
i
0
20
i
k i
, uk i }) d ti
Applications to Real Radiobioassay Data
Data
Analyte
90
Sr (mBq/day)
137
Cs (mBq/kg)
m
x
N
ss(x
(x ii))
r '2
128
3.61
5.08
0.151
5,337
50.6
112.0
0.350
239-240
Pu (µBq)
3,270
1.24
61
0.471
239-240
Pu (µBq)
3,268
0.040
34
-0.67
Impossible!
For Pu measurements, either the uncertainties ui are
overestimated, or a covariance term has been neglected.
21
Variability
Fractions
r′2
137Cs
r´2=0.35
uRMS
s(mi )
ˆ (ti )
s(mi )
22
137Cs
Variability
Fractions
r′2
90Sr
90Sr
r´2=0.15
137Cs
r´2=0.35
uRMS
s(mi )
ˆ (ti )
s(mi )
23
137Cs
Variability
Fractions
r′2
239Pu
239Pu
r´2~0
90Sr
90Sr
r´2=0.15
137Cs
r´2=0.35
uRMS
s(mi )
ˆ (ti )
s(mi )
24
137Cs
90Sr
Results for 4 Individuals
Uncensored Data Are Critical!
Measurand
Negative Result
Result ≈ 0
Measurement
Likelihood PDF
Prior
Result ≈ Average
Result = Large Positive
25
A Movie of 128
90Sr
Results
• Short Dashes (Green): Likelihood (Data)
• Long Dashes (Red): Everybody Else Prior
• Solid (Blue): Posterior
26
90Sr Measurand
Mean of90
Arithmetic
90
90
of Arithmetic
etic Mean
Measurand
Sr
of
Mean
Measurand
Arithmetic
Sr
of (mBq/day)
Mean
Sr Measurand
90Sr
of(mBq/day)
(mBq/day)(mB
Mean
Measurand
Arithmetic
(mBq/day)
25
90
/ 4242..68
68
33;3..31
uu
RMS
uu
RMS
uRMS

.31
31;;
RMS
u

31
RMS //// 2
uuu
2

3
.
31
RMS
2

3
.
RMS
/
2

3
.
31
RMS /// 22
.3.34
;; ;
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2
.31
uRMS
2
3
.
31
RMS
0.707u
RMS
uuuuu
/
2

2
.
34
/
2

2
34
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/
2

3
.
31
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222
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//22
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2...34
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34
;;
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uuu

34
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2

2
34
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u
2

2
.
;
0.5u
RMS
242
212
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..34
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
.
34
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uuuu
1
.
/
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
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34
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4

1
17
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4
1
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1112..17
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34;
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4
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
17
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u
4

1
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u
4

1
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17
;
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s
(
data
)

5
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s
(
data)

5.08
u

4
.
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/
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
1
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17
u
/
4

1
.
17
;;
s
(
data)

5.08
Measurand
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u=0
ss(((RMS
data
))

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68
.
4

data)
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/
4

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;
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
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68
data
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data)
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;
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4

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
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x
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
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data
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
xx
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68

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
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(
data
)
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;
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
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31
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
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;
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=
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.
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;;
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
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
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
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
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;
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17
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.
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data
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sussu
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data)
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1
u
/
4

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;;
data)

5.08
u=0
08
.
5

)
data
(
s
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data)

5.08
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
1
.
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;
08;
5.08
data ))5.08
data)
5
data

x
x
yysyssy
=(((x
data)

5.08
data
x )  5.08;
rr2  00..15
58
22  0.58

r
rrr222  000...79
58
r22  0.79
58
79
 00..95
79
rrrr2222 

79
 000...79
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r 22 
95
22  0.95

r

r

0
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000
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
0
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
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
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
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 000..15

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 00.58
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
 000...15
rrr222 
58
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
r
rrr222  000...79
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r22  0.79
58
79
79
 00..95
rr 22 
r 
79
95
 000...79
rr2222 
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 00.95
rr22 
r 
95
 000..95
rr222 
r2  0
r 0
r 2  0
Sr Measurands v Measurements
20
25
2525
20
1520
20
20 15
15
15
10
10
15
10
10

yysyy(
xxxx )  5.08;
data


x
yyy 
 xx
yx
5 5
1055
0
-10
-5
0
5
10
15
20
000 -15
Arithmetic Mean of 90Sr Likelihood (mBq/day)
-10
-5-5
00
5 5 10 10 15 15 20 20 25
-15
-10
-15
-10
5 -15
90Sr Likelihood (mBq/day)
Arithmetic
Mean
of
90Sr Likelihood (mBq/day)
Arithmetic
Mean
of
90
Arithmetic Mean of Sr
27 Likelihood (mBq/day)
25
25
90Sr Measurand
Mean90of90
Arithmetic
90
90
90
of ofSrArithmetic
(mBq/day)
Mean
Mean
ic
etic
Measurand
Measurand
Sr
of
of
(mBq/
(mB
Mean
Mean
Measurand
Measurand
Arithmetic
Arithmetic
SrSr
of (mBq/day)
Mean
Sr Measurand
90Sr
of(mBq/day)
(mBq/day)
Mean
Measurand
Arithmetic
(mBq/day)
25
90
20
2025
25
25
25 20
1520
20
15
20
15
20
15
15
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68
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 4422422...68
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uyyyssyyRMS
68
RMS
31
308

2....68
RMS

x

31

RMS
31
3

2
2
;
(
data
)
5

x
RMS
u
/
2

3
.
31
u

4
68
u

4
68
;
RMS
/
2

3
.
31
;
RMS
0.707u
uuuuyRMS
== xxx/x////222
RMS
2
.
34
RMS
;
34
.
2

2

3
.
31
yy
RMS
y
RMS
2

3
.
31
u
2

3
.
31
;
RMS

2
.
34
y =u
x
;;
34
.34
2...3334

RMS
uuyRMS
..31
 xx////2222222
222

;
RMS

34
RMS

31
;
34
.
2

2
3
.
31
RMS
0.5u
0.707u
RMS
y

uuuuu
/
2

2
.
34
;
34
.
2

2
/
RMS
..334
;;; ;
.31
RMS
RMS
RMS
222
112211
2....17
.17
34
RMS
/////424244
u
334
.31
RMS
RMS
17
u
RMS
34
RMS
u
17
RMS
u
/
2

2
.
;
34
uuRMS
///424244
112211.....17
.25u
17
0.5u
RMS

17
RMS
34
RMS

17
;
RMS
u
/
4

1
.
17
17
42)5.08
u(RMS
.08
34;;;;
RMS
RMS
24
25
(RMS
data
...34
RMS
17
111522...17
suussu
data)
/////4
1
RMS
4
17
u
2

.
34
RMS
(
data)

5.08
4

1
17
u=0
.25u
RMS
;
08
.

)
data
(
s
RMS
u
///4

1
..17
data)

5.08
RMS
4

1
.
17
;
RMS
ussus((RMS
4

1
17
08
5
)
data
(
RMS
RMS
data)

5.08
data
..08
uyssyussy=(((RMS
//44)))

15
08;;;
5.17
5.08
data
(RMS
1
17

x

x
data)
yu=0
x
08
.
5

data
(
RMS

x
data)

5.08
s(data )  5.08;
 000...58
15
58
rrrrr222222 
15

0
.
58

58
rrr222 
15
 000...79
58
79
2
r

0
.
58

79
rr222  00..79
58
79
79
rrr222 
0
.
95

0
.
79
58
 00..79
95

95
rrr2222222 
0
.
95

0
.
79
95

.
95
r

0
rrrr22222 
000
...15
95
79

0
15
2
95
222 

r
0
.


15
 0000..15
rrrrr2222222 
95
15


15
.
0
58

0
95
rrr22222 
0
.
15
58

 0000...15
rrrr2222 

58
0
.
58
r

0
222 
15
58
.
58

rrrr2222 
 0000...79
15

58
79
22  0.58

r
79
rrr222 
000...79
79

58
79
95
rr2222 
0
.
79
58
95
 00..79

95
rrrr222 
0
.
95
95

0
.
79

0..79
95

95
rrr22222 
0
0


0
.
95
 00.95
rr2222 

0.95

rr 22 

0
r2  0
r  0
r 2  0
Sr Measurands v Measurements
10
5.08
155..17
ssyussyy((((RMS
data)


xxxx/ 4))
08;;
data
10 10
data)

5.08
data

08

y

y

x
y

x
ysyy
= x
15
 xxxxx )5.08
data
5..08
08;;

yyssyyy(((data)


x
data
)

5

x
10
10
15
yyy 
xxx

yx
5
5
555
10
10 0
-10
-5
0
5
10
15
20
000 -15
Arithmetic Mean of 90Sr Likelihood (mBq/day)
0
-10
-5-5
00
5 5 10 10 15 15 20 20 25
-15
-10
-15
-10
5 -15
90Sr Likelihood (mBq/day)
Mean
of
90
-10 Arithmetic
-5
0
5 Sr Likelihood
10
15
20
25
5 -15
Arithmetic
Mean
of
(mBq/day)
90
Arithmetic Mean of Sr
28 Likelihood (mBq/day)
25
25
90Sr
90Sr
90Sr
90Sr
of of
(mBq/day)
(mBq/day)
Mean
Mean
ic
etic
Measurand
Measurand
of
of
(mBq/
(mB
Mean
Mean
Measurand
Measurand
Arithmetic
Arithmetic
90
Arithmetic Mean of Sr Measurand (mBq/day)
20
2025
25
25
 000...58
15
58
rrrrr222222 
15

0
.
58

58
rrr222 
15
 000...79
58
79
2
r

0
.
58

79
rr222  00..79
58
79
79
rrr222 
0
.
95

0
.
79
58
 00..79
95

95
rrr2222222 
0
.
95

0
.
79
95

.
95
r

0
rrrr22222 
000
...15
95
79

0
15
2
95
222 

r
0
.


15
 0000..15
rrrrr2222222 
95
15


15
.
0
58

0
95
rrr22222 
0
.
15
58

 0000...15
rrrr2222 

58
0
.
58
r

0
222 
15
58
.
58

rrrr2222 
 0000...79
15

58
79
22  0.58

r
79
rrr222 
000...79
79

58
79
95
rr2222 
0
.
79
58
95
 00..79

95
rrrr222 
0
.
95
95

0
.
79

0..79
95

95
rrr22222 
0
0


0
.
95
 00.95
rr2222 

0.95

rr 22 

0
r2  0
r  0
r 2  0
Effect of Reducing Uncertainty
1520
15
20
20
15
//
68
33;3..31
uu
 4422422...68
RMS
uu
RMS
uRMS
/

.31
31;;;
RMS
RMS

68
u

31
/

3
.
RMS
uuu
/
2

3
.
31
u

4
.
68
;
RMS

4
.
68
/
2

3
.
31
RMS
RMS
RMS
/
2

3
.
31
22..68
33.31
..31
2242
2
..;34
;; ;;
/////
68
334
RMS
uRMS
4
RMS

31
RMS
uu
0.707u
RMS
RMS
uuuuu
2
2
.

2
34
2

3
.
31
RMS
/
RMS
..31
2

334
.31
RMS
//2222

2
2....33334
.34
; ;;
RMS
2

31
uuu

2
34
RMS

2
;
RMS ////2
2

2
34
RMS
2

.
31
2

3
.
31
u
2

2
.
34
;
RMS
0.707u
0.5u
RMS
uuuuRMS

2
22...17
..17
34
RMS/////2
42
11
.34
;;;; ;
2
334
.31
RMS
RMS
2

34
RMS
4

1
2
3
.
31
2
2
RMS
u
/
RMS
u
/
4

.
17
RMS
2
2
.
34
RMS
4
1
.
17
RMS
RMS
1122112.....17
.17
34;;
uuRMS
//2
34
44242
.25u
0.5u
4
17
RMS
RMS

17
34
RMS////4
RMS
2
2
.
34;;;
RMS
usuu
1
.
17
u
4

1
.
17
RMS
ussu
/
24
.5.08
25
.17
34
RMS
/
1
17
(
data
)
.
08
RMS
(
data)

4
1
.
4
68
;
RMS
RMS
4
1
.
17
;;
u
/
2
2
.
34
RMS
(
data)

5.08
u
/
4

1
.
17
RMS
u=0
.25u
((RMS
data
))

55..17
.;17
08
RMS
///4

1
68
.
4
((RMS
data)
5.08
RMS
4

1
.
;
RMS
uuussuussRMS
4
.
68
4

1
17
data
08
RMS
68
4
44))44
RMS
data)
5.08
(RMS
data
..;;;08
;;
RMS
RMS
....68
68
uysu
.
/
15
.17
sy=(RMS
(RMS
data

5
08
//
1
17
udata)

5.08

x

x
68
4

u
yu=0
x
s
(
data
)
5
.
08
;;
y

x
RMS
;
68
4
2.268

317
..31
ssyRMS
((RMS
data
)
5
08
/
4
1
.
;
ssu
((RMS
data)

5.08
u

4

x
u

4
.
68
;
y

x
RMS
RMS
data)

5.08
RMS
31
3

/
data
5
; ;;
RMS
Uncertainty
Assigned
yy=yyRMS

xxx/xx/)424
5.08
33.;08
..31
...68
usu
68
RMS

uyu
x
RMS
31
.
3

2
u
(
data)

RMS

RMS


4
68
u
/
2

31
RMS
///2)42242

...31
data
xxx//
(RMS
5.33.;334
08
; ;;;
uuu

..31

uyyyssyyRMS
68
RMS
31
308

2....68
RMS

x

31

RMS
31
3

2
2
;
(
data
)
5

x
RMS
u
/
2

3
.
31
u

4
68
u

4
68
;
RMS
/
2

3
.
31
;
RMS
0.707u
uuuuyRMS
= xx/x////222
RMS
2
.
34
RMS
;
34
.
2

2

3
.
31
y
RMS
y
RMS
2

3
.
31
u
2

3
.
31
;
RMS
u

2
.
34
;;
34
.34
2...3334

RMS
uuyRMS
..31
 xx////2222222
222

;
RMS

34
RMS

31
;
34
.
2

2
3
.
31
RMS
0.5u
0.707u
RMS
y

uuuuu
/
2

2
.
34
;
34
.
2

2
/
RMS
..334
;;; ;
.31
RMS
RMS
RMS
222
112211
2....17
.17
34
RMS
/////424244
u
334
.31
RMS
RMS
17
u
RMS
34
RMS
u
17
RMS
u
/
2

2
.
;
34
uuRMS
///424244
112211.....17
.25u
17
0.5u
RMS

17
RMS
34
RMS

17
;
RMS
u
/
4

1
.
17
17
42)5.08
u(RMS
.08
34;;;;
RMS
RMS
24
25
(RMS
data
...34
RMS
17
111522...17
suussu
data)
/////4
1
RMS
4
17
u
2

.
34
RMS
(
data)

5.08
4

1
17
u=0
.25u
RMS
;
08
.

)
data
(
s
RMS
u
///4

1
..17
data)

5.08
RMS
4

1
.
17
;
RMS
ussus((RMS
4

1
17
08
5
)
data
(
RMS
RMS
data)

5.08
data
..08
uyssyussy=(((RMS
//44)))

15
08;;;
5.17
5.08
data
(RMS
1
17

x

x
data)
yu=0
x
08
.
5

data
(
RMS

x
data)

5.08
s(data )  5.08;
10
10
15
1510
5
55
10
10
00
0
-15
5 -15
5 -15
5.08
155..17
ssyussyy((((RMS
data)


xxxx/ 4))
08;;
data
data)

5.08
data

08

y

y

x
y

x
ysyy
= x
 xxxxx )5.08
data
5..08
08;;

yyssyyy(((data)


x
data
)

5

x
yyy 
xxx

yx
uRMS  4.68; 2 r 2  0.15
uRMS  4.68;
r   0.15
uRMS / 2  3.312; r2  0.58
uRMS / 2  3.31; r  0.258
uRMS / 2  2.34; 2 r   0.79
uRMS / 2  2.34;
r   0.79
uRMS / 4  1.17; 2 r 2  0.95
uRMS / 4  1.17;
r   0.95
-10-10 -5-5
00
5 5 10 10 s(data
15 )15 5.08
20; 220 r2 25
 025
90Sr Likelihood
s(data ) 
5.08; 20
r  0 25
Mean
of
(mBq/day)
-10 Arithmetic
-5
0
5
10
15
yx
Arithmetic Mean of 90Sr
Likelihood
(mBq/day)
29
yx
90Sr
90Sr
90Sr
90Sr
of of
(mBq/day)
(mBq/day)
Mean
Mean
ic
etic
Measurand
Measurand
of
of
(mBq/
(mB
Mean
Mean
Measurand
Measurand
Arithmetic
Arithmetic
90
Arithmetic Mean of Sr Measurand (mBq/day)
20
2025
25
25
//
68
33;3..31
uu
 4422422...68
RMS
uu
RMS
uRMS
/

.31
31;;;
RMS
RMS

68
u

31
/

3
.
RMS
uuu
/
2

3
.
31
u

4
.
68
;
RMS
/
2

3
.
31

4
.
68
RMS
RMS
/
2

3
.
31
RMS
RMS
22..68
33.31
..31
2242
2
..;34
;; ;;
RMS
/////
68
334
uRMS
4
RMS

31
RMS
0.707u
uu
RMS
uuuuu
2
2
.
RMS

2
34
2

3
.
31
/
RMS
RMS
..31
2

334
.31
RMS
//2222

2
2....33334
.34
; ;;
RMS
2

31
uuu

2
34
RMS

2
;
RMS ////2
2

2
34
RMS
2

.
31
2

3
.
31
u
2

2
.
34
;
RMS
0.5u
0.707u
RMS
uuuuRMS

2
22...17
..17
34
RMS/////2
42
11
.34
;;;; ;
2
334
.31
RMS
RMS
2

34
RMS
4

1
2
3
.
31
2
2
RMS
u
/
RMS
u
/
4

.
17
RMS
2
2
.
34
RMS
4
1
.
17
RMS
RMS
1122112.....17
.17
34;;
uuRMS
//2
34
44242
.25u
4
17
0.5u
RMS
RMS

17
34
RMS////4
RMS
2
2
.
34;;;
u
4

1
.
17
RMS
usuu
1
.
17
RMS
ussu
/
24
.5.08
25
.17
34
RMS
/
1
17
(
data
)
.
08
RMS
(
data)

4
1
.
4
68
;
RMS
RMS
4
1
.
17
;;
u
/
2
2
.
34
RMS
(
data)

5.08
u
/
4

1
.
17
RMS
u=0
.25u
((RMS
data
))

55..17
.;17
08
RMS
/
4

1
68
.
4

((RMS
data)
5.08
RMS
/
4

1
.
;
RMS
uuussuussRMS

4
.
68
/
4

1
17
data
08
u
RMS
68
4
44))44
RMS
data)
5.08
(RMS
data
..;;;08
;;
RMS
RMS
....68
68
uysu
.
/
15
.17
sy=(RMS
(RMS
data

5
08
/
1
17
data)

5.08

x

x
68
4

u
yu=0
x
s
(
data
)
5
.
08
;;
y

x
RMS
;
68
4

///44)
2.268

317
..31
ssyRMS
((RMS
data
5
08
1
.
;
ssu
((RMS
data)
5.08
u


x
u
4
.
68
;
y

x
RMS
RMS
data)

5.08
RMS
31
3

data
5
; ;;
RMS
Uncertainty
Assigned
yy=yyRMS

xxx/xx/)424
5.08
33.;08
..31
...68
usu
68
RMS

uyu
x
RMS
31
.
3

2
u
0.707u
(
data)

RMS

RMS


4
68
u
/
2

31
RMS
///2)42242

...31
data
xxx//
(RMS
5.33.;334
08
; ;;;
uuu

..31

uyyyssyyRMS
68
RMS
31
308

2....68
RMS

x

31

RMS
31
3

2
2
;
(
data
)
5

x
RMS
u
/
2

3
.
31
u

4
68
u

4
68
;
RMS
/
2

3
.
31
;
RMS
0.707u
uuuuyRMS
 x/x////222
RMS
2
.
34
RMS
;
34
.
2

2

3
.
31
RMS
y

RMS
2

3
.
31
u
2

3
.
31
;
RMS
u

2
.
34
;;
34
.34
2...3334

222
uuyRMS
////2
..31

yRMS
= xx
222

;
RMS
22

34
RMS

31
;
34
.
2

2
2
3
.
31
RMS
0.5u
0.707u
RMS
y

x
uuuuu
/
2

2
.
34
;
34
.
2

2
/
RMS
..334
;;; ;
.31
RMS
RMS
RMS
222
112211
2....17
.17
34
RMS
/////424244
u
334
.31
RMS
RMS
17
u
RMS
34
RMS
u
17
RMS
u
/
2

2
.
;
34
uuRMS
///424244
112211.....17
.25u
17
0.5u
RMS

17
RMS
34
RMS

17
;
RMS
u
/
4

1
.
17
17
42)5.08
u(RMS
.08
34;;;;
RMS
RMS
24
25
(RMS
data
...34
17
111522...17
RMS
suussu
data)
/////4
1
RMS
4
17
u
2

.
34
RMS
(
data)

5.08
4

1
17
u=0
.25u
RMS
;
08
.

)
data
(
s
RMS
u
///4

1
..17
data)

5.08
RMS
4

1
.
17
;
RMS
ussus((RMS
4

1
17
08
5
)
data
(
RMS
RMS
data)

5.08
data
..08
uyssyussy=(((RMS
//44)))

15
08;;;
5.17
5.08
data
(RMS
1
17

x
data)

x
yu=0
x
08
.
5

data
(
RMS

x
data)

5.08
data
5.08
08;;
155..17

yyussyy(((RMS

xxxx/ 4))
s
data)
data
08


y

x
y

x
ysyy

 xxx )5.08
data
s=y((xdata)
5.08;
  00..15
58
rrrrr222222 
0
.
58
15

0
.
58

58
rrr222 
15
 000...79
58
79
2
r

0
.
58

79
rrr222 
000...79
58

79
79

95
2
rr222 
0
.
79
58
 00..79
95

95
rrrr222222 
0
.
95

0
.
79
95

.
95

0
rrrr22222 
000
...15
95
79

0
15
2
95
222 

r
0
.


15
 0000..15
rrrrr2222222 
95
15


15
.
0
58

0
95
rrr22222 
0
.
15
58

 0000...15
rrrr2222 

58
0
.
58
r

0
222 
15
58
.
58

rrrr2222 
 0000...79
15

58
79
22  0.58

r
79
rrr222 
000...79
79

58
79
95
rr2222 
0
.
79
58
95
 00..79

95
rrrr2222 
0
.
95
95

0
.
79

0..79
95

95
rrr22222 
0
0


0
.
95
 00.95
rr2222 

0.95

rr  22 

0
r2  0
r  0
r 2  0
Effect of Reducing Uncertainty
1520
15
20
20
15
10
10
15
1510
5
55
10
10
00
0
-15
5 -15
5 -15

yysyy(
xxxx )  5.08;

data

x
yyy 
 xx
yx
uRMS  4.68;
r 2  0.15
uRMS / 2  3.31; r2  0.58
uRMS / 2  2.34;
r 2  0.79
uRMS / 4  1.17;
r 2  0.95
s(data )  5.08;
r 2  0
yx
uRMS  4.68; 2 r 2  0.15
uRMS  4.68;
r   0.15
uRMS / 2  3.312; r2  0.58
uRMS / 2  3.31; r  0.258
uRMS / 2  2.34; 2 r   0.79
uRMS / 2  2.34;
r   0.79
uRMS / 4  1.17; 2 r 2  0.95
uRMS / 4  1.17;
r   0.95
-10-10 -5-5
00
5 5 10 10 s(data
15 )15 5.08
20; 220 r2 25
 025
90Sr Likelihood
s(data ) 
5.08; 20
r  0 25
Mean
of
(mBq/day)
-10 Arithmetic
-5
0
5
10
15
yx
Arithmetic Mean of 90Sr
Likelihood
(mBq/day)
30
yx
90Sr
90Sr
90Sr
90Sr
of of
(mBq/day)
(mBq/day)
Mean
Mean
ic
etic
Measurand
Measurand
of
of
(mBq/
(mB
Mean
Mean
Measurand
Measurand
Arithmetic
Arithmetic
90
Arithmetic Mean of Sr Measurand (mBq/day)
20
2025
25
25
//
68
33;3..31
uu
 4422422...68
RMS
uu
RMS
uRMS
/

.31
31;;;
RMS
RMS

68
u

31
/

3
.
RMS
uuu
/
2

3
.
31
u

4
.
68
;
RMS
/
2

3
.
31

4
.
68
RMS
RMS
/
2

3
.
31
RMS
RMS
22..68
33.31
..31
2242
2
..;34
;; ;;
RMS
/////
68
334
uRMS
4
RMS

31
RMS
0.707u
uu
RMS
uuuuu
2
2
.
RMS

2
34
2

3
.
31
/
RMS
RMS
..31
2

334
.31
RMS
//2222

2
2....33334
.34
; ;;
RMS
2

31
uuu

2
34
RMS

2
;
RMS ////2
2

2
34
RMS
2

.
31
u
2

2
.
34
;
2

3
.
31
RMS
0.5u
0.707u
RMS
uuuuRMS

22...17
..17
34
RMS/////2
11
.34
;;;; ;
2
334
.31
RMS
RMS
2

34
RMS
424242

112
2
3
.
31
2
RMS
u
/
RMS
u
/
4

.
17
RMS
2
.
34
RMS
.
17
RMS
1122112.....17
.17
34;;
uuRMS
//2
34
44242
.25u
4
17
0.5u
RMS
RMS

17
34
RMS////4
RMS
u
4

1
.
17
2
2
.
34;;;
RMS
usuu
1
.
17
RMS
u
/
2

2
.
34
RMS
u
/
4
1
17
s
(
data
)

5
.
08
RMS
(
data)

5.08
4
1
.
17

4
.
68
;
RMS
RMS
4
1
.
17
;;
u
/
2

2
.
34
RMS
s
(
data)

5.08
u
/
4

1
.
17
RMS
u=0
.25u
data
))

55..17
.;17
08
RMS
/
4

1
68
.
4

((RMS
data)
5.08
RMS
u(
/
4

1
.
;
RMS
uuussuussRMS

4
.
68
RMS
/
4

1
17
(
data
08
RMS
68
4
44))44
RMS
data)
5.08
(RMS
data
..;;;08
;;
RMS
RMS
....68
68
uysu
.
/
15
.17
sy=(RMS
(RMS
data

5
08
/
1
17
data)

5.08

x

x
68
4

u
yu=0
x
s
(
data
)
5
.
08
;;
y

x
RMS
;
68
4

///44)
2.268

317
..31
ssyRMS
((RMS
data
5
08
1
.
;
ssu
((RMS
data)
5.08
u


x
u
4
.
68
;
y

x
0.707u
RMS
RMS
data)

5.08
RMS
31
3

data
5
; ;;
RMS
Uncertainty
Assigned
yy=yyRMS

xxx/xx/)424
5.08
33.;08
..31
...68
usu
68
RMS

uyu
x
RMS
31
.
3

2
u
(
data)

RMS

RMS


4
68
u
/
2

31
RMS
///2)42242

...31
data
xxx//
(RMS
5.33.;334
08
; ;;;
uuu

..31

uyyyssyyRMS
68
RMS
31
308

2....68
RMS

x

31

0.5u
RMS
31
3

2
2
;
(
data
)
5

x
RMS
u
/
2

3
.
31
u

4
68
u

4
68
;
RMS
/
2

3
.
31
;
RMS
0.707u
uuuuyRMS
 x/x////222
RMS
2
.
34
RMS
;
34
.
2

2

3
.
31
RMS
y

RMS
2

3
.
31
u
2

3
.
31
;
RMS
u

2
.
34
;;
34
.34
2...3334

uuyRMS
..31

x////2222222
222

;
RMS
yRMS
= xx

34
RMS

31
;
34
.
2

2
3
.
31
RMS
0.5u
0.707u
RMS
y

uuuuu
/
2

2
.
34
;
34
.
2

2
/
RMS
..334
;;; ;
.31
RMS
RMS
RMS
222
112211
2....17
.17
34
RMS
/////424244
u
334
.31
RMS
RMS
17
u
RMS
34
RMS
u
17
RMS
u
/
2

2
.
;
34
uuRMS
///424244
112211.....17
.25u
17
0.5u
RMS

17
34
RMS
RMS

17
;
RMS
u
/
4

1
.
17
17
42)5.08
u(RMS
.08
34;;;
RMS
RMS
24
25
(RMS
data
...34
17
111522...17
RMS
suussu
data)
/////4
1
RMS
4
17
;;
u
2

.
34
RMS
(
data)

5.08
4

1
17
u=0
.25u
RMS
08
.

)
data
(
s
RMS
u
/
4

1
.
17
data)

5.08
RMS
/
4

1
.
17
;
RMS
ussus((RMS
/
4

1
.
17
08;;
data
data)
(RMS
data
)))
5.08
5155.17
..08
RMS
uyssyussy=((RMS
/
4

1
08
data
((RMS
/
4
17

x
data)

5.08

x
yu=0
x
;
08
.
5

)
data
(
RMS

xxx/ 4)5.08
data)
data
5
08
1
.
17
;

yyussyy(((RMS

s
data)

5.08
data
5.08;
 xxxxx )5.08


ysyy



x )  5.08;
s=y((xdata)
data
  00..15
58
rrrrrr222222 

0
.
58
15

0
.
58
58
15
 000...79
58
rrr22222 
79
0
.
58
79
rrr22 
000...79

79
58
79

95
2
rr222 
0
.
79
58
 00..79
95

95
rrrr222222 
0
.
95
95

0
.
79

.
95

0
rrrr22222 
000
...15
95
79

0
15
2
95
222 

r
0
.


15
 0000..15
rrrrr2222222 
95
15


15
.
0
58

0
95
rrr22222 
0
.
15
58

 0000...15
rrrr2222 

58
0
.
58
r

0
222 
15
58
.
58

rrrr2222 
 0000...79
15

58
79
22  0.58

r
79
rrr222 
000...79
79

58
79
95
rr2222 
0
.
79
58
95
 00..79

95
rrrr2222 
0
.
95
95

0
.
79

0..79
95

95
rrr22222 
0
0


0
.
95
 00.95
rr2222 

0.95

rr  22 

0
r2  0
r  0
r 2  0
Effect of Reducing Uncertainty
1520
15
20
20
15
10
10
15
1510
5
55
10
10
00
0
-15
5 -15
5 -15
uRMS  4.68;
r 2  0.15
uRMS / 2  3.31; r22  0.58
uRMS  4.68;
r  2  0.15
r   0.79
uRMS / 2  2.34;
uRMS / 2  3.31; r22  0.58
r   0.95
uRMS / 4  1.17;
uRMS / 2  2.34;
r 22  0.79
r  0
s(data )  5.08;
uRMS / 4  1.17;
r 2  0.95
yx
2

s
(
data
)

5
.
08
;
r
0
x

yysyy(
x

x
data
)

5
.
08
;

x
yyy 
xxx
yx

yx
uRMS  4.68; 2 r 2  0.15
uRMS  4.68;
r   0.15
uRMS / 2  3.312; r2  0.58
uRMS / 2  3.31; r  0.258
uRMS / 2  2.34; 2 r   0.79
uRMS / 2  2.34;
r   0.79
uRMS / 4  1.17; 2 r 2  0.95
uRMS / 4  1.17;
r   0.95
-10-10 -5-5
00
5 5 10 10 s(data
15 )15 5.08
20; 220 r2 25
 025
90Sr Likelihood
s(data ) 
5.08; 20
r  0 25
Mean
of
(mBq/day)
-10 Arithmetic
-5
0
5
10
15
yx
Arithmetic Mean of 90Sr
Likelihood
(mBq/day)
31
yx
90Sr
90Sr
90Sr
90Sr
of of
(mBq/day)
(mBq/day)
Mean
Mean
ic
etic
Measurand
Measurand
of
of
(mBq/
(mB
Mean
Mean
Measurand
Measurand
Arithmetic
Arithmetic
90
Arithmetic Mean of Sr Measurand (mBq/day)
20
2025
25
25
  00..15
58
rrrrr222222 
0
.
58
15

0
.
58

58
rrr222 
15
 000...79
58
79
2
r

0
.
58

79
rr222  00..79
58
79
79
rrr222 
0
.
95

0
.
79
58
 00..79
95

95
rrrr2222222 
0
.
95

0
.
79
95

.
95

0
rrrr22222 
000
...15
95
79

0
15
2
95
222 

r
0
.


15
 0000..15
rrrrr2222222 
95
15


15
.
0
58

0
95
rrr22222 
0
.
15
58

 0000...15
rrrr2222 

58
0
.
58
r

0
222 
15
58
.
58

rrrr2222 
 0000...79
15

58
79
22  0.58

r
79
rrr222 
000...79
79

58
79
95
rr2222 
0
.
79
58
95
 00..79

95
rrrr2222 
0
.
95
95

0
.
79

0..79
95

95
rrr22222 
0
0


0
.
95
 00.95
rr2222 

0.95

rr  22 

0
r2  0
r  0
r 2  0
uRMS  4.68;
r 2  0.15
Effect of Reducing Uncertainty
1520
15
20
20
15
//
68
33;3..31
uu
 4422422...68
RMS
uu
RMS
uRMS
/

.31
31;;;
RMS
RMS

68
u

31
/

3
.
RMS
uuu
/
2

3
.
31
u

4
.
68
;
RMS

4
.
68
/
2

3
.
31
RMS
RMS
/
2

3
.
31
RMS
RMS
22..68
33.31
..31
2242
2
..;34
;; ;;
RMS
/////
68
334
uRMS
4
RMS

31
RMS
0.707u
uu
RMS
uuuuu
2
2
.
RMS

2
34
2

3
.
31
RMS
/
RMS
..31
2

334
.31
RMS
//2222

2
2....33334
.34
; ;;
RMS
2

31
uuu

2
34
RMS

2
;
RMS ////2
2

2
34
RMS
2

.
31
2

3
.
31
u
2

2
.
34
;
RMS
0.5u
0.707u
RMS
uuuuRMS

2
22...17
..17
34
RMS/////2
42
11
.34
;;;; ;
2
334
.31
RMS
RMS
2

34
RMS
4

1
2
3
.
31
2
2
RMS
u
/
RMS
u
/
4

.
17
RMS
2
2
.
34
RMS
4
1
.
17
RMS
RMS
1122112.....17
.17
34;;
uuRMS
//2
34
44242
.25u
4
17
0.5u
RMS
RMS

17
34
RMS////4
RMS
2
2
.
34;;;
RMS
u
4

1
.
17
usuu
1
.
17
RMS
ussu
/
24
.5.08
25
.17
34
RMS
/
1
17
(
data
)
.
08
RMS
(
data)

4
1
.
4
68
;
RMS
RMS
4
1
.
17
;;
u
/
2
2
.
34
RMS
(
data)

5.08
u
/
4

1
.
17
RMS
u=0
.25u
data
))

55..17
.;17
08
RMS
/
4

1
68
.
4

u(
((RMS
data)
5.08
RMS
/
4

1
.
;
RMS
uuussuussRMS

4
.
68
RMS
/
4

1
17
(
data
08
RMS
68
4
44))44
RMS
data)
5.08
(RMS
data
..;;;08
;;
RMS
RMS
....68
68
uysu
.
/
15
.17
sy=(RMS
(RMS
data

5
08
//
1
17
data)

5.08

x

x
68
4

u
yu=0
x
s
(
data
)
5
.
08
;;
0.707u
y

x
RMS
;
68
4
2.268

317
..31
ssyRMS
((RMS
data
)
5
08
/
4
1
.
;
ssu
((RMS
data)

5.08
u

4

x
u

4
.
68
;
y

x
RMS
RMS
data)

5.08
RMS
31
3

/
data
5
; ;;
RMS
Uncertainty
Assigned
yy=yyRMS

xxx/xx/)424
5.08
33.;08
..31
...68
usu
68
RMS

uyu
x
RMS
31
.
3

2
u
0.5u
(
data)

RMS

RMS


4
68
u
/
2

31
RMS
///2)42242

...31
data
xxx//
(RMS
5.33.;334
08
; ;;;
uuu

..31

uyyyssyyRMS
68
RMS
31
308

2....68
RMS

x

31

RMS
31
3

2
2
;
(
data
)
5

x
RMS
u
/
2

3
.
31
u

4
68
u

4
68
;
RMS
/
2

3
.
31
;
RMS
0.707u
uuuuyRMS
.25u
 x/x////222
RMS
2
.
34
RMS
;
34
.
2

2

3
.
31
RMS
y

RMS
2

3
.
31
u
2

3
.
31
;
RMS
u

2
.
34
;;
34
.34
2...3334

RMS
uuyRMS
..31

x////2222222
222

;
RMS

34
RMS

31
;
34
.
2

2
3
.
31
RMS
yRMS
= x/x
0.5u
0.707u
RMS
y

uuuuu
2

2
.
34
;
34
.
2

2
/
..334
;;; ;
.31
RMS
RMS
RMS
222
112211
2....17
.17
34
RMS
/////424244
u
334
.31
RMS
RMS
17
u
RMS
34
RMS
u
17
RMS
u
/
2

2
.
;
34
uuRMS
///424244
112211.....17
.25u
17
0.5u
RMS

17
RMS
34
RMS

17
;
RMS
u
/
4

1
.
17
17
42)5.08
u(RMS
.08
34;;;;
RMS
RMS
24
25
(RMS
data
...34
17
111522...17
RMS
suussu
data)
/////4
1
RMS
4
17
u
2

.
34
RMS
(
data)

5.08
4

1
17
u=0
.25u
RMS
;
08
.

)
data
(
s
RMS
u
///4

1
..17
data)

5.08
RMS
4

1
.
17
;
RMS
ussus((RMS
4

1
17
08
5
)
data
(
RMS
RMS
data)

5.08
data
..08
uyssyussy=(((RMS
//44)))

15
08;;;
5.17
5.08
data
(RMS
1
17

x
data)

x
yu=0
x
08
.
5

data
(
RMS

x
data)

5.08
08;
us(data
/ 4)  15.17
10
10
15
1510
5
55
10
10
00
0
-15
5 -15
5 -15
uRMS / 2  3.31; r22  0.58
r 2  0.15
uRMS  4.68;
r  2  0.79
uRMS / 2  2.34;
uRMS / 2  3.31; r2  0.58
95
r 2  0.15
1.;17;
uRMS /44.68
r  2  0.79
uRMS / 2  2.34;
 0.58
; ; r22 
data
us(RMS
/ ) 2 5.08
3.31
r   0.95
uRMS / 4  1.17;
 x/ 2  2.34;
uyRMS
r 22  0.79
r  0
s(data )  5.08;
uRMS / 4  1.17;
r 2  0.95
yx
ssyssy((((RMS
data)


xxxx ))
08;;
5..08
5.08
data
data)

5.08
data

5

y
y

2

xxxxx )5.08
yysyy



s
(
data
)

5
.
08
;
r
0

sysy=yyy(((xdata)
data
5
.
08
;
x

xxx )  5.08;

yy 
data
yy 
xxx
yx

yx
uRMS  4.68; 2 r 2  0.15
uRMS  4.68;
r   0.15
uRMS / 2  3.312; r2  0.58
uRMS / 2  3.31; r  0.258
uRMS / 2  2.34; 2 r   0.79
uRMS / 2  2.34;
r   0.79
uRMS / 4  1.17; 2 r 2  0.95
uRMS / 4  1.17;
r   0.95
-10-10 -5-5
00
5 5 10 10 s(data
15 )15 5.08
20; 220 r2 25
 025
90Sr Likelihood
s(data ) 
5.08; 20
r  0 25
Mean
of
(mBq/day)
-10 Arithmetic
-5
0
5
10
15
yx
Arithmetic Mean of 90Sr
Likelihood
(mBq/day)
32
yx
90Sr
90Sr
90Sr
90Sr
of of
(mBq/day)
(mBq/day)
Mean
Mean
ic
etic
Measurand
Measurand
of
of
(mBq/
(mB
Mean
Mean
Measurand
Measurand
Arithmetic
Arithmetic
90
Arithmetic Mean of Sr Measurand (mBq/day)
20
2025
25
25
//
68
33;3..31
uu
 4422422...68
RMS
uu
RMS
uRMS
/

.31
31;;;
RMS
RMS

68
u

31
/

3
.
RMS
uuu
/
2

3
.
31
u

4
.
68
;
RMS
/
2

3
.
31

4
.
68
RMS
RMS
/
2

3
.
31
RMS
RMS
22..68
33.31
..31
2242
2
..;34
;; ;;
RMS
/////
68
334
uRMS
4
RMS

31
RMS
0.707u
uu
RMS
uuuuu
2
2
.
RMS

2
34
2

3
.
31
/
RMS
RMS
..31
2

334
.31
RMS
//2222

2
2....33334
.34
; ;;
RMS
2

31
uuu

2
34
RMS

2
;
RMS ////2
2

2
34
RMS
2

.
31
2

3
.
31
u
2

2
.
34
;
RMS
0.5u
0.707u
RMS
uuuuRMS

2
22...17
..17
34
RMS/////2
42
11
.34
;;;; ;
2
334
.31
RMS
RMS
2

34
RMS
4

1
2
3
.
31
2
2
RMS
u
/
RMS
u
/
4

.
17
RMS
2
2
.
34
RMS
4
1
.
17
RMS
RMS
1122112.....17
.17
34;;
uuRMS
//2
34
44242
.25u
4
17
0.5u
RMS
RMS

17
34
RMS////4
RMS
u
4

1
.
17
2
2
.
34;;;
RMS
usuu
1
.
17
RMS
u
/
2

2
.
34
RMS
u
/
4
1
17
s
(
data
)

5
.
08
RMS
(
data)

5.08
4
1
.
17

4
.
68
;
RMS
RMS
4
1
.
17
;;
u
/
2

2
.
34
RMS
s
(
data)

5.08
u
/
4

1
.
17
RMS
u=0
.25u
data
))

55..17
.;17
08
u(RMS
/
4

1
68
.
4

((RMS
data)
5.08
RMS
/
4

1
.
;
RMS
uuussuussRMS

4
.
68
RMS
/
4

1
17
(
data
08
RMS
68
4
44))44
RMS
data)
5.08
(RMS
data
..;;;08
;;
RMS
RMS
....68
68
uysu
.
/
15
.17
sy=(RMS
(RMS
data

5
08
/
1
17
0.707u
data)

5.08

x

x
68
4

u
yu=0
x
s
(
data
)
5
.
08
;;
y

x
RMS
;
68
4

///44)
2.268

317
..31
ssyRMS
((RMS
data
5
08
1
.
;
ssu
((RMS
data)
5.08
u


x
u
4
.
68
;
y

x
RMS
RMS
data)

5.08
0.5u
RMS
31
3

data
5
; ;;
RMS
Uncertainty
Assigned
yy=yyRMS

xxx/xx/)424
5.08
33.;08
..31
...68
usu
68
RMS

uyu
x
RMS
31
.
3

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(
data)

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
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

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68
u
/
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
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///2)42242

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5.33.;334
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uuu

..31

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68
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31
308
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RMS

x

31

RMS
31
3

2
2
;
(
data
)
5

x
RMS
u
/
2

3
.
31
u

4
68
u

4
68
;
RMS
/
2

3
.
31
;
RMS
0.707u
uuuuyRMS
 x/x////222
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2
.
34
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;
34
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2

2

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31
u=0
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y

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
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.
31
u
2

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.
31
;
RMS
u

2
.
34
;;
34
.34
2...3334

RMS
uuyRMS
..31

x////2222222
222

;
RMS

34
RMS

31
;
34
.
2

2
3
.
31
RMS
0.5u
0.707u
yRMS
= x/x
RMS
y

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2

2
.
34
;
34
.
2

2
/
..334
;;; ;
.31
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RMS
222
112211
2....17
.17
34
RMS
/////424244
u
334
.31
RMS
RMS
17
u
RMS
34
RMS
u
17
RMS
u
/
2

2
.
;
34
uuRMS
///424244
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.25u
17
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
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34
RMS
RMS

17
;
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u
/
4

1
.
17
17
42)5.08
u(RMS
.08
34;;;
RMS
RMS
24
25
(RMS
data
...34
17
111522...17
RMS
suussu
data)
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1
RMS
4
17
;;
u
2

.
34
RMS
(
data)

5.08
4

1
17
u=0
.25u
RMS
08
.

)
data
(
s
RMS
u
/
4

1
.
17
data)

5.08
RMS
/
4

1
.
17
;
RMS
ussus((RMS
/
4

1
.
17
08;;
data
data)
(RMS
data
)))
5.08
5155.17
..08
RMS
uyssyussy=((RMS
/
4

1
08
data
((RMS
/
4
17

x
data)

5.08

x
yu=0
x
;
08
.
5

)
data
(
RMS

xxx/ 4)5.08
data)
data
5
08
1
.
17
;

yyussyy(((RMS

s
data)

5.08
data
5.08;
 xxxxx )5.08


ysyy



x )  5.08;
s=y((xdata)
data
  00..15
58
rrrrrr222222 

0
.
58
15

0
.
58
58
15
 000...79
58
rrr22222 
79
0
.
58
79
rrr22 
000...79
58

79
79

95
2
rr222 
0
.
79
58
 00..79
95

95
rrrr222222 
0
.
95
95

0
.
79

.
95

0
rrrr22222 
000
...15
95
79

0
15
2
95
222 

r
0
.


15
 0000..15
rrrrr2222222 
95
15


15
.
0
58

0
95
rrr22222 
0
.
15
58

 0000...15
rrrr2222 

58
0
.
58
r

0
222 
15
58
.
58

rrrr2222 
 0000...79
15

58
79
22  0.58

r
79
rrr222 
000...79
79

58
79
95
rr2222 
0
.
79
58
95
 00..79

95
rrrr2222 
0
.
95
95

0
.
79

0..79
95

95
rrr22222 
0
0


0
.
95
 00.95
rr2222 

0.95

rr  22 

0
r2  0
r  0
r 2  0
uRMS  4.68;
r 2  0.15
Effect of Reducing Uncertainty
1520
15
20
20
15
10
10
15
1510
5
55
10
10
00
0
-15
5 -15
5 -15
uRMS / 2  3.31; r22  0.58
r 2  0.15
uRMS  4.68;
r  2  0.79
uRMS / 2  2.34;
uRMS / 2  3.31; r2  0.58
95
r 2  0.15
1.;17;
uRMS /44.68
r  2  0.79
uRMS / 2  2.34;
 0.58
; ; r22 
data
us(RMS
/ ) 2 5.08
3.31
r   0.95
uRMS / 4  1.17;
 x/ 2  2.34;
uyRMS
r 22  0.79
r  0
s(data )  5.08;
uRMS / 4  1.17;
r 2 2 0.95
x  4.68;
y uRMS
r  0.15
2

s
(
data
)

5
.
08
;
r
2 0
x

yysyy(
x

x
data
)

5
.
08
;


x
uRMS / 2  3.31; r  0.58
x
yyy 
x
yx
 xx
y
uRMS / 2  2.34;
r 2  02 .79
uRMS  4.68; 2 r   0.15
uRMS
0.015
uRMS /44.68
 1; .17; r  r 2 
.95
uRMS / 2  3.312; 2 r2  0.58
uRMS
/ 2)  53.08
31; r r 
0.20
58
s(data
uRMS / 2  2.34; 2 r   0.79
y / x2  2.34;
uRMS
r   0.79
uRMS / 4  1.17; 2 r 2  0.95
uRMS / 4  1.17;
r   0.95
-10-10 -5-5
00
5 5 10 10 s(data
15 )15 5.08
20; 220 r2 25
 025
90Sr Likelihood
s(data ) 
5.08; 20
r  0 25
Mean
of
(mBq/day)
-10 Arithmetic
-5
0
5
10
15
yx
Arithmetic Mean of 90Sr
Likelihood
(mBq/day)
33
yx
Visualizing Uncertainty Reduction
uRMS
r´2 = 0.15
‹
σ(νi)
34
Visualizing Uncertainty Reduction
uRMS
r´2 = 0.15
r´2 = 0.57
‹
σ(νi)
35
Visualizing Uncertainty Reduction
uRMS
r´2 = 0.15
r´2 = 0.57
r´2 = 0.78
‹
σ(νi)
36
Visualizing Uncertainty Reduction
uRMS
r´2 = 0.15
r´2 = 0.57
r´2 = 0.78
r´2 = 0.94
‹
σ(νi)
37
Visualizing Uncertainty Reduction
uRMS
r´2 ≈ 0
r´2 = 0.15
r´2 = 0.57
r´2 = 0.78
r´2 = 0.94
‹
σ(νi)
38
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
The Common View:
The Measurement Is the Measurand
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
Oh, no! Results are
below some level
(DL, DT, LOD, etc.).
Might not be real!
500
0
-500
0
500
1000
1500
Oops! Activity < 0 is meaningless.
-500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
39
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
The Bayesian View: The Measurement
and the Prior Give the Measurand
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
40
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
The Bayesian View: The Measurement
and the Prior Give the Measurand
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
41
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
42
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
43
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
44
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
45
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
46
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
47
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
48
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
49
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
50
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
51
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
52
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
53
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
54
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
55
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
56
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
57
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
58
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
59
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
60
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
61
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
62
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
63
2000
1500
500
1000
000
500
500
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
000
The Bayesian View: The Measurement
and the Prior Give the Measurand
2000
Measurand
Measurement
y= x
Measurand
Measurand
Measurand
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
64
2000
000
1500
500
1000
000
500
500
137137
of of
Mean
Measurand
Arithmetic
CsCs
(mBq/kg)
Mean
Measurand(mBq/kg)
Arithmetic
5,337 137Cs Measurements Showing
2000
Uncertainty in Measurements & Measurands
2000
Measurand
Measurand
Measurand
Measurement
Measurand
Measurand
Measurand
y=
y =x x
yyy=
==xxx
1500
y= x
1000
500
0
-500
-500
0
500
1000
1500
Arithmetic Mean of 137Cs Likelihood (mBq/kg)
(mBq/kg)
65
2000
90Sr
90Sr
90Sr
90Sr
90
of of
(mBq/day)
Mean
Mean
ic
etic
Measurand
Measurand
of of
(mBq/
(mB
Mean
Mean
Measurand
Measurand
Arithmetic
Arithmetic
of (mBq/day)
Mean
Arithmetic
Sr Measurand
(mBq/day)
//
68
33;3..31
uu
 4422422...68
RMS
uu
RMS
uRMS
/

.31
31;;;
RMS
RMS

68
u

31
/

3
.
RMS
uuu
/
2

3
.
31
u

4
.
68
;
RMS

4
.
68
/
2

3
.
31
RMS
RMS
RMS
/
2

3
.
31
22..68
33.31
..31
2242
2
..;34
;; ;;
/////
68
334
RMS
uRMS
4
RMS

31
RMS
uu
0.707u
RMS
RMS
uuuuu
2
2
.

2
34
2

3
.
31
RMS
/
RMS
..31
2

334
.31
RMS
//2222

2
2....33334
.34
; ;;
RMS
2

31
uuu

2
34
RMS

2
;
RMS ////2
2

2
34
RMS
2

.
31
2

3
.
31
u
2

2
.
34
;
RMS
0.707u
0.5u
RMS
uuuuRMS

2
22...17
..17
34
RMS/////2
42
11
.34
;;;; ;
2
334
.31
RMS
RMS
2

34
RMS
4

1
2
3
.
31
2
2
RMS
u
/
RMS
u
/
4

.
17
RMS
2
2
.
34
RMS
4
1
.
17
RMS
RMS
1122112.....17
.17
34;;
uuRMS
//2
34
44242
.25u
0.5u
4
17
RMS
RMS

17
34
RMS////4
RMS
2
2
.
34;;;
RMS
usuu
1
.
17
u
4

1
.
17
RMS
ussu
/
24
.5.08
25
.17
34
RMS
/
1
17
(
data
)
.
08
RMS
(
data)

4
1
.
4
68
;
RMS
RMS
4
1
.
17
;;
u
/
2
2
.
34
RMS
(
data)

5.08
Measurand
u
/
4

1
.
17
RMS
u=0
.25u
((RMS
data
))

55..17
.;17
08
RMS
///4

1
68
.
4
((RMS
data)
5.08
RMS
4

1
.
;
RMS
uuussuussRMS
4
.
68
4

1
17
data
08
RMS
68
4
44))44
RMS
data)
5.08
(RMS
data
..;;;08
;;
RMS
RMS
....68
68
uysu
.
/
15
.17
sy=(RMS
(RMS
data

5
08
//
1
17
data)

5.08

x

x
68
4

u
yu=0
x
s
(
data
)
5
.
08
;;
y

x
RMS
;
68
4
2.268

317
..31
ssyRMS
((RMS
data
)
5
08
/
4
1
.
;
ssu
((RMS
data)

5.08
u

4

x
u

4
.
68
;
y

x
RMS
RMS
y
=
x
data)

5.08
RMS
31
3

/
data
5
; ;;
RMS
Uncertainty
Assigned
yy=yyRMS

xxx/xx/)424
5.08
33.;08
..31
...68
usu
68
RMS

uyu
x
RMS
Measurand
31
.
3

2
u
(
data)

RMS

RMS


4
68
u
/
2

31
RMS
///2)42242

...31
data
xxx//
(RMS
5.33.;334
08
; ;;;
uuu

..31

uyyyssyyRMS
68
RMS
31
308

2....68
RMS

x

31

RMS
31
3

2
2
;
(
data
)
5

x
RMS
u
/
2

3
.
31
u

4
68
u

4
68
;
RMS
/
2

3
.
31
;
RMS
0.707u
uuuuyRMS
 x/x////222
RMS
2
.
34
RMS
;
34
.
2

2

3
.
31
RMS
y

RMS
2

3
.
31
u
2

3
.
31
;
RMS
u

2
.
34
;;
34
.34
2...3334

y= x
RMS
uuyRMS
..31
 xx////2222222
222

;
RMS

34
RMS

31
;
34
.
2

2
3
.
31
RMS
0.5u
0.707u
RMS
y

uuuuu
/
2

2
.
34
;
34
.
2

2
/
RMS
..334
;;; ;
.31
RMS
RMS
RMS
222
112211
2....17
.17
34
RMS
/////424244
u
334
.31
RMS
RMS
17
u
RMS
34
RMS
u
17
RMS
u
/
2

2
.
;
34
uuRMS
///424244
112211.....17
.25u
17
0.5u
RMS

17
RMS
34
RMS

17
;
RMS
u
/
4

1
.
17
17
42)5.08
u(RMS
.08
34;;;;
RMS
RMS
24
25
(RMS
data
...34
RMS
17
111522...17
suussu
data)
/////4
1
RMS
4
17
u
2

.
34
RMS
(
data)

5.08
4

1
17
u=0
.25u
RMS
;
08
.

)
data
(
s
RMS
u
///4

1
..17
data)

5.08
RMS
4

1
.
17
;
RMS
ussus((RMS
4

1
17
08
5
)
data
(
RMS
RMS
data)

5.08
data
..08
uyssyussy=(((RMS
//44)))

15
08;;;
5.17
5.08
data
(RMS
1
17

x

x
data)
yu=0
x
08
.
5

data
(
RMS

x
data)

5.08
s(data )  5.08;
 000...58
15
58
rrrrr222222 
15

0
.
58

58
rrr222 
15
 000...79
58
79
2
r

0
.
58

79
rr222  00..79
58
79
79
rrr222 
0
.
95

0
.
79
58
 00..79
95

95
rrr2222222 
0
.
95

0
.
79
95

.
95
r

0
rrrr22222 
000
...15
95
79

0
15
2
95
222 

r
0
.


15
 0000..15
rrrrr2222222 
95
15


15
.
0
58

0
95
rrr22222 
0
.
15
58

 0000...15
rrrr2222 

58
0
.
58
r

0
222 
15
58
.
58

rrrr2222 
 0000...79
15

58
79
22  0.58

r
79
rrr222 
000...79
79

58
79
95
rr2222 
0
.
79
58
95
 00..79

95
rrrr222 
0
.
95
95

0
.
79

0..79
95

95
rrr22222 
0
0


0
.
95
 00.95
rr2222 

0.95

rr 22 

0
r2  0
r  0
r 2  0
2000
2000
20
25
25 20
u
1500
Arithmetic Mean of 137 Cs Measurand (mBq/kg)
Arithmetic Mean of 137Cs Measurand (mBq/kg)
25
5,337 137Cs20 Results
with Same Uncertainty
1500
1000
1000
500500
00
-500
-500
15
15
20
y = x 15
20
10
5.08
155..17
ssyussyy((((RMS
data)


xxxx/ 4))
08;;
data
10 10
data)

5.08
data

08

y

y

x
y

x
ysyy
= x
15
 xxxxx )5.08
data
5..08
08;;

yyssyyy(((data)


x
data
)

5

x
15
yyy 
xxx

yx
5
5
5
10
10 0
-10
-5
0
5
10
15
0 -15
90Sr Likelihood (mBq/d
Arithmetic
Mean of
0 0
500
1000
1500
-10
-5
0
5
10 2000 15
5 -15
Arithmetic
Mean of 137
(mBq/kg)
-10
-5 Cs Likelihood
0
5
10
15
5 -15
90
Arithmetic
Mean of Sr Likelihood (mBq/day)
66
0
500
1000
Arithmetic Mean of 137 Cs Likelihood (mBq/kg)
1500
2000
90Sr
90Sr
90Sr
90Sr
90
of of
(mBq/day)
Mean
Mean
ic
etic
Measurand
Measurand
of of
(mBq/
(mB
Mean
Mean
Measurand
Measurand
Arithmetic
Arithmetic
of (mBq/day)
Mean
Arithmetic
Sr Measurand
(mBq/day)
1500
1500
1000
1000
500500
00
-500
-500
 000...58
15
58
rrrrr222222 
15

0
.
58

58
rrr222 
15
 000...79
58
79
2
r

0
.
58

79
rr222  00..79
58
79
79
rrr222 
0
.
95

0
.
79
58
 00..79
95

95
rrr2222222 
0
.
95

0
.
79
95

.
95
r

0
rrrr22222 
000
...15
95
79

0
15
2
95
222 

r
0
.


15
 0000..15
rrrrr2222222 
95
15


15
.
0
58

0
95
rrr22222 
0
.
15
58

 0000...15
rrrr2222 

58
0
.
58
r

0
222 
15
58
.
58

rrrr2222 
 0000...79
15

58
79
22  0.58

r
79
rrr222 
000...79
79

58
79
95
rr2222 
0
.
79
58
95
 00..79

95
rrrr222 
0
.
95
95

0
.
79

0..79
95

95
rrr22222 
0
0


0
.
95
 00.95
rr2222 

0.95

rr 22 

0
r2  0
r  0
r 2  0
137Cs Results
25
5,331
20
20
25
25 20
u
Arithmetic Mean of 137 Cs Measurand (mBq/kg)
Arithmetic Mean of 137Cs Measurand (mBq/kg)
2000
2000
//
68
33;3..31
uu
 4422422...68
RMS
uu
RMS
uRMS
/

.31
31;;;
RMS
RMS

68
u

31
/

3
.
RMS
uuu
/
2

3
.
31
u

4
.
68
;
RMS

4
.
68
/
2

3
.
31
RMS
RMS
RMS
/
2

3
.
31
22..68
33.31
..31
2242
2
..;34
;; ;;
/////
68
334
RMS
uRMS
4
RMS

31
RMS
uu
0.707u
RMS
RMS
uuuuu
2
2
.

2
34
2

3
.
31
RMS
/
RMS
..31
2

334
.31
RMS
//2222

2
2....33334
.34
; ;;
RMS
2

31
uuu

2
34
RMS

2
;
RMS ////2
2

2
34
RMS
2

.
31
2

3
.
31
u
2

2
.
34
;
RMS
0.707u
0.5u
RMS
uuuuRMS

2
22...17
..17
34
RMS/////2
42
11
.34
;;;; ;
2
334
.31
RMS
RMS
2

34
RMS
4

1
2
3
.
31
2
2
RMS
u
/
RMS
u
/
4

.
17
RMS
2
2
.
34
RMS
4
1
.
17
RMS
RMS
1122112.....17
.17
34;;
uuRMS
//2
34
44242
.25u
0.5u
4
17
RMS
RMS

17
34
RMS////4
RMS
2
2
.
34;;;
RMS
usuu
1
.
17
u
4

1
.
17
RMS
ussu
/
24
.5.08
25
.17
34
RMS
/
1
17
(
data
)
.
08
RMS
(
data)

4
1
.
4
68
;
RMS
RMS
4
1
.
17
;;
u
/
2
2
.
34
RMS
(
data)

5.08
Measurand
u
/
4

1
.
17
RMS
u=0
.25u
((RMS
data
))

55..17
.;17
08
RMS
///4

1
68
.
4
((RMS
data)
5.08
RMS
4

1
.
;
RMS
uuussuussRMS
4
.
68
4

1
17
data
08
RMS
68
4
44))44
RMS
data)
5.08
(RMS
data
..;;;08
;;
RMS
RMS
....68
68
uysu
.
/
15
.17
sy=(RMS
(RMS
data

5
08
//
1
17
data)

5.08

x

x
68
4

u
yu=0
x
s
(
data
)
5
.
08
;;
y

x
RMS
;
68
4
2.268

317
..31
ssyRMS
((RMS
data
)
5
08
/
4
1
.
;
ssu
((RMS
data)

5.08
u

4

x
u

4
.
68
;
y

x
RMS
RMS
y
=
x
data)

5.08
RMS
31
3

/
data
5
; ;;
RMS
Uncertainty
Assigned
yy=yyRMS

xxx/xx/)424
5.08
33.;08
..31
...68
usu
68
RMS

uyu
x
RMS
Measurand
31
.
3

2
u
(
data)

RMS

RMS


4
68
u
/
2

31
RMS
///2)42242

...31
data
xxx//
(RMS
5.33.;334
08
; ;;;
uuu

..31

uyyyssyyRMS
68
RMS
31
308

2....68
RMS

x

31

RMS
31
3

2
2
;
(
data
)
5

x
RMS
u
/
2

3
.
31
u

4
68
u

4
68
;
RMS
/
2

3
.
31
;
RMS
0.707u
uuuuyRMS
 x/x////222
RMS
2
.
34
RMS
;
34
.
2

2

3
.
31
RMS
y

RMS
2

3
.
31
u
2

3
.
31
;
RMS
u

2
.
34
;;
34
.34
2...3334

y= x
RMS
uuyRMS
..31
 xx////2222222
222

;
RMS

34
RMS

31
;
34
.
2

2
3
.
31
RMS
0.5u
0.707u
RMS
y

uuuuu
/
2

2
.
34
;
34
.
2

2
/
RMS
..334
;;; ;
.31
RMS
RMS
RMS
222
112211
2....17
.17
34
RMS
/////424244
u
334
.31
RMS
RMS
17
u
RMS
34
RMS
u
17
RMS
u
/
2

2
.
;
34
uuRMS
///424244
112211.....17
.25u
17
0.5u
RMS

17
RMS
34
RMS

17
;
RMS
u
/
4

1
.
17
17
42)5.08
u(RMS
.08
34;;;;
RMS
RMS
24
25
(RMS
data
...34
RMS
17
111522...17
suussu
data)
/////4
1
RMS
4
17
u
2

.
34
RMS
(
data)

5.08
4

1
17
u=0
.25u
RMS
;
08
.

)
data
(
s
RMS
u
///4

1
..17
data)

5.08
RMS
4

1
.
17
;
RMS
ussus((RMS
4

1
17
08
5
)
data
(
RMS
RMS
data)

5.08
data
..08
uyssyussy=(((RMS
//44)))

15
08;;;
5.17
5.08
data
(RMS
1
17

x

x
data)
yu=0
x
08
.
5

data
(
RMS

x
data)

5.08
s(data )  5.08;
15
15
20
y = x 15
20
10
5.08
155..17
ssyussyy((((RMS
data)


xxxx/ 4))
08;;
data
10 10
data)

5.08
data

08

y

y

x
y

x
ysyy
= x
15
 xxxxx )5.08
data
5..08
08;;

yyssyyy(((data)


x
data
)

5

x
15
yyy 
xxx

yx
5
5
5
10
10 0
-10
-5
0
5
10
15
0 -15
90Sr Likelihood (mBq/d
Arithmetic
Mean of
0 0
500
1000
1500
-10
-5
0
5
10 2000 15
5 -15
Arithmetic
Mean of 137
(mBq/kg)
-10
-5 Cs Likelihood
0
5
10
15
5 -15
90
Arithmetic
Mean of Sr Likelihood (mBq/day)
67
0
500
1000
Arithmetic Mean of 137 Cs Likelihood (mBq/kg)
1500
2000
The
Importance of
Accurate
Uncertainty
Upper (red) point
everybody else prior
likelihood (data)
posterior (measurand)
• Nearly the same
measurement result
• s(lower) ≈ 2.5s(upper)
• Upper posterior
resembles likelihood
(i.e., measurement)
• Lower posterior
resembles prior
68
Lower (yellow) point
everybody else prior
posterior
(measurand)
likelihood (data)
n of
137
of
(mBq/
Mean
Arithmetic
Cs Measurand
137
(mBq/kg)
Arithmetic Mean of Cs Measurand
500
137Cs
5,331
2000
100
1000
1500
u
Likelihood (mBq/kg)
1500
Measurand
137
y=x
zero line
Cs Results
2000
25
90Sr (mBq/day)
easurand
of
ean
Measurand
of 90
Mean
metic
Sr Measurand(mBq/day)
(mBq/day)
Arithmetic Mean of 137 Cs Measurand (mBq/kg)
222  0.15
uuRMS  44..68
;
r
68
;;
rr 2  00..15
uuRMS
.68
RMS 44
.
68
15
RMS
u

4
.
68
22  0.15

y= x
u

4
.
68
;
r
RMS
RMS
RMS
u

// 44422...68
 ;33..31
;; rr222  00..58
uRMS
68
15
uuu
68
RMS
RMS/
RMS
31
58
u
2

3
.
31


4
.
68
u

4
.
68
;
r

0
.
15
RMS
uuuRMS

RMS
/
2

3
.
31
;
r

0
.
58
RMS

3
.
31
RMS /// 2
222  0.58
uuuRMS
2

3
.
31
RMS

2

3
.
31
;
r

..31
RMS //// 22
.33.34
;; ; rrr222  000...79
RMS
2
2
31
uRMS
2
3
.
31
58
RMS
0.707u
RMS
uuuuu
/
2

2
.
34
25
/
2

2
34
79
RMS
/
2

3
.
31

RMS
u
/
2

3
.
31
;
r

0
.
58
/
2

2
.
34
RMS
2
2222..34
.34
34;; r 22  0.79
79
RMS
uuuRMS
RMS ///2
20
2

34
RMS
2

.
0.5u
1000
2
RMS
u
/
2

2
.
34

RMS
/ 2442
..34
000...95
RMS
uRMS
112112...17
.17
34;;; rrr222 

79
RMS
uuu
17

95
RMS/////4
4

17
RMS
RMS
u
2

2
.
34
;
r

0
.
79
RMS
u
/
4

1
.
17
.25u
4
1
17
95
2
RMS
u
/
4

1
.
17
500
1000
1500
2000
10
RMS
RMS
22  0.95
RMS //44
25
2

u
1
.
17
u

1
.
17
;
r
RMS
data
..;08
;;

RMS
sMeasurand
data)
ususus(((RMS
4
68
rrrrr22222 
00000...15
/

15
/44))
..5.08
15.17
17

95
data)
5.08
u=0
(
data

.
08
;

RMS
15
;
68
4
137
s
(
data)

5.08
/
4

1
.
17
;
95
u

4
.
68
RMS

an of Cs Likelihood (mBq/kg)
data
5.;08
08;
 000.15
20
RMS

rrr2222 
68
4...68
))4
ussyRMS
su
(((RMS
data)
5.08
u

data
5

x
y

x
15
00..58
r
;
68
4

y
=
x
2
RMS
s
(
data)

5.08
y

x
RMS
RMS
u

4
.
68


/
2

3
.
31
;
r

0
2
s
(
data
)

5
.
08
;
r

Measurand
2
y

x
u

4
.
68
;
15
RMS
y

x
u

4
.
68
22  0.58
RMS
yu
=
x x/x
RMS

RMS
r
;
31
.
3

2
/
uuuysyRMS
(
data
)

5
.
08
;



u
2

3
.
31

4
.
68

4
.
68
;
r

0
.
15
RMS
u
RMS
58
 00..58
31;; rr22 
.31
3333..31
RMS
15
500
 xx///// 22222
RMS

uuuyyyRMS

31
y= x u
.

2
RMS

3
.
31

x

RMS /// 22
.3.34
;; ; rrr222  000...79
RMS
.31
uyRMS
222
334
.31
58
RMS
0.707u

x
RMS
uuuuu
/
2

.
34
79
2
2
/
RMS
/
2
3
.
31

RMS
u
/
2

3
.
31
;

0
.
58
/
2

2
.
34
RMS
79
r
;
34
.
2

2
2
RMS
uuuRMS

222..34
RMS ///2
20 15
2

34
RMS
79
 00..95
34;; rr222 
.17

2
0.5u
RMS
u
/
2

212
.17
34
RMS
/
4

.
RMS
uRMS
/
2

.
34
;
r

0
.
79
RMS
uuu
/
4

1
.
2

/
2

2
.
34
2
95
.
0

r
;
17
.
1

RMS //44
1
.
17
RMS

RMS
u
2

2
.
34
;
r

0
.
79
RMS
u
/
4

1
.
17
.25u
95
17
1
4
2
RMS
u
/
4

1
.
17
RMS
RMS
2
RMS //44

u
1
.
17
95
.
0

r
;
17
.
1

u
2
RMS

s
(
data
)

5
.
08
;
r

0
RMS
susus((RMS
data)
//44)

115.17
5.08
..17
;;
rr222 
95
data)
5.08
u=0
 000..95
08
data
RMS
ssus((((RMS
data)

5.08
1010-500
/
4

1
.
17
;
r

500
1000
15000
2000500 10
08
5
)
data
2
data)
1000
1500
5.08; 2000 r2  0
)5.08
data

x
x
yysyssy
=(((x
data)
5.08

x
10
data
)

50..08
082000
;
r100
 00

yysyy(1500

xxxx )  5
-500
0
500
1000
Arithmetic
Mean of Cs
Likelihood
(mBq/kg)
ean of 137Cs Likelihood
(mBq/kg)
-500
-400
-300
-200
-100
2 
data
;
r


15
yyy 
xxx
137

Arithmetic Mean of Arithmetic
Cs Likelihood
y(mBq/kg)
 of
x 137Cs Likelihood (m
Mean
137
69
n of
137
of
(mBq/
Mean
Arithmetic
Cs Measurand
137
(mBq/kg)
Arithmetic Mean of Cs Measurand
500
137Cs
Measurand
y=x
zero line
5,331 137Cs Results – Log Scale
2000
100
1000
1500
u
Likelihood (mBq/kg)
1500
2000
25
25
90Sr
90Sr
90
(mBq/day)
asurand
easurand
ofof
(mBq/day)
(mBq/day)
an
ean
Measurand
Measurand
of (mBq/day)
Mean
metic
Sr Measurand
(mBq/day)
Arithmetic Mean of 137 Cs Measurand (mBq/kg)
222  0.15
uuRMS  44..68
;
r
68
;;
rr 22  00..15
uuRMS
..68
RMS 44
.
68
15
2222  00..15
RMS

4
68
;
r
u

4
.
68
y= x
u

4
.
68
;
r
15
RMS
RMS
RMS
u

4
.
68
;
r

0
.
15
RMS
//4442.2.68
 ;33..31
;; rr222 
uuu

RMS
uRMS
.68
68
00..58
15
RMS
RMS
RMS
RMS
31

58
RMS
u
/
2

3
.
31


4
.
68
u
u

4
.
68
;
r

0
.
15
2
RMS
uuuRMS

2
RMS
/
2

3
.
31
;
r

0
.
58
RMS
RMS
4
.
68
2

3
.
31

RMS //
/
2

3
.
31
;
r
0
.
58
222 
uuu
2

3
.
31
u

4
.
68
;
15
RMS

/
2

3
.
31
;
r

0
.
58

4
.
68
RMS
RMS
2
2
31
RMS
22..68

33..31
..31

58
RMS///
2242

.3.;34
;; ;; rrrr2222 
000...79
RMS
68
334

4
15
RMS
/

31

58
RMS
0.707u
uu
RMS
u
RMS
uuuuu
/
2
2
.
RMS
25
/

2
34
79
2

3
.
31
/
RMS
2

RMS
2

3
.
31
u
/
2

3
.
31
;
r

0
.
58
RMS
/
2

2
.
34
222

2...3334
.34
34
; ; rr22  00..79
79
RMS
uuRMS
..31
222

;
RMS ////2
20
22

34
RMS

31
2

2
.
34
;
79
2
3
.
31
58
RMS
0.5u
1000
0.707u
2
RMS
uuuuu
/
2

2
.
34

2
/
2

2
.
34
;
r

0
.
79
RMS
4
1
.
17
;
95
/
2

3
.
31
RMS
RMS

RMS
2

2
.
34
;
r

0
.
79
RMS
/
4

1
.
17
u
/
2

3
.
31
;
58
2
2
2
.
34
RMS
u

.34
17
 00..79
95
RMS ////4
2244
RMS
u
20
RMS
RMS
uRMS
24
1112221112.......17
.17
34;;;
rr2222 

u

34
RMS
17
.25u
17
95
0.5u
RMS
u
/
4

17
500
1000
1500
2000
10
2
34
RMS
RMS
/
4

r
0
.
95
RMS
25
2

u
/
4

1
.
17
2
2
.
34
;
r

0
.
79
RMS
u
/
4

1
.
17
;
95
2
2
RMS
25
//442424)
112...17
17

95
su
(RMS
data
.5.08
..34

RMS
sMeasurand
data)
u
//

1
4
68
rrrrr222222 
00000....15
RMS
17

95
us(RMS
.;;08
34;;;
79
ssu
((RMS
data)
5.08
u

1
17
u=0
.25u
RMS
(
data
)

5
.
08

RMS
u
/
4

1
.
17
15
68
.
4

RMS
137
data)

5.08
RMS
2
/
4

1
.
17
;
95
RMS
25
u

4
.
68
2

RMS
u
/
4

1
.
17
u
r

0
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70
y
137
Innovation 1
• this work addresses the situation in which each measurement is
accompanied by a very good estimate of its uncertainty
– not described in the literature reviewed
– which occurs routinely in radiochemical and radiobioassay measurements
71
Innovation 2
• This work provides a solution to the vexing problem of making
sense of negative measurement results for a quantity, such as
activity in becquerels, which physically must be nonnegative
– none of the literature addresses negative values
• The method makes sense of uncertain low-level measurements
– without injecting a bias into the dataset by left-censoring
– by implicitly recognizing that spurious negative results are accompanied by
an equal amount of spurious positive signal
72
Innovation 3
• This work provides posterior estimates, in the form of probability
distributions, of the true value of each measurand
– while the literature is concerned with correcting estimates of slopes of doseresponse relationships for the effects of classical measurement error
73
Innovation 4
• This work shows that accurate estimates of uncertainty are as
important as the values of the measurement results
– overestimates of uncertainty can lead to nonsense results
74
Innovation 5
• This work provides the ability to explore the impact of the
magnitude of uncertainty on the posterior distribution of
measurands by thought experiments involving substitution of the
mean square measurement uncertainty, or some multiple or
submultiple of it, for the individual uncertainties
75
Innovation 6
• The method is shown to closely correspond to classical
(frequentist) methods when uncertainty is relatively small
76
Innovation 7
• This work answers the questions, conditional on plausible
assumptions,
– “What true state of nature gave rise to this set of observations?”
– “For each individual measurement result, what are the probable values of
the measurand that led to this measurement result?”
• The authors believe that the method represents a significant step
forward in the making sense of groups of uncertain, low-level
radioactivity measurements
77
Conclusions
• Sample variance of a set of measurements is disaggregated into
– measurement uncertainty
– population variability
• A reasonable, possible distribution of measurands for a
population is the result
• When x  0 , positive posterior PDFs of the measurand are
computed using “everybody else” priors for each individual
– negative values are eliminated
– mean of measurements is preserved
• When there is essentially no variance in the data due to
population variability, the method cannot be expected to work,
and it does not work
78
Conclusions: Utility and Correspondence
• The method eliminates negative measurement results in an
uncensored data set and preserves the arithmetic mean of the
data set
– If measurement results have a relatively large uncertainty, the posterior
PDF of the measurand resembles the prior
– If the measurement results have a relatively small uncertainty, the posterior
PDF of the measurand resembles the likelihood, that is, it is relatively
close to the measurement result before application of the Bayesian methods
• Best estimate of uncertainty is just as important as measurement!
• As required by Bohr’s correspondence principle, results
produced by the methods introduced here correspond to results of
traditional statistical inference in the domain in which that
inference is known to be correct
79
Authors
Strom
MacLellan
Joyce
Watson
Lynch
Antonio
Zharov
(Mayak PA)
Birchall
(UK HPA)
(Scherpelz, Vasilenko)
Acknowledgments
• PNNL: Kevin Anderson, Gene Carbaugh, Michelle
Johnson, Bruce Napier, Bob Scherpelz, Paul Stansbury,
Rick Traub
• SUBI: Vadim Vostrotin
80