Battelle Team 

Dose Reconstruction 

Project for NIOSH

Document Title: 

Default Assumptions and Methods for Atomic 

Weapons Employer Dose Reconstructions 

Document Number: Battelle-TIB-5000 

 PNWD-3741 

Revision: 00 

Effective Date: 02 Apr. 2007 

Type of Document: Technical Information 

Bulletin (TIB) 

Supersedes: None 

Subject Experts: Daniel J. Strom 

Document Owner 

Approval: 

Signature on file 

Approval Date: 20 Dec 2006 

Daniel J. Strom, Staff Scientist 

Approval: Signature on file Approval Date: 20 Dec 2006

Jay A. MacLellan, Battelle PNWD Task Manager 

Concurrence: Signature on file Concurrence Date: 20 Dec 2006

Richard J. Traub, Staff Scientist 

Approval: Signature on file Approval Date: 02 Apr 2007 

James W. Neton, Associate Director of Science 

 New Total Rewrite Revision Page Change 

FOR DOCUMENTS MARKED AS A TOTAL REWRITE, REVISION, OR PAGE CHANGE, REPLACE THE PRIOR 

REVISION AND DISCARD / DESTROY ALL COPIES OF THE PRIOR REVISION. 
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Publication Record 

EFFECTIVE 

DATE 

REVISION 

NUMBER DESCRIPTION 

27 March 2006 00 - A Document initiated by Daniel J. Strom 

4 April 2006 00 – A.01 Corrections to Eisenbud 1950 data, other errors; revision of 3.0 

24-28 April 06 00 – A.02 Additions 

May 2006 00 – A.03 Additions 

May 2006 00 – A.04 Additions 

May 2006 00 – A.06 Additions 

July 2006 00 – A.07 Added 3.17 

31 July 2006 00 – A.10 Review draft sent to NIOSH 

11 Sept 2006 00 – A.11 Response to NIOSH Comments 

Radon and Thoron section completed 

Added Section 3.23, Use of time-dependent data 

21 Sept 2006 00 – A.12 Fixed List of Tables 

13 Oct 2006 00 – A.13 Responding to comments from OCAS 

7 Nov 2006 00 – E.01 Completed comment responses to OCAS 

20 Dec 2006 00 – F Completed comment responses to OGC 

 
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Contents 

Publication Record.......................................................................................................................................iii

Acronyms and Abbreviations ....................................................................................................................viii

1.0 Introduction ...................................................................................................................................... 10

2.0 Fitting Statistical Distributions to Data ............................................................................................ 11

2.1 Lognormal Distributions ......................................................................................................... 11

2.1.1 Uncensored Individual Observations ......................................................................... 12

2.1.2 Summary Statistics..................................................................................................... 13

2.1.2.1 Example Using Two Data Points ............................................................... 14

2.1.2.2 Using Minimum, Mean, and Maximum Values with Number of 

Observations to Determine the Parameters of a Lognormal Distribution.. 15

2.1.2.3 Using Range and Mean Value without Number of Observations to 

Determine the Parameters of a Lognormal Distribution............................ 15

2.1.2.4 Use of Range and Average Value Data that Are Inconsistent with a 

Lognormal Distribution ............................................................................. 16

2.1.2.5 Use of a Single Measurement Value.......................................................... 17

2.1.3 Censored Individual Observations ............................................................................. 17

2.1.3.1 Left-Censored Data.................................................................................... 17

2.1.3.2 Finney Weighting Factors.......................................................................... 18

2.1.3.3 Right-censored Data................................................................................... 19

2.1.4 Grouped, Censored Observations............................................................................... 19

2.1.4.1 Frequency Weighting for Grouped Data.................................................... 20

2.1.5 “Reasonableness” of a Lognormal Distribution......................................................... 22

2.1.6 Summary of Default Assumptions for Fitting Lognormal Distributions ................... 23

2.2 Triangular Distributions .......................................................................................................... 23

2.3 Normal Distributions............................................................................................................... 26

2.3.1 Normally-distributed Measurement Uncertainty and an Underlying Lognormallydistributed Measurand: Mirror Image Method........................................................... 26

2.3.2 Normally-distributed Measurement Uncertainty and an Underlying Lognormallydistributed Measurand: Preserved Mean and Variance Method................................. 27

2.3.2.1 Test of the Preserved Mean and Variance Method .................................... 28

2.4 Rectangular Distributions........................................................................................................ 32

2.5 Constant “Distributions” ......................................................................................................... 32

3.0 Default Assumptions ........................................................................................................................ 33

3.1 Introduction ............................................................................................................................. 33

3.2 External Irradiation Geometry................................................................................................. 33

3.3 The 95%ile and “Constant” Uncertainty Distribution for Limited Data Sets ......................... 34

3.4 Uncertainty in Biokinetic Models ........................................................................................... 34

3.5 Aerosol Particle Size and Respirable Fraction ........................................................................ 35
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3.6 Use of Time-period-specific, Process-based GSDs for Published Mean Aerosol 

Concentration Data.................................................................................................................. 36

3.7 Use of Distributions to Describe Multiple Populations........................................................... 37

3.8 Use of Time-Weighted Averages, Breathing Zone (BZ) Air Samples, and General Area 

(GA) Air Samples, Process (P) Air Samples, and Considerations of Sample Duration.......... 38

3.9 Particle Solubility (ICRP 66 Transportability Classes F, M, S) and f1 (Gastrointestinal 

Absorption Fractions).............................................................................................................. 41

3.10 Exposure Time and Intake Calculations.................................................................................. 42

3.11 Ingestion .................................................................................................................................. 42

3.12 Occupational Medical Doses................................................................................................... 42

3.13 External Dose Conversion Factors .......................................................................................... 43

3.14 External Missed Dose When There Was Monitoring.............................................................. 43

3.15 Internal Missed Dose When There Was Monitoring............................................................... 43

3.16 Environmental Dose................................................................................................................ 43

3.17 Radon and Thoron and Their Short-lived Decay Products...................................................... 43

3.17.1 Radon and Thoron...................................................................................................... 44

3.17.2 Potential Alpha Energy Exposure and Concentration ................................................ 44

3.17.3 Equilibrium Factors.................................................................................................... 45

3.17.4 Summary of radon and thoron quantities and conversion factors .............................. 47

3.18 Radium Monitoring by Breath Radon Analysis ...................................................................... 48

3.19 Determination of the Uncertainty Distribution for Annual Organ Doses Summed Over 

Multiple Intakes....................................................................................................................... 48

3.20 Representativeness of Air Samples ......................................................................................... 52

3.20.1 Inferring Representativeness by Comparing BZ with GA Samples........................... 52

3.20.2 Inferring Representativeness by Comparing Excretion Rates Predicted from Air 

Samples with Measured Excretion Rates .................................................................. 55

3.21 Propagation of Medians and Uncertainties for Lognormal Distributions ............................... 56

3.21.1 Propagation of Medians (not Means) for Products of Lognormal Distributions........ 56

3.21.2 Propagation of Uncertainties for Lognormal Distributions........................................ 57

3.22 Adding Doses with Differing Distributions ............................................................................ 57

3.23 Adjusting Process-Specific Dose Rates or Air Concentrations for Time Trends over 

Periods of Years ...................................................................................................................... 58

4.0 Glossary............................................................................................................................................ 63

5.0 References ........................................................................................................................................ 67
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Figures 

Figure 1. Finney weights (y), standard normal distribution, and ratio. 18

Figure 2. Lognormal fits to 194 air monitoring results from one site, with the first 39 results being 0 or 

“less-thans”. 19

Figure 3. Examples of fitting a lognormal distribution to grouped, left- and right-censored data (from 

Table 7, 1949, Eisenbud and Quigley 1956) 20

Figure 4. Mean, median, and upper 95th %ile for the airborne uranium data from Table 7 of Eisenbud and 

Quigley (1956) 21

Figure 5. Lognormal fits to grouped annual deep dose equivalent measurements for 458 persons. Note the 

significantly different slopes for different weighting. 22

Figure 6. The probability density function (pdf), cumulative density function (cdf), and parameters of a 

triangular distribution 24

Figure 7. Analysis of Y-12 uranium in urine measurements (Strom 1984, Figure 16). a. the product of a 

lognormal and a normal distribution producing a negative tail. b. histogram of uranium excretion values 

from Y-12 in 1971. 27

Figure 8. Normalized probability density functions (pdfs) for Hanford in vivo 137Cs measurements on 

unexposed workers 29

Figure 9. Cumulative probability density functions (cdfs), observed (rough red line) and predicted 

(smooth line) using GSD = 1.4 29

Figure 10. PDFs with uncertainty model (smooth thin line), lognormal "true state of nature" model 

(dashed line), and predicted value (heavy black line) 31

Figure 11. Residuals for predicted cdf. 31

Figure 12. Lognormal plot of mean airborne U concentrations for 136 different processes in uranium 

refining (Christofano and Harris 1960). 37

Figure 13. Time-Weighted Average (TWA) Radon Concentration in Air, WL (if 100% equilibrium) 40

Figure 14. Distribution of results from Monte Carlo simulation of TWA radon data. 41

Figure 15. Graph of 10,000 Monte Carlo trials adding 10 samples from lognormal distributions with x50 = 

1 and GSD = 4. 51

Figure 16. Graph from Caldwell (1972) digitized in Table 3.10. The straight line refers to predictions of 

the ICRP Publication 2 lung model 53

Figure 17. Comparision of lapel (BZ) air samples to general area air samples (Fig. 4 from Caldwell et al. 

1967) 54

Figure 18. Lognormal fit to the ratio of daily uranium excretion divided by the average airborne uranium 

concentration for maintenance workers, from Yu and Sherwood (1996). 55

Figure 19. Original Figure 1 from Christofano and Harris (1960) showing time-weighted average 

exposure trends over the years 1948-1956. 58

Figure 20. Analysis of time-weighted average air concentrations from Christofano and Harris (1960; Fig. 

1). Variability bars are the geometric mean times or divided by one GSD. 60

Figure 21. Use of year-specific correction factors to adjust air concentrations for processes over time. 61

Tables 

Table 2.1. Uncertainty distributions and their parameters for input to IREP 11

Table 2.2. Symbols, parameters, and relationships for the lognormal distribution 12

Table 2.3. Fifteen distinct ways of determining a lognormal distribution from minimal information 13

Table 2.4. “Exposure to Soluble Uranium Compounds,” reproduced from Eisenbud and Quigley (1956).

14

Table 2.5. Deriving a lognormal distribution from two data points. 14
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Table 2.6. Regression values and statistics for the 1949 example. x , x50, Std. Dev., and Upper 95th %ile 

are all in mg/m3

. 21

Table 2.7. Comparison of attempts to fit lognormal and triangular distributions to mean and range data 

from Christofano (1960) 25

Table 2.8. 90, 95, and 99 %iles of lognormal "true state of nature" distributions with GSDs of 1.2, 1.4, 

and 1.6. 32

Table 3.1. Reliability categories for selected results of ICRP Publication 30 biokinetic and dosimetric 

models 34

Table 3.2. Estimated reliability, for selected radionuclides, of the effective dose coefficient values 

recommended in ICRP Publication 30 (from NCRP 1998 Table 8.2) 35

Table 3.3. Daily Weighted Rn. Exposure to Tower Workers (Table I from NIOSH Ref ID 8921 p.13, May 

1951, Lake Ontario Ordinance Works) 39

Table 3.4. Summary statistics from TWA analysis. 40

Table 3.5. Default exposure time assumptions as a function of date 42

Table 3.6. Uncertainty distributions for equilibrium factors for converting radon and thoron gas 

measurements to working levels (WL) 47

Table 3.7. Summary of numerical conversions for radon and thoron quantities, regardless of the precision 

of measurements 48

Table 3.8. Results of adding random samples from 10 lognormal distributions with the same GSD. 50

Table 3.9. Parameters of the lognormal uncertainty distribution due to lack of representativeness of an air 

sample distribution. 52

Table 3.10. Breathing zone (BZ or Lapel) air sampling and general area (GA) air sampling (data digitized 

from Caldwell 1972; adapted from (Strom, Watson, and Stansbury 2002)). 53

Table 3.11. Summary statistics for the lognormal fit to the Pu distribution shown in Figure 17. 54

Table 3.12. Results of digitizing data from Figure 19. All concentrations (columns 3-7 and 9) are in 

dpm/m3 59

Table 3.13. Year-specific correction factors for tabulated process-specific air concentrations, derived from 

analysis of Christofano and Harris (1960) Fig. 1. 62
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Acronyms and Abbreviations 

μR microroentgen 

ABRWH Advisory Board on Radiation and Worker Health 

AP anterior-posterior 

AWE Atomic Weapons Employer 

Bq becquerel 

BZ breathing zone 

CATI Computer-Assisted Telephone Interview 

CV coefficient of variation 

d/m/m3 disintegrations per minute per cubic meter 

DCF dose conversion factor 

DOL Department of Labor 

dpm disintegrations per minute 

DR dose reconstruction 

Dx diagnosis 

EEOICPA Energy Employees Occupational Illness Compensation Program Act 

F ICRP Respiratory Tract Transportability Type Fast

FGR-12 Federal Guidance Report 12 

FUSRAP Formerly Utilized Sites Remedial Action Program 

GA general area 

GSD geometric standard deviation 

HP health physicist 

ICD-9 International Classification of Diseases Revision 9 

ICRP International Commission on Radiological Protection 

IH industrial hygienist 

IMBA Integrated Modules for Bioassay Analysis 

IREP Interactive Radioepidemiological Program 

ISO isotropic 

keV kiloelectronvolt 

LAT lateral 

LOD limit of detection 

LOGNORM4 computer program 

mR milliroentgen 

M ICRP Respiratory Tract Transportability Type Moderate

Max maximum 

MeV megaelectronvolt 

Min minimum 

N/A not applicable 

NIOSH National Institute for Occupational Safety and Health 

NOCTS NIOSH Occupational Claims Tracking System 

NORMSDIST() standard Normal distribution function in Microsoft Excel 

NORMSINV() inverse standard Normal distribution function in Microsoft Excel 

OCAS Office of Compensation Analysis and Support 
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PA posterior-anterior 

PC probability of causation 

Q quality factor 

R roentgen 

ROT rotational 

S ICRP Respiratory Tract Transportability Type S

SD standard deviation (arithmetic) 

SEC Special Exposure Cohort 

SQRI Site Query Research Interface 

TBD Technical Basis Document 

TIB Technical Information Bulletin 

TLD thermoluminescent dosimeter 

TLV® Threshold Limit Value (® American Conference of Governmental Industrial 

Hygienists) 

Tn thoron (220Rn) 

TWA time-weighted average 

UMTRCA Uranium Mill Tailings Radiation Control Act 

WLM working level month (a unit of potential alpha energy concentration) 

wR ICRP radiation weighting factor 

 
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1.0 Introduction 

Technical Information Bulletins (TIBs) are general working documents that provide guidance concerning 

the preparation of dose reconstructions at particular sites or categories of sites. They will be revised if 

additional relevant information is obtained. TIBs may be used to assist the National Institute for 

Occupational Safety and Health (NIOSH) in the completion of individual dose reconstructions. 

In this document the word “facility” is used as a general term for an area, building, or group of buildings 

that served a specific purpose at a site. It does not necessarily connote an “atomic weapons employer 

facility” or a “Department of Energy [DOE] facility” as defined in the Energy Employees Occupational 

Illness Compensation Program Act of 2000 [42 U.S.C. § 7384l(5) and (12)]. 

There are several areas in which default assumptions are useful in conducting dose reconstructions under 

the Energy Employees Occupational Illness Compensation Program Act (EEOICPA; 42 U.S.C. § 7384 et 

seq. This TIB provides a technical justification and basis for assumptions in several areas needed for dose 

reconstruction for claimants from Atomic Weapons Employers (AWEs). 

Section 2 treats fitting statistical distributions to data, often when the data are sparse or summarized. 

Lognormal, normal, triangular, and rectangular distributions are covered. 

Section 3 addresses a variety of problems in dose reconstruction for which science policy choices need to 

be elucidated. Some are mathematical or statistical, while others are simply default assumptions based on 

limited data. 
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2.0 Fitting Statistical Distributions to Data 

Data that are entered into the Interactive Radioepidemiological Program (IREP) are characterized by one 

of 7 kinds of uncertainty distribution, as shown in Table 2.1. Each kind of uncertainty is characterized by 

1, 2, or 3 parameters. 

Table 2.1. Uncertainty distributions and their parameters for input to IREP 

Uncertainty Distribution Parameter 1 Parameter 2 Parameter 3 

Lognormal median geometric standard deviation N/A 

Normal mean standard deviation N/A 

Triangular or LogTriangular minimum mode maximum 

Uniform or LogUniform minimum maximum N/A 

Constant value N/A N/A 

It must be emphasized that merely fitting a statistical distribution to data does not eliminate some critical 

science policy decisions and expert judgments. When data are inconsistent with a statistical distribution, 

additional science policy questions and the need for expert judgment arise. Discussions and examples of 

science policy questions and expert judgments are given in Section 3.0. 

2.1 Lognormal Distributions 

Many kinds of occupational and environmental measurements are found to be lognormally-distributed. In 

particular, “[a] lognormal process is one in which the random variable of interest results from the product 

of many independent random variables multiplied together” (Ott 1995). Ott further states, “In processes 

observed in the environment, the number of independent random variables multiplied together usually 

does not have to be very great before characteristic lognormal properties emerge. Because environmental 

concentrations usually depend on the number of molecules of a pollutant present per unit volume, they 

ordinarily are positive random variables (Ott 1995). 

The lognormal distribution is characterized by two parameters (Aitchison and Brown 1981), the 

geometric mean (which is the median and here denoted by x50) and the geometric standard deviation, 

GSD. Many other properties of interest are expressed in terms of these. Unlike the normal distribution, the 

mean, median, and mode are not equal for a lognormal distribution. If only one value from a lognormal 

distribution is to be used in a risk calculation, it is the arithmetic mean (also known as the expectation 

value), not the geometric mean. This is true of any distribution, not just the lognormal. The arithmetic 

mean is always larger than the geometric mean. 

Table 2.2 lists the key parameters of a lognormal distribution. 
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Table 2.2. Symbols, parameters, and relationships for the lognormal distribution 

Symbol Parameter 

μ logarithm of the geometric mean 50 μ = ln x

σ logarithm of the geometric standard deviation σ = lnGSD

x50 geometric mean (median) x50 = exp( ) μ

GSD geometric standard deviation GSD = exp( ) σ

x arithmetic mean exp( / 2) 2

x = μ +σ

xmode mode ( ) 2

mode x = exp μ −σ

CV coefficient of variation exp( ) 1 2 2 CV = σ −

var(x) variance ( )2 var(x) = xCV

SD arithmetic standard deviation SD(x) = xCV

x95 95th %ile exp( 1.645 ) x95 = μ + σ

x1, x2 values 1 and 2 

z1, z2 standard normal deviates 1 and 2 

p1, p2 percentiles for x1, x2 

f1, f2 fractiles for x1, x2 

2.1.1 Uncensored Individual Observations 

For lognormally-distributed data in which all data points are positive values, the maximum likelihood 

estimator of the geometric mean (median), x50, of the data is 

ln , 1 exp( ) exp

1

50 ⎟

⎠

⎞ ⎜

⎝

⎛ = = ∑=

n

i

i x

n

x μ 1 

where μ is the natural logarithm of the geometric mean. The maximum likelihood estimator of the 

variance of the logarithms of the data, σ

2

, is 

( ) ln , 1

1

1

2 2

∑=

− − =

n

i

i x

n

σ μ 2 

and the geometric standard deviation, GSD, is 

( ) ln . 1

1 exp( ) exp

1

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ − − = = ∑=

n

i

i x

n

GSD σ μ 3 

Any given percentile or fractile xf of the lognormal distribution is given by 

exp(μ σ ). f f x = + z 4 

For the 95th percentile, z0.95 = 1.645, so 
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exp( 1.645 ). x95 = μ + σ 5 

2.1.2 Summary Statistics 

There are many combinations of summary statistics from which a lognormal distribution can be uniquely 

defined. Strom and Stansbury (2000) described 15 methods, and a freeware program called LOGNORM4 

can be downloaded1

 to perform the calculations. The first 15 methods in Table 2.3 can be found using 

LOGNORM4, while method 16 is described in Section 2.1.2.3. 

Table 2.3. Fifteen distinct ways of determining a lognormal distribution from minimal information 

Method Determining Parameters 

1 the mean and median (or their natural logs) 

2 the mean and mode (or their natural logs) 

3 the median and mode (or their natural logs) 

4 the median (or its natural log) and the GSD or sigma = ln(GSD) 

5 the mean (or its natural log) and the GSD or sigma = ln(GSD) 

6 the mode (or its natural log) and the GSD or sigma = ln(GSD) 

7 a value and its percentile OR fractile OR std norm deviate and GSD or sigma=ln(GSD) 

8 the median and a value with its percentile OR fractile OR std normal deviate 

9 the mean and a value with its percentile OR fractile OR std normal deviate 

10 the mode and a value with its percentile OR fractile OR std normal deviate 

11 the median and [arithmetic] standard deviation OR coefficient of variation 

12 the mean and [arithmetic] standard deviation OR coefficient of variation 

13 the mode and [arithmetic] standard deviation OR coefficient of variation 

14 a value and its percentile OR fractile OR std norm deviate and [arithmetic] SD or CV

15 a pair of values and their percentiles OR fractiles OR std normal deviates 

16 minimum, maximum, and mean values (see 2.1.2.3) 

Sometimes data are reported in groups, e.g., airborne uranium concentration measurements from 

Eisenbud and Quigley (1956) shown in Table 2.4. 

 

1 http://qecc.pnl.gov/LOGNORM4.htm 
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Table 2.4. “Exposure to Soluble Uranium Compounds,” reproduced from Eisenbud and Quigley 

(1956). 

Exposure, 

mg/m3 1948 1949 1950 1951 1952 

0 - 0.1 9 13 38 33 55 

0.1 - 0.5 13 14 62 55 48 

0.5 - 2.5 44 31 -- 30 22 

>2.5 34 61 32 8 -- 

2.1.2.1 Example Using Two Data Points 

In one of these cases, the year 1950, there are effectively only 2 usable data points for fitting a lognormal 

distribution: 38 observations at or below 0.1, and 62 observations between 0.1 and 0.5. Using the 

LOGNORM4 program gives the results in Table 2.5, as does the use of the following equations: 

ln ,

ln and

2 2

1 1

μ σ

μ σ

x z

x z

= +

= +

 6 

which can be solved for μ and σ : 

. 2

ln ln ( )

and ln ln

2 1 2 1

2 1

2 1

x x z z

z z

x x

+ − + =

−

− =

σ

μ

σ

7

To use Equation (7), the fractiles fi = (i − 0.5)/n must be transformed into standard normal deviates zi, 

which is done automatically by LOGNORM4 and can be done in Microsoft Excel using z = 

NORMSINV(f). 

Table 2.5. Deriving a lognormal distribution from two data points. Concentration, C (mg/m3) Frequency Cumulative Frequency 

fi z ln C σ = 

Δ [ln(

C)] / 

Δz

μ GSD x50 x95 x

0.1 38 38 0.28409 -0.5707 -2.3026 

0.5 62 100 0.75379 0.68646 -0.6932 1.280 -1.572 3.597 0.208 1.705 0.438 

2.5 0 100 0.75379 0.68646 0.9163 2.560 -0.841 12.94 0.431 29.08 11.43

> 2.5 32 132 0.99621 2.67041 

The row labeled 2.5 mg/m3 provides a plausible fit, with plausible values for x50, x95, and x . 
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When only 2 data points are used to determine a lognormal distribution, there is no way to compute the 

uncertainty in the parameters. 

2.1.2.2 Using Minimum, Mean, and Maximum Values with Number of Observations to 

Determine the Parameters of a Lognormal Distribution 

When only minimum, mean, and maximum values are quoted but the number of observations is given, 

there are four possibilities. 

• The minimum, maximum, and number of observations can be used with Equation (7) or 

LOGNORM4 to determine a lognormal distribution. However, in this case, one must examine the 

mean value predicted by the distribution and compare it to the observed mean. Note that this 

method even works with as few as two data points, which become the 25th and 75th percentiles of 

the resultant lognormal distribution. 

• One can perform two additional fits using LOGNORM4 method 9 (Table 2.3), determining the 

lognormal parameters using the mean and a value with its percentile, fractile, or standard normal 

deviate. These two fits result in two more sets of lognormal distribution parameters. The analyst 

should examine these, and if they are reasonably similar, average them. If they are not reasonably 

similar, then the data were probably not lognormally distributed. 

• Ignoring the number of observations, the minimum, maximum, and mean values can be used with 

the method in Section 2.1.2.3. The estimate of the number of data points produced by this method 

should be compared with the given number of observations for reasonableness. 

If there is left-censoring, that is, if the minimum is quoted as “less than” some value, then only the second 

method will work and then only with the mean and maximum value. 

2.1.2.3 Using Range and Mean Value without Number of Observations to Determine the 

Parameters of a Lognormal Distribution 

When only a range and mean value are quoted, as in Christofano and Harris (1960), but the number of 

observations is not given, inference may become more difficult. 

In principle, if the xmin and xmax values are symmetric about the geometric mean x50, then the 3 values 

uniquely determine a lognormal distribution. Under this assumption, fmin = 1 − fmax, so that −zmin = zmax, 

and it is easily shown that 

.

or

2

ln ln

50 min max

min max

x x x

x x

=

+

μ = 8 

From the relationship exp( / 2) 2

x = μ +σ in Table 2.2, we find 

2ln ln ln . min max

2 σ = x − x − x 9

If the right hand side of equation 9 is negative or zero, a lognormal relationship is ruled out. If it is 

positive, then 
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2ln ln ln . min max σ = x − x − x 10 

A quality check can be done by computing 

σ

− μ = min

min

ln x

z 11 

and transforming it back into fmin, which can be done in Microsoft Excel using fmin = NORMSDIST(zmin). 

If there were only 2 data points, fmin = 0.25 and fmax = 0.75. Thus, fmin must be ≤ 0.25 and fmax must be ≥

0.75 for this fit to make any sense. A rough estimate of the number of data points that were used to derive 

the mean is 

. 2

1

min f

n = 12 

In this method, there is no way to compute uncertainty in parameters. 

In Christofano and Harris (1960), 126 lines in tables had ranges and means that contained no zeroes. In 

103 cases (81.7%), the method above gave a plausible lognormal fit with 1.3 ≤ n < 200 and GSD < 15. Of 

the attempts, none failed because fmin was 0 or too small; 18 (13.5%) failed because the mean was too 

close to the bottom of the range (that is, xmax was an outlier inconsistent with xmin and the mean); none 

failed because the GSD was > 15 (not plausible); and none failed because the GSD = 1.000000. In 

addition, there were 10 lines with zeroes at the bottom of the range, which is inconsistent with a 

lognormal distribution in which values must be strictly positive. 

Of the 103 successful lognormal fits, 69 (67.0%) had x95 > xmax, indicating that the assumption of a 

triangular distribution in these cases is definitely not favorable to the claimant. In addition, using xmax as 

the upper limit of a triangular distribution presupposes a value of x95 that is less than xmax, compounding 

the lack of favorability to the claimant of the assumption of the triangular distribution. There is more 

discussion of these data in Section 2.2. 

Methods to deal with the remaining 18.3% are presented below. 

2.1.2.4 Use of Range and Average Value Data that Are Inconsistent with a Lognormal 

Distribution 

If the range (minimum and maximum values) and average value data are inconsistent with a lognormal 

distribution, a possibility is to use a triangular distribution with the minimum and maximum values being 

parameters 1 and 3 in Table 2.1, but this eliminates the possibility that the underlying population 

exhibited values outside of the quoted range. The latter possibility is quite significant if the number of 

measurements was few. 

An assumption that is more favorable to the claimant than the triangular distribution might be to assume 

that the low and high values are the 5th and 95th %iles, i.e., that there were 10 measurements. Again, 

however, this eliminates the possibility that the 95th %ile was actually greater than the largest observation. 

The average value is generally a more robust statistic than minimum or maximum values, which can be 

outliers. The preferred default assumption is to ignore the minimum and maximum values, use the 

average (arithmetic mean) value, and assume a lognormal distribution with 
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• a GSD of 5 for data describing a single process (e.g., a series of air samples), or 

• a GSD of 10 for data describing an entire site, plant, or factory. 

The basis for assuming these GSDs derives from analyzing data from many facilities. The median 

(geometric mean) x50 is computed from the average (arithmetic mean) x using 

exp( / 2), 2x50 = x −σ 13

where σ = ln(GSD). 

Methods to deal with the remaining 18.3% are presented below. 

2.1.2.5 Use of a Single Measurement Value 

A single measurement is taken by metrologists to be the average or expectation value. Use the single 

result as the average (arithmetic mean) value, and assume a lognormal distribution with 

• a GSD of 5 for data describing a single process (e.g., a series of air samples), or 

• a GSD of 10 for data describing an entire site, plant, or factory. 

The median (geometric mean) x50 is computed from the average (arithmetic mean) x using Eq. 13. 

2.1.3 Censored Individual Observations 

The simple method in 2.1.1 is not always available, especially in cases of left-censoring, grouping, and 

right-censoring. 

2.1.3.1 Left-Censored Data 

Sometimes values are reported as “less-than” some number or as zero. This is referred to as leftcensoring. Since one cannot take the logarithm of zero or a less-than value, the method in 2.1.1 cannot be 

used. An alternative method consists of 

• sorting the data in ascending order 

• assigning fractiles fi = (i − 0.5)/n to each data point 

• transforming the fractiles into standard normal deviates, zi

• taking the natural logarithm of the non-zero, non-censored values, ln xi

• using only the logs of non-zero, non-censored observations, perform a uniformly-weighted (i.e., 

unweighted) linear regression of ln xi as a function of zi

The slope of the linear regression is σ , and the intercept of the linear regression is μ . This method is 

described by Strom (1986) and probably many others. 
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2.1.3.2 Finney Weighting Factors 

Because extreme (i.e., high-z) values are broadly spaced at the upper tail, they have an inordinate leverage 

on the regression line. One approach is to use weights that emphasize data near the center of the 

distribution. The weighting factors of Finney (Finney 1971) are 

, (1 )

( ) 2

i i

i

i f f

N f w − = 14 

where N(fi) is the probability density of the normal distribution. These are shown in Figure 1. While 

Finney's weights are based on the binomial distribution for quantal response data, their use here results in 

a fit which emphasizes the center, not the tails, of the distribution, and often gives a line which is closer to 

an “eyeball fit.” The user may choose either the Finney-weighted or the uniformly-weighted fit as desired, 

based on inspection of plots of observed and predicted values of data versus z. 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-4 -3 -2 -1 0 1 2 3 4

Standard Normal Deviate, z

y

y = Finney Weight

y = N(z)

y = N(z) / Finney Weight

Figure 1. Finney weights (y), standard normal distribution, and ratio. 

An example of this kind of fit is given for 194 air monitoring results from one site in which 39 results 

were less-thans. These fits are shown in Figure 2. The Frequency and Finney-Frequency weighted fits are 

explained below. Note that some measurements were actually above 100 mg/m3

; the current Threshold 

Limit Value (TLV) is 0.2 mg/m3

, which is about the median for these fits. 
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0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

-4 -3 -2 -1 0 1 2 3 4

Standard Normal Deviate (z)

Uranium Concentration (mg/m^3)

Data

Finney

Uniform

Finney-Freq.

Frequency

Figure 2. Lognormal fits to 194 air monitoring results from one site, with the first 39 results being 0 

or “less-thans”. 

The various curves have medians around 0.2 mg/m3

, and GSDs of 18 (Finney and Uniform weighting) to 

25 (Finney-Frequency and Frequency weighting). In this case, the latter differ from the former by the 

effect of weighting the lowest point at (39 + 1) = 40× to account for the 39 missing less-than values. 

Arguably, with individual data points above the left-censoring value, Frequency weighting of the lowest 

point does not improve the “eyeball fit.” 

2.1.3.3 Right-censored Data 

Sometimes observations may be reported as “greater than” some value. Such data points must be sorted to 

the end of the list and assigned fractiles based on their position on the list. They are not used in the linear 

regression. 

2.1.4 Grouped, Censored Observations 

Sometimes data are reported in groups, e.g., airborne uranium concentration measurements from 

Eisenbud (1956) shown in Table 2.4. In this case, there is grouping as well as left- and right-censoring. 

While it is true that “two points determine a straight line,” at least 3 data points are required by some data 
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fitting routines that calculate statistics other than the slope and the intercept. This method is described by 

Strom (1986) and probably many others. 

2.1.4.1 Frequency Weighting for Grouped Data 

Grouped, censored observations will require additional weighting considerations. The first data point in 

1949 represents 13 of the 119 total observations; the second, 14; the third, 31, and the final point, 64. The 

fit will generally be improved if each point is weighed by the number of observations it represents. In 

addition, a weighting factor that accounts for both frequency and weights data near the median can be 

created by multiplying the frequency by the Finney weight described in Section 2.1.3.2. 

An example of fitting a lognormal distribution to grouped, left- and right-censored data is shown in 

Figure 3, the 1949 data from Table 2.4. With more than half of the observations greater than the median, 

the upper 95th %ile is very large. 

0.001

0.01

0.1

1

10

100

1000

10000

-3 -2 -1 0 1 2 3

Standard Normal Deviate (z)

Uranium Concentration (mg/m^3)

Data

Finney

Uniform

Finney-Freq.

Frequency

Figure 3. Examples of fitting a lognormal distribution to grouped, left- and right-censored data 

(from Table 7, 1949, Eisenbud and Quigley 1956) 
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Table 2.6. Regression values and statistics for the 1949 example. x , x50, Std. Dev., and Upper 95th

%ile are all in mg/m3

. 

n x x50 GSD μ σ

Std. 

Dev. CV r

2

x95

Finney Reg 3 81.4 2.96 13.1 1.09 2.57 2236 27.5 0.988 204 

Uniform Reg 3 95.2 3.03 13.8 1.11 2.63 2990 31.4 0.988 228 

Finney-Freq 3 71.0 2.86 12.6 1.05 2.53 1759 24.8 0.991 185 

Frequency 3 80.3 2.90 13.2 1.06 2.58 2227 27.7 0.991 201 

In the example in Figure 3, there is little difference between the fits, since all fits are excellent, as shown 

by the r

2

 values in Table 2.6. However, the huge GSD results in upper 95th %ile values that are very large 

but within observed ranges elsewhere. Results for 1948 had GSDs in the range of 4.9-5.8 and upper 95th

%iles of 19-25 mg/m3

, while results for 1951 had GSDs in the range of 4.35-4.47 and upper 95th %iles of 

2.80-2.99. While the fits had r

2

 statistics on the order of 0.99, the data analyst needs to critically examine 

the data. The arithmetic mean, median (geometric mean) derived from frequency-weighted regressions for 

five years of data are shown in Figure 4. Whether these numbers can be deemed representative of 

workplace exposures is not implicit in the fits. 

0.1

1

10

100

1000

1947 1948 1949 1950 1951 1952 1953

Year

Airborne Uranium Concentration (mg/m3

)

Mean

Median

Upper 95th %ile

Figure 4. Mean, median, and upper 95th %ile for the airborne uranium data from Table 7 of 

Eisenbud and Quigley (1956) 

Another example of left-censored, grouped data involves annual thermoluminescent dosimeter (TLD) 

results for deep dose equivalent2

, shown in Figure 5. The differently-weighted lines have different slopes, 

although in this case, very similar 95th %iles. Here the lowest recorded dose was 100 μSv (10 millirems). 

There is clearly a down-turn in the data near the high end of the distribution. This effect has been 

documented in many occupational dose distributions as being caused by more stringent radiation controls 

 

2

 Data in 1990 from a U.S. university which remains anonymous. 
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as doses approach a limit; see, e.g., Kumazawa (1981; 1991). Fitting a lognormal distribution using 

frequency or Finney-weighting will be favorable to the claimant in this instance by overestimating the 

95th %ile. 

0.001

0.01

0.1

1

10

100

1000

10000

100000

-4 -3 -2 -1 0 1 2 3 4

Standard Normal Deviate (z)

Annual Deep Dose Equivalent (μSv)

Data

Finney

Uniform

Finney-Freq.

Frequency

Figure 5. Lognormal fits to grouped annual deep dose equivalent measurements for 458 persons. 

Note the significantly different slopes for different weighting. 

2.1.5 “Reasonableness” of a Lognormal Distribution 

Generally, lognormal distributions with GSDs above 10 are not plausibly drawn from a single population. 

In fact, any GSD > 5 should probably be investigated if possible. Additional considerations are given in 
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Section 3.7 and 3.8. A lognormal may still be a better choice than other distributions accommodated by 

IREP. 

2.1.6 Summary of Default Assumptions for Fitting Lognormal Distributions 

In summary, different methods of determining the geometric mean, geometric standard deviation, and 

upper 95th %ile are needed for 4 different kinds of data. 

If there is no censoring, estimate μ and σ from the average and standard deviation of the natural 

logarithms of the data, and compute any needed statistics from there as shown in Section 2.1.1. It still 

may be wise to perform weighted linear regressions and examine plots of the data with the predictions. 

If only summary statistics are available, such as the mean and standard deviation of observations, use the 

methods in Strom and Stansbury (2000), which are implemented in freeware LOGNORM4, or the 

equations in Section 2.1.2. It is sometimes possible to generate a lognormal from a range and a mean 

value, but the analyst must still look at the reasonableness of the resulting distribution. Fitting a 

lognormal to a minimum-mean-maximum data triad will almost always produce a higher 95%ile value 

than fitting a triangular distribution. 

If individual observations are available, but there is left-censoring, right censoring, or both, perform 

weighted linear regressions of the uncensored observations xi against their known standard normal 

deviates zi, and examine various weighting schemes, e.g., uniform-, Finney-, and frequency-weighting, as 

described in Section 2.1.3. 

If only grouped data are available, which are inherently left-censored and may or may not be rightcensored, perform a frequency-weighted linear regression of the uncensored group upper limits xi against 

the known standard normal deviates zi, using the methods described in Section 2.1.4. The analyst is 

encouraged to examine various other weighting schemes, e.g., uniform-, Finney-, and frequencyweighting. 

In each of the four cases, the data analyst should qualitatively determine whether the fitted lognormal 

distribution makes reasonable predictions. 

2.2 Triangular Distributions 

There is little information available on the use of triangular distributions from NIOSH/OCAS, although 

they are available in IREP. Like the uniform distribution, the triangular distribution has absolute limits 

below or above which the distribution is zero (Brighton Webs Ltd. 2006; Weisstein 2006; Wikipedia 

2006). Such is not the case for the normal and lognormal distributions, although they have effective upper 

and lower bounds. 

IREP accommodates any triangular distribution using minimum xmin, mode xmode, and maximum values 

xmax as input parameters. An example of a triangular distribution is shown in Figure 6. 

Most features of the triangular distribution have different functional forms for different ranges of x. The 

pdf for a triangular distribution is 
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( )( )

2( ) : pdf

( )( )

2( ) : pdf

max min max mode

max

mode max

max min mode min

min

min mode

x x x x

x x

x x x

x x x x

x x

x x x

− −

− ≤ ≤ =

− −

− ≤ ≤ =

 15 

x

pdf or cdf

x50

x

xmode

xmin xmax

1

0

0

cdf

pdf

Figure 6. The probability density function (pdf), cumulative density function (cdf), and parameters 

of a triangular distribution 

The cdf for a triangular distribution is 

. ( )( )

( ) : cdf 1

; ( )( )

( ) : cdf

max min max mode

2

max

mode max

max min mode min

2

min

min mode

x x x x

x x

x x x

x x x x

x x

x x x

− −

− ≤ ≤ = −

− −

− ≤ ≤ =

 16 

 The median for a triangular distribution also has two functional forms: 

. 2

( )( ) , 2

if

; 2

( )( ) , 2

If

max min max mode

50 min

max min

mode

max min mode min

50 max

max min

mode

x x x x

x x

x x

x

x x x x

x x

x x

x

− − = + − ≥

− − = − − ≤

 17 
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The derivation of the three general parameters of the triangular distribution from data values such as mean 

and range is straightforward. Since the mean of a triangular distribution is 

, 3

min max mode x x x

x

+ + = 18 

the mode can be obtained by 

3 . mode min max x = x − x − x 19 

The inequality 

min mode max x ≤ x ≤ x 20 

must hold or a triangular distribution is inconsistent with the data. 

Analyzing the data in Christofano and Harris (1960) assuming that xmin = 0 when less-than values were 

given, this method produces plausible triangular distributions, for 68 of 136 cases (50.0%), as shown in 

Table 2.7. For the remaining 68 cases, one or the other part of inequality 20 was violated. For 66 cases, 

xmode < xmin, and for 2 cases, xmode > xmax. Of the 68 successes, 3 triangular distributions succeeded in cases 

where the mean was too small to be consistent with a lognormal. Of the 10 data triads (minimum, mean, 

and maximum) which were left-censored, triangular fits succeeded in only 3 cases. Of 136 data sets (10 of 

which were left-censored), 109 could be fit by a lognormal, a triangular, or both; 27 could be fit by 

neither a triangular nor a lognormal. Lognormal fits could be found for 41 data triads that couldn’t be fit 

by triangular distributions, while triangular fits could be found for only 6 data triads that couldn’t be fit by 

lognormal distributions, including 3 that were left-censored. 

Table 2.7. Comparison of attempts to fit lognormal and triangular distributions to mean and range 

data from Christofano (1960) 

 Triangular Fit

Lognormal Fit 

Failed: 

Mode 

Too 

High

Failed: 

Mode 

Too 

Low

Plausible 

Triangular

Grand 

Total

Total w/o 

Censored Percent

Failed: Improbable n 5 5 5 4.0%

Failed: Left Censored 1 6 3 10 

Failed: Mean Too Close to xmin 15 3 18 18 14.3%

Plausible Lognormal 1 40 62 103 103 81.7%

Grand Total 2 66 68 136 126 100.0%

Percent 1.5% 48.5% 50.0% 100% 

All things considered, the lognormal distributions generally are more successful in describing this kind of 

data than are triangular distributions. 

Triangular distributions will be used where dictated by OCAS practices, such as in the inference of 

missed dose. 
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2.3 Normal Distributions 

Because of Ott’s argument, stated in Section 2.1, it is unlikely that most exposure parameters have 

normally-distributed uncertainties. Exceptions are bioassay and dose measurements associated with 

individuals. In these cases, often information is available concerning the mean value, and sometimes the 

standard deviation. The uncertainty of standard measurements such as film dosimetry can often be 

inferred from the literature. 

2.3.1 Normally-distributed Measurement Uncertainty and an Underlying Lognormallydistributed Measurand: Mirror Image Method 

In cases where the standard deviation is unknown, and negative values have been observed and recorded, 

it is possible to analyze the negative tails using the “mirror image” procedure suggested by Strom (1984; 

1984). In that method, the negative tail of the observed distribution is reflected about the ordinate, and the 

standard deviation of the resulting distribution is used to estimate the measurement uncertainty. A 

preliminary version of the method was used in 1983 to deduce measurements uncertainty in uranium 

urinalysis results at the Y-12 plant in Oak Ridge, Tennessee (Strom 1984). An example for 1971 results is 

shown in Figure 7. 
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Figure 7. Analysis of Y-12 uranium in urine measurements (Strom 1984, Figure 16). a. the product 

of a lognormal and a normal distribution producing a negative tail. b. histogram of 

uranium excretion values from Y-12 in 1971. 

2.3.2 Normally-distributed Measurement Uncertainty and an Underlying Lognormallydistributed Measurand: Preserved Mean and Variance Method 

A more sophisticated alternative to the crude “mirror image” technique includes constraints on the mean 

and variance of the measurand3

 distribution (the underlying "true state of Nature") that preserve the 

variance of the observed data in the predicted function. The method recommended here is based on 

several assumptions: 

 

3

 “Measurand” is the ISO term (International Organization for Standardization (ISO) 1995; Taylor and Kuyatt 1994) 

for the true but unknown value of “the specific quantity subject to measurement”. 
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1. The observed probability density function (pdf) is the result of combining a normally-distributed 

measurement uncertainty with a lognormally-distributed measurand. 

2. The mean of the lognormal “true state of nature” is equal to the mean of the observations. 

3. Measurements are unbiased, so the mean of the measurement uncertainty distribution is zero. 

4. The variance of the observed values is equal to the sum of the variance of the measurement 

uncertainty plus the variance of the lognormal “true state of nature.” 

Since the uncertainty distribution is characterized by a mean and standard deviation (or variance), and the 

lognormal “true state of nature” is characterized by a median and a geometric standard deviation (GSD), 

there are only 4 adjustable parameters. The assumptions listed above constrain the problem so that only 

one parameter can be freely adjusted, either the standard deviation of the uncertainty distribution or the 

GSD of the lognormal. For this analysis, it is recommended that the GSD of the lognormal be chosen as 

the varying parameter. 

The best fit for this purpose cannot be arrived at by a single “goodness of fit” statistical test, such as 

Kolmogov-Smirnoff, Cramer-Von Mises, or a runs test. Examining of the residuals for the fits reveals 

systematic but not large differences in the observations from the assumptions above. 

2.3.2.1 Test of the Preserved Mean and Variance Method 

A description of 137Cs baseline in vivo monitoring data measured at the Pacific Northwest National 

Laboratory’s (PNNL’s) In Vivo Radioassay Research Facility for 409 persons with no known 

occupational exposure or workplace indicators of exposure. Measurements were made in 2000, 2001, and 

2002. Measurements were reported as positive, zero, or negative values expressed in nanocuries (nCi). 

An observed probability density function (pdf) for these data is shown in Figure 8, with each 

result sorted into bins of 0.0025-nCi width. The very rough curve is the raw pdf, while the 

smooth curve is the smoothed PDF using the model-free visualization method of 

(Chomentowski, Kellerer, and Pierce 2000). The smoothing is similar to spectrum smoothing, 

and it preserves the larger trends in the dataset while reducing the noise. These and all curves 

below are normalized to an area of 1. 

The observed cumulative density function (CDF) is the integral of the observed PDF. This is shown in 

Figure 9. 
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0

1

2

3

4

5

6

7

-1 -0.5 0 0.5 1 1.5 2

137Cs (nCi)

Histogram or Smoothed pdf (nCi-1)

pdf

Smoothed pdf

Figure 8. Normalized probability density functions (pdfs) for Hanford in vivo 137Cs measurements 

on unexposed workers 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

137Cs (nCi)

cdf

Predicted cdf

Observed cdf

Figure 9. Cumulative probability density functions (cdfs), observed (rough red line) and predicted 

(smooth line) using GSD = 1.4 
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The best fits were lognormals with mean = 0.0936 nCi (the mean of the data), GSDs of 1.2, 1.4, 

or 1.6 (medians of 0.0921, 0.0885, and 0.0838 nCi, respectively), and uncertainty distributions 

with standard deviations of 0.180 to 0.185 nCi. The results for GSD = 1.4 are shown in Figure 10

and Figure 11. These three were essentially indistinguishable, given the observed 137Cs values. 

The model developed here assumes that observations arise from a lognormal “true state of 

nature” (dashed curve in Figure 10) filtered through the lens of a normal measurement 

uncertainty. This model provides an adequate fit to the data, although the choice of GSD (and 

related parameters) is not obvious but lies in the range of 1.2 to 1.6. 

Figure 11 shows the residuals for the predicted cdf shown in Figure 9. Examining of the residuals 

for the fits reveals systematic but not large differences in the observed 137Cs population from the 

assumptions above. The mean uncertainty of the observations was 0.191 nCi, slightly larger than 

any uncertainty consistent with the assumptions above. Clearly there are clusters of points here 

and there in the observed distribution, but overall, the predicted cdf in Figure 9 is close to the 

observed cdf. 
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0

2

4

6

8

10

12

14

16

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

137Cs (nCi)

Histogram or pdf (nCi-1)

pdf

Uncertainty

Lognormal

 Predicted pdf

Smoothed pdf

Figure 10. PDFs with uncertainty model (smooth thin line), lognormal "true state of nature" model 

(dashed line), and predicted value (heavy black line) 

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

-1 -0.5 0 0.5 1 1.5 2

137Cs (nCi)

Predicted cdf - Observed cdf

Figure 11. Residuals for predicted cdf. 
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If a non-occupationally exposed population has retained quantities of 137Cs that are lognormally 

distributed with means of 0.0936 and GSDs of 1.2, 1.4, or 1.6 as shown in the table, then the specified 

percent of them will have retained quantities equal to or exceeding the Table 2.8 values in nCi.

Table 2.8. 90, 95, and 99 %iles of lognormal "true state of nature" distributions with GSDs of 1.2, 

1.4, and 1.6. 

%ile 

% 

exceeding 1.2 1.4 1.6 

90 10 0.116 0.136 0.153

95 5 0.124 0.154 0.182

99 1 0.141 0.194 0.250

It is clear that the 95 %ile values are on the order of the standard deviation of the measurements, 

typically 0.19 nCi. 

In measuring 409 persons at Hanford with no reason to expect an occupational intake, one 

observes that the results are at or above 0.35 nCi 5% of the time. The fact that this value is nearly 

double the largest value in Table 2.8, and about twice the average measurement uncertainty of 

0.19 nCi is due to the fact that the measurement uncertainty is just over twice as large as the 

average measured value. 

In the absence of workplace indicators, one concludes that a 137Cs measurement result of 0.35 

nCi has a more than 5% chance of being due to environmental exposure and measurement 

uncertainty. 

The method illustrated here can be used to determine the likelihood of an individual exceeding a 

particular exposure or dose level when uncensored data are available.

2.4 Rectangular Distributions 

IREP can accept a rectangular distribution as input. A rectangular distribution consists of 2 values, and 

upper and a lower bound that are believed to bracket the range of plausible values of concentration, 

intake, dose, or dose rate. Rectangular distributions are non-physical, but can be used to represent a 

limited state of knowledge. 

2.5 Constant “Distributions” 

IREP can accept a constant distribution as input. A constant distribution is a single number, generally 

taken as an overestimate that is favorable to the claimant of a concentration, intake, dose, or dose rate. See 

section 3.3. 
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3.0 Default Assumptions 

3.1 Introduction 

There are science policy and expert judgment issues that must be considered in dose reconstruction. 

This section contains assumptions about 

• external irradiation geometry 

• use of the 95th %ile for air sample distributions 

• uncertainty in biokinetic models 

• aerosol particle size and respirable fraction 

• use of time-period-specific, process-based GSDs for published mean aerosol concentration data 

• use of time-weighted averages, breathing-zone (BZ) air samples, and general area (GA) air 

samples, and considerations of sample duration 

• particle solubility (ICRP 66 transportability classes F, M, S) and f1 (gastrointestinal absorption 

fractions) 

• exposure time 

• ingestion 

• occupational medical doses 

• external dose conversion factors 

• external missed dose when there was monitoring 

• internal missed dose when there was monitoring 

• environmental dose 

• radon and thoron and their short-lived decay products 

3.2 External Irradiation Geometry 

Default assumptions of irradiation geometry may be reasonably justified, as described in Table 4.2 on 

page 53 of NIOSH OCAS-IG-001 (2002). The “job” entries in that table for uranium facilities have been 

generalized for all of the job categories found among claimants in the cases covered by this TIB. These 

are detailed in a 326-line spreadsheet entitled “Irradiation_Geometry_by_Job_Title.xls”. To make these 

choices, and experienced IH assigned job titles to 

• the “general (roamer)” category denoted in IG-001 as “General Laborer” 

• the “operator (doer)” category denoted in IG-001 as “Machinist” 

• the “white collar (supervisor)” category denoted in IG-001 as “Supervisor” or 

• the “not assigned” category (44 job titles) that have to be elaborated from additional information 

or, failing that, will be assigned the maximum DCF values. 
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3.3 The 95%ile and “Constant” Uncertainty Distribution for Limited Data 

Sets 

Sometimes one must infer exposures from a small set of measurements such as a few air samples taken at 

various locations at a facility. The inference is that the distribution these samples represent applies to the 

entire facility. For people who move around, such as crafts and maintenance personnel, the average may 

be appropriate. However, if it is credible that a claimant routinely worked in one of the higher airborne 

areas, this distribution does not fairly represent the individual’s exposure. To account for this, one infers 

that the individual was exposed to the 95th percentile of the distribution. 

Another example is a distribution of a few film badges. If these are the result of only a few representative 

monitored workers, it is credible that any particular claimant could really represent the upper end of that 

distribution. 

According to a communication from NIOSH, use of the 95th %ile assumption has expanded recently and 

is used in more and more situations. In any case, the 95th percentile is considered to represent the upper 

bound of the exposure. Any exposures calculated from this are entered into IREP as a constant. 

3.4 Uncertainty in Biokinetic Models 

The National Council on Radiation Protection and Measurements (NCRP) used an expert group of 

internal dosimetrists to create a subjective quantification of the reliability of ICRP Publication 30 

biokinetic and dosimetric models (National Council on Radiation Protection and Measurements (NCRP) 

1998). While IMBA uses the newer ICRP Publication 66 respiratory tract model and newer biokinetic 

models, the results of these models may not be that much better than the ICRP 30 models for some 

radionuclides in cases where f1 is the dominant uncertainty. NCRP developed “Reliability Categories” A 

through D, as described in Table 3.1. The two right hand columns were developed in this work for 

lognormal distributions consistent with the NCRP criteria. 

Table 3.1. Reliability categories for selected results of ICRP Publication 30 biokinetic and 

dosimetric models 

Reliability categories 

For at least 90% of a Group, does the 

effective dose coefficient E(i) lie 95 50 x / x

GSD if 

lognormal 

"well known" A Between E(i)/3 and E(i)*3 3 ≤1.95 

"reasonably well 

known" 

B outside the previous range, but between 

E(i)/5 and E(i)*5 

5 1.95≤ GSD 

≤ 2.66 

"poorly known" C outside the previous range, but between 

E(i)/10 and E(i)*10 

10 2.66 ≤ GSD 

≤ 4.05 

"very poorly known" D outside the previous range >10 GSD > 4.05 

 

The NCRP experts ranked each of 26 radionuclides for a) healthy adult males, and b) special populations 

of infants, diseased people, (and presumably women, who are not “healthy adult males”). Some 

radionuclides were also ranked separately for ingestion intakes and inhalation intakes. Rankings for 

thorium and uranium are given in Table 3.2. 
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Table 3.2. Estimated reliability, for selected radionuclides, of the effective dose coefficient values 

recommended in ICRP Publication 30 (from NCRP 1998 Table 8.2) 

 Reliability 

Category 

Radionuclide 

Mode of 

Intake 

Adult 

Male 

Special 

Group Comments 

230Th Ingestion C D 

(infants) 

Values of f1 are low and uncertain and bone dosimetry 

230Th Inhalation B C Values of f1 and bone dosimetry 

234U Ingestion C D Uncertainty in the value of f1, mass dependency for f1, 

biokinetics not studied in children 

234U Inhalation B C Absorption from the lung and biokinetics poorly known 

Boecker and colleagues (Boecker et al. 1991) give a GSD for interpreting 239,240Pu urine bioassay data as 

3.4 (p. 99). 

The NCRP and Boecker et al. analyses do not explicitly address how many bioassay samples are taken, 

and what the uncertainties in the measurement results themselves are. 

Following NIOSH OCAS precedent (Bihl, Brackett, and Toohey 2006), a lognormal distribution with a 

GSD of 3 will be used. This is reasonably consistent with the above findings. 

3.5 Aerosol Particle Size and Respirable Fraction 

Some very high concentrations, e.g., > 10 mg/m3, are not credible as being entirely “respirable” or 

adequately described as having a 5 μm activity median aerodynamic diameter (AMAD). A consensus for 

various air sampling scenarios will be developed using expert IHs and HPs. 

By the early 1950s, the staff at the AEC Health and Safety Laboratory (HASL) were aware that their air 

samplers were sampling more airborne radioactive material than would be deposited “in the non-ciliated 

portion of the lung” (Lippmann and Harris 1962), what we would now term the alveolar-interstitial region 

(International Commission on Radiological Protection (ICRP) 1994). The Human Respiratory Tract 

Model (HRTM) “supposes that the ‘total’ ambient aerosol is sampled.” (International Commission on 

Radiological Protection (ICRP) 2003). The ICRP provides extensive discussions of the HRTM in Annex 

B of Supporting Guidance 3 (ICRP 2003), and explains how to correct for particle size selection aerosol 

samplers such as those recommended by the American Conference of Governmental Industrial 

Hygienists, the Comité Européen de Normalisation, and the International Organization for 

Standardization. “These organizations specify that three aerosol fractions must be measured to assess 

exposure of workers to aerosols in an industrial environment: 

• the inhalable fraction which is the mass fraction of total airborne particles which is inhaled 

through the nose or the mouth, 

• the thoracic fraction which is the mass fraction of inhaled particles which penetrate beyond the 

larynx, and 

• the respirable fraction which is the mass fraction of inhaled particles which penetrates to nonciliated airways (alveoli).” (ICRP 2003, p. 148). 
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The HRTM is only correct if the true AMAD of the aerosol is used and if the entire aerosol is sampled. If 

an aerosol has a true AMAD of 30, 20 or 10 μm but is modeled as having an AMAD of 5 μm, the 

resultant dose estimates for uranium will be high by factors of x, y, and z, respectively. 

Default assumptions of ICRP Pub. 66, i.e., 5 μm AMAD, will be used in the absence of other information. 

3.6 Use of Time-period-specific, Process-based GSDs for Published Mean 

Aerosol Concentration Data 

Pending review and additional justification, the current default assumption when no information is 

available on uncertainty in aerosol measurements is that they are lognormally-distributed with a GSD of 5 

for a single process or activity, and 10 for an entire site, plant, or factory. These choices are based on an 

analysis of data from Christofano and Harris (1960). 

In that paper, as mentioned in Section 2.1.2.4, there were 108 instances when a lognormal distribution 

could be fit to tabulated data. The GSDs of those lognormal distributions ranged from 1.13 to 12.8, with 

an average of 2.89 + 1.58 (at 1 standard deviation), a geometric mean of 2.59 ×÷ 1.56, and an upper 95th

percentile of ~5.2. Based on this analysis, for single processes, it is reasonably favorable to the claimant 

to choose a GSD of 5 as a default when there is no other way to estimate the GSD. 

Looking at all 136 process-specific mean airborne U concentrations reported in Christofano and Harris 

(1960), one arrives as the analysis shown in Figure 12. These values are roughly lognormally distributed, 

with GSDs ranging from 9.0 to 10.4, depending on the weighting method chosen. Based on this analysis, 

for site-wide data, it is reasonably favorable to the claimant to choose a GSD of 10 as a default when there 

is no other way to estimate the GSD. 
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0.1

1

10

100

1000

10000

100000

1000000

-3 -2 -1 0 1 2 3

Standard Normal Deviate, z

Airborne U Concentration (dpm/m3

)

Sorted Data

ln(Data) Pred. GM= 523. GSD= 9.16

Uniform Pred. GM= 523. GSD= 9.01

Finney Pred. GM= 475. GSD= 10.4

Figure 12. Lognormal plot of mean airborne U concentrations for 136 different processes in uranium 

refining (Christofano and Harris 1960). 

Currently, when a single value is available, it is assumed to be the mean ( x ; also known as the 

expectation value, the arithmetic mean, and the average) of the resultant lognormal distribution. The 

median (x50; also known as the geometric mean), which is always less than the mean for lognormallydistributed data, is calculated from the mean and the GSD using Eq. 31. The median and the GSD are the 

parameters required by IREP for any lognormally-distributed quantity. 

3.7 Use of Distributions to Describe Multiple Populations 

Statistical distributions such as the lognormal are used to describe observations of single populations, e.g., 

the distribution of body mass of adult people. One does not generally use distributions to model combined 

populations, e.g., body masses of infants and body masses of adults, or body masses of microbes, insects 

and mammals. However, it may be necessary to combine all air samples for a single plant to describe the 

exposures of workers who are “roamers” (Section 3.2) or for whom no description of job duties or 

locations is available. 

It is possible to combine breathing zone (BZ) air sample results for different processes and describe the 

combined data with a statistical distribution, but it is difficult to imagine what such a distribution signifies 
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unless one has a specific end in mind. One could, for example combine BZ results for “furnace operators” 

and “fork lift operators” and use the result to describe exposures to “operators” of an unspecified type. 

It is possible to combine air samples that were taken for a fraction of a minute during the “dirtiest” part of 

batch processes, e.g., dumping ore from a drum into a process bin or chipping out a crucible, with air 

samples that were taken over a period of hours during continuous processes. Again, it is difficult to know 

what such a distribution of combined results would represent. The short-duration air samples were 

probably taken as worst-case values, while the longer-term samples may have represented average values 

to which workers may have had prolonged exposures. 

One data set, shown in Figure 2, is not, in the judgment of a panel of health physicists and industrial 

hygienists, taken from the same population. They may be a combination of incommensurate values. 

It is the policy of the Battelle Dose Reconstruction Team to minimize the combining of populations into a 

single distribution. When possible, job- or task-specific data are to be used in constructing time-weighted 

averages. 

3.8 Use of Time-Weighted Averages, Breathing Zone (BZ) Air Samples, and 

General Area (GA) Air Samples, Process (P) Air Samples, and 

Considerations of Sample Duration 

In light of the problems described above, the preferred (although not always possible) approach is to use 

time-weighted averages (TWAs) of airborne concentrations to assess worker exposures, and assess 

uncertainty of the TWA. 

Table 3.3 shows a time-weighted average “Rn. Exposure” to tower workers at the Lake Ontario 

Ordinance Works (LOOW) where high-226Ra content K-65 tailings were stored (NIOSH SQRI Ref ID 

8921, pp. 13-14 of .pdf, R.C. Heatherton to W.B. Harris, 18 May 1951). Study of this and related 

documents shows that these measurements represent “equilibrium-equivalent radon concentrations” as 

defined by ICRP publications 32 and 65. Thus, 10−10 Ci/L represents 100 pCi/L or 1 working level (WL) 

if an equilibrium factor of 1.00 is assumed. 

This TWA covers a shift in which the worker was exposed for 300 minutes (5 hours). The author of this 

report states, “tower operations might show life time variations on any day from the estimated average 

times assigned to teach operation or operating area in Table I. The error inherent in this fact would 

probably far outweight (sic) any sampling or measurement error.” 
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Table 3.3. Daily Weighted Rn. Exposure to Tower Workers (Table I from NIOSH Ref ID 8921 p.13, 

May 1951, Lake Ontario Ordinance Works) 

Concentration (C) 

10-10 C/L 

Operation or Operating Area 

Time 

Per 

Shift 

(min) 

(T) 

No. of 

Samples Low High Aver. 

Con'c. 

Times 

Total Time 

(T × C)

1. B.Z. Removing covers from Drums 24 6 1.1 2370 851 20424.0

2. G.A. Unloading Platform 60 5 <1.0 9.5 4.9 294.0

3. GA in Thawhouse 10 4 10.6 68.5 34.6 346.0

4. GA Elevator Shaft and in Dumper House 20 4 24.0 31.7 29.0 580.0

5. GA Boiler House 156 1 -- -- 0.6 93.6

6. GA Outside 30 2 0.56 1.1 0.8 24.0

300 ∑(T×C) 21761.6

The overall uncertainty of this TWA can be evaluated by 1) fitting lognormal distributions to these data 

summaries and doing a Monte Carlo simulation using Crystal Ball. Alternatively, since complete, 

uncensored original data are available, try 2) doing a Monte Carlo simulation directly from original data; 

or 3) fit lognormals to each of the complete data sets (assume GSD = 3 for the set with 1 measurement) 

by taking the average and standard deviation of the natural logarithms and do a Monte Carlo simulation 

using Crystal Ball. 

Option 2) has 960 distinct outcomes (6 × 5 × 4 × 4 × 1 × 2), and does not allow for the possibility that 

additional measurements may have been greater than the largest of the observations. 

The 6 individual results for “removing covers from drums” in Table 3.3 are clearly not from the same 

population: 3 were in the range of 1.1 to 17 and 3 were in the range 450 to 2370. Separating the two data 

triplets, plausible GSDs were found for each and for the other data sets by simply finding the average and 

standard deviations of the natural logs of each result as described in Section 2.1.1. Allocating 12 minutes 

exposure time to each of the two lognormal distributions derived for “removing covers from drums,” and 

using the Shift (min) values for the other distributions, a mean TWA was computed from 10,000 Monte 

Carlo trials using Crystal Ball (Figure 13). 

C/L 72.5 Times the Maximum AllowableConcentration.

300

21761 10 10

= × = × −

∑

∑

T

T C
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Time-Weighted Average Air Concentration

0

100

200

300

400

500

600

700

800

900

6.9 48.5 90.1 131.7 173.3

Frequency

Figure 13. Time-Weighted Average (TWA) Radon Concentration in Air, WL (if 100% equilibrium) 

The distribution in Figure 13 is derived from sums of lognormals, and is not expected to be lognormal. 

Indeed, as shown in Figure 14, it deviates systematically from lognormality, and the lognormal 

distribution slightly underestimates the high-end values, as shown in Figure 14 and Table 3.4. Note that 

the mean of these simulations is less than the mean of 72.5 derived by the original analyst because the 

average for the “drum cover removal” was biased high by averaging two separate distributions. Finally, if 

all 22 observations were fit to a lognormal distribution, and that distribution was claimed to be 

representative of exposure to tower workers, one has the results in the right hand column of Table 3.4. 

The mean is 3 times higher, the GSD of 9.8 is barely plausible, the 95th %ile is nearly 4 times higher, and 

the 99th %ile is over 8 times higher. This proves the point that pooling measurements from different 

populations doesn’t produce technically defensible results, but rather dramatically overestimates workers’ 

exposures. 

Table 3.4. Summary statistics from TWA analysis. 

parameter Simulation Fit to 10,000 simulation 

results 

Merging all 22 observations 

into one lognormal 

mean 55.6 same 168 

standard deviation 56.2 same 514 

median, x50 39.1 40.7 12.9 

GSD 2.31 2.14 9.8 

95th %ile 152 142 546 

99th %ile 288 239 2309 

If the exposure times were allowed to vary, perhaps by choosing a random number from a rectangular 

distribution between 0.5 and 1.5 for each time, and normalizing the sum to 1, more realistic variability 

would be introduced and a more realistic uncertainty, that is, a larger GSD, would be generated. 
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LOOW May 8-9, 1951

Radon TWA for Tower Workers, 10-10 Ci/L

1

10

100

1000

10000

-4 -3 -2 -1 0 1 2 3 4

Standard Normal Deviate (z)

TWA Radon Concentration (100pCi/L)

Time-Weighted Average Air

Concentration

Fit

Figure 14. Distribution of results from Monte Carlo simulation of TWA radon data. 

3.9 Particle Solubility (ICRP 66 Transportability Classes F, M, S) and f1

(Gastrointestinal Absorption Fractions) 

The ICRP respiratory tract model (Publication 66) gives three default values for transportability types, S 

(slow), M (moderate), and F (fast). In reality, no material is ever going to be described perfectly by any 

one of these defaults. Most materials fall somewhere between two defaults. The NIOSH approach has 

been to use all credible defaults. For example, intakes from handling uranium metal usually means 

oxidized uranium. These oxides generally fall between type M and type S transportability types. Use 

whichever generates the higher probability of causation (PC) for any particular case. It is often the case 

that type S most favors the claimant for respiratory tract tissues and type M for systemic organs. If it is 

determined from urine samples, type S should be the most favorable to the claimant for all the organs. 

If type F is credible at a particular site, then that is also considered. 
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3.10 Exposure Time and Intake Calculations 

In the absence of specific information, assume exposure times as shown in Table 3.5. 

Table 3.5. Default exposure time assumptions as a function of date 

Dates 

Default Work 

Week 

Weeks per 

Year 

Hours per 

Year 

170-hour “Months” per 

yeara

through 12/31/1950 48 hours 50 2400 14.12 

1/1/1951-12/31/1955 44 hours 50 2200 12.94 

1/1/1956 onward 40 hours 50 2000 11.76 a

For computing working level months (WLM) from time-weighted average potential alpha energy 

concentrations of radon or thoron progeny in working levels (WL). 

It is important to note that when IMBA performs a dose calculation based on a chronic intake, the intake 

is assumed to occur 7 days per week, 52.14 (= 365÷7) or 52.29 (= 366÷7) weeks per year. Thus, IMBA’s 

intake rate in Bq/d should be calculated by dividing the worker’s annual intake rate by 365 (dropping the 

0.25 day is favorable to the claimant). 

The worker’s annual intake is computed by using 

(1.2m /h)(actual or default hours per year)(concentration). 3 I annual = 21

The worker’s chronic daily intake rate for input to IMBA is 

. 365 days/year

(pCi/year) (pCi/day) annual

chronic daily

I

I = 22

3.11 Ingestion 

In the 1940s and early 1950s, uranium and its compounds seem to have been treated like any other metal 

and ore, unlike the later decades of the 20th century when it was treated as a radiological hazard. In 

particular, contamination of the workplace and of workers’ clothing and hands was commonplace and 

evidently acceptable (Bailey and Rohr 1953b; Bailey and Rohr 1953a; Bailey, Becher, and Henry 1956). 

Some ingestion intakes are presumed to have occurred, especially among smokers, based on Bailey’s 

measurements of hand contamination and its transfer to cigarettes (Bailey and Rohr 1953b) and 

measurements of clothing contamination and its resuspension (Bailey, Becher, and Henry 1956). Some 

studies have attempted to quantify hand contamination in the workplace (Brouwer, Kroese, and Van 

Hemmen 1999; Yu and Sherwood 1996). All that is needed for ingestion of contamination is the presence 

of eating, drinking, smoking, chewing of gum or tobacco, storage or preparation of food, or application of 

cosmetics without hand washing. 

In the interim, ingestion intakes are determined following the OCAS method (Neton 2004). 

3.12 Occupational Medical Doses 

The default assumptions in OTIB-0006 (Kathren and Shockley 2005) will be used if no other information 

is available. Photofluorography will be assumed for chest x-rays at du Pont sites for the same years as at 

Hanford unless information to the contrary is discovered (Gehrmann 1944). 
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3.13 External Dose Conversion Factors 

Correction of radiation survey instrument readings, dosimeter readings, and conversion of recorded 

neutron doses to correct neutron doses using a wR/Q ratio will be determined using existing OTIBs and 

IG-001 (Office of Compensation Analysis and Support (OCAS) 2002a). 

3.14 External Missed Dose When There Was Monitoring 

“Missed” doses are assigned for zero, “minimal,” or null dosimeter results readings to account doses 

below the “limit of detection” (LOD) being received by a monitored worker but not recorded or reported 

(Office of Compensation Analysis and Support (OCAS) 2002a). Missed dose can also be assigned for a 

lost or damaged dosimeter. 

One approach is to use co-worker data following the methods of Merwin (2005). In this case, the worker 

is assigned the same dose as a worker with a similar job title for the period in question, assuming that the 

co-worker’s dose is a mean value. The imputed dose is taken as a lognormal distribution with a median 

(IREP Parameter 1) = (Appropriate Coworker’s Dose) × 0.5469 (= exp(−ln2

(3)/2)) and a GSD (IREP 

Parameter 2) = 3. Correcting the Appropriate Coworker’s Dose by the median-to-mean ratio preserves the 

arithmetic mean (average) value. If the coworker’s dose is the median of a lognormal distribution, then it 

is used directly as IREP Parameter 1. 

Another approach is to substitute a value for each dosimeter reading. The approach here is to assign a 

triangular distribution with minimum = 0, mode = 0.5×LOD, and maximum = LOD. 

3.15 Internal Missed Dose When There Was Monitoring 

[Reserved] 

3.16 Environmental Dose 

At uranium plants, environmental dose from five sources is to be evaluated: 

• Submersion in a cloud of a radioactive gas or aerosol 

• Direct irradiation from a contaminated surface 

• Inhalation of a radioactive gas or aerosol 

• Irradiation of hands and forearms from contaminated surfaces 

• Irradiation of the skin of the whole body. 

3.17 Radon and Thoron and Their Short-lived Decay Products 

In general, the practices described by Daer et al. (2006) will be followed. Some of what follows is 

borrowed verbatim from the DOE Standard for Internal Dosimetry (U.S. Department of Energy (DOE) 

1999). 
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3.17.1 Radon and Thoron 

The chemical element radon has two radiologically important isotopes that occur in nature: 220Rn and 

222Rn. Following popular usage, this document refers to the former as “thoron” and the latter as “radon.” 

Radon and its short-lived progeny (decay products) are continuously produced by decay of 226Ra, a 

member of the naturally occurring 238U series. Airborne concentrations of radon’s short-lived progeny 

(

218Po, 214Pb, 214Bi, and 214Po) are of interest due to their potential for deposition in the lung, leading to 

subsequent irradiation of lung tissue by alpha emissions from 218Po and 214Po. 

Thoron and its short-lived progeny are continuously produced by the decay of 224Ra, a member of the 

naturally occurring 232Th series. Thoron and 216Po have short half-lives: 56 s and 0.145 s, respectively. 

Lead-212 and 212Bi are of interest due to the possibility of their being deposited in the lung and irradiating 

tissue with alpha emissions. 

3.17.2 Potential Alpha Energy Exposure and Concentration 

The basis for protection from airborne short-lived decay products of radon and thoron is explained in 

ICRP Publication 32 (International Commission on Radiological Protection (ICRP) 1981). Exposure to 

airborne short-lived decay products of radon and thoron is given the special name potential alpha energy 

exposure (PAEE) for two reasons: 

• The relevant ionizing energy is delivered to the bronchial epithelium by alpha particles from 

218Po and 214Po in the case of 222Rn and from 212Bi and 212Po in the case of 220Rn (thoron). 

• The decay-product aerosol often contains an unknown mixture of the various radon and/or 

thoron progeny. 

For radon and thoron progeny, PAEE can be expressed as the product of average potential alpha energy

concentration (PAEC) and worker stay time and divided by the assigned respiratory protection factor, if 

any. The traditional unit of PAEC is the working level (WL), and traditionally, stay times have been 

measured in occupational “Months” of 170 hours. Thus, the traditional unit of PAEE is the working level 

month, or WLM. 

For routine monitoring of workers who are chronically exposed, weekly average air concentrations can be 

used for workers whose stay times are less than 40 hours in a given week. 

PAEC can be computed from concentration measurements of the short-lived radon progeny in air 

(National Council on Radiation Protection and Measurements (NCRP) 1990): 

218 214 214

218 214 214

(0.105 0.516 0.379 ) (WL) 100pCi/L of radon per WL at equilibrium

(0.00105 0.00516 0.00379 ) ,

Po Pb Bi

Rn

Po Pb Bi

CC C PAEC

CCC

− −−

− −−

+ + =

= ++

23

where 

 CPo-218 denotes the concentration of 218Po in pCi/L; 

 CPb-214 denotes the concentration of 214Pb in pCi/L; and 
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 CBi-214 denotes the concentration of 214Bi in pCi/L. 

PAEC can be computed from concentration measurements of the short-lived thoron progeny in air 

(United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) 1993): 

( )

( )

Pb-212 212

212 212

0.913 0.087

(WL) 7.43pCi/L of thoron per WL at Equilibrium

0.12288 0.011709 ,

Bi

Tn

Pb Bi

C C

PAEC

C C

−

− −

+ =

= +

24

where 

 CPb-212 denotes the concentration of 212Po in pCi/L; and 

CBi-212 denotes the concentration of 212Bi in pCi/L. 

3.17.3 Equilibrium Factors 

When radon and thoron concentration measurements are given, equilibrium factors are needed to convert 

these to potential alpha energy concentrations (PAEC) in working levels (WL). Another acceptable 

method for workplace monitoring of exposure to radon progeny is to measure the 222Rn itself, and convert 

it to PAEC using known equilibrium factors. One may calculate equilibrium equivalent concentration, 

EEC, from radon concentration measurements, C, based on knowledge or assumption of an equilibrium 

factor, F: 

EEC (pCi/L) = C (pCi/L) ⋅ F 25

If F has not been measured, it is acceptable under some circumstances to assume a default indoor value of 

FRn = 0.4 (International Commission on Radiological Protection (ICRP) 1993; United Nations Scientific 

Committee on the Effects of Atomic Radiation (UNSCEAR) 1988; United Nations Scientific Committee 

on the Effects of Atomic Radiation (UNSCEAR) 1993). If C is in units of μCi/mL, then EEC will also be 

in units of μCi/mL (note: 1 pCi/L = 10-9 μCi/mL). 

For 222Rn, FRn is defined as: 

222

218 214 379 214 0.105 0.516 0.

−

− + − + − =

Rn

Pb Pb Bi

Rn C

C C C F 26

where 

 CPo-218 = the concentration of 218Po; 

CPb-214 = the concentration of 214Pb 

CBi-214 = the concentration of 214Bi; and 

 CRn-222 = the concentration of 222Rn. 

For 220Rn, FTn is defined as 
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220

212 212 0.913 0.087

−

− + − =

Rn

Pb Bi

Tn C

C C F 27

where 

 CPb-212 = the concentration of 212Pb; 

CBi-212 = the concentration of 212Bi 

CRn-220 = the concentration of 220Rn (thoron). 

If it is not practical to measure equilibrium factors, a default 222Rn equilibrium factor of 0.4 (International 

Commission on Radiological Protection (ICRP) 1993; United Nations Scientific Committee on the 

Effects of Atomic Radiation (UNSCEAR) 1993) may be used for indoor areas with normal ventilation 

rates and outdoor areas with radon sources no closer than 400 m (≈ 1/4 mile). Average indoor 

equilibrium factors increase with increasing particle concentration in air, and decrease with increased air 

exchange rate (James et al. 1988; James 1994; National Research Council 1991; Nuclear Energy Agency 

(NEA) 1986; United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) 

1993). For outdoor areas with local sources of radon and highly ventilated indoor areas, the appropriate 

equilibrium factor should be determined by concurrent radon and radon progeny measurements made 

over a set of conditions that present the range of equilibrium factors to be encountered when workers are 

present. These measurements and the rationale for their application in the inference of radon progeny 

concentration should be documented in the internal dosimetry technical basis documentation. 

Triangular distributions may be assumed, as given in Table 3.6; however, for computational simplicity, 

lognormal distributions may be assumed. While the lognormal is unbounded on the high side, and an 

equilibrium factor greater than 1 is impossible, the probability of having an equilibrium factor greater 

than 1 is negligible for these distributions. 

The average equilibrium factor value of 0.4 for 222Rn is well established (Harley and Chittaporn 2006; 

U.S. Department of Energy (DOE) 1999; International Commission on Radiological Protection (ICRP) 

1993). An average equilibrium factor value of 0.04 for 220Rn (thoron) has been suggested (U.S. 

Department of Energy (DOE) 1999), but more recent and thorough work shows that it should be more 

like 0.02 (Harley and Chittaporn 2006). To date, no thoron measurements have been encountered, so the 

thoron equilibrium factor is included merely for completeness. 
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Table 3.6. Uncertainty distributions for equilibrium factors for converting radon and thoron gas 

measurements to working levels (WL) 

Gas Measured 

Mean of 

Distribution 

Lognormal 

Distribution Parameter 

1 (median) 

Lognormal 

Distribution Parameter 

2 (GSD) 99th %ile of lognormal 

220Rn (thoron) 0.02 0.0184 1.5 0.0473 

222Rn (radon) 0.4 0.368 1.5 0.946 

Triangular 

Distribution Parameter 

1 (minimum) 

Triangular 

Distribution Parameter 

2 (mode) 

Triangular 

Distribution Parameter 

3 (maximum) 

220Rn (thoron) 0.02 0 0.01 0.05 

222Rn (radon) 0.4 0 0.2 1 

3.17.4 Summary of radon and thoron quantities and conversion factors 

Numerical conversions for 222Rn and 220Rn quantities are given in Table 3.7. 
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Table 3.7. Summary of numerical conversions for radon and thoron quantities, regardless of the precision 

of measurements 

Multiply 

In Units 

Of By To Obtain 

In Units 

Of 

Concentration, C pCi/L 10−9 Concentration, C μCi/mL

Ambient 222Rn or 220Rn 

concentration, C

pCi/L F* Equilibrium equivalent 222Rn or 

220Rn concentration, EEC

pCi/L 

222Rn EEC pCi/L 1/100 = 0.01 Potential alpha energy 

concentration, PAEC

WL 

220Rn EEC pCi/L 1/(7.43) = 

0.13459 PAEC WL 

222Rn or 220Rn progeny PAEC WL Exposure time, t

(hours) ÷170 

Potential alpha energy exposure, 

PAEE

WLM 

Integrated 222Rn 

concentration, NV (ambient) 

pCi·d/L F× 1.4118E-3 PAEE WLM 

Integrated 222Rn 

concentration, NV (ambient) 

pCi·d/L 5.6471E-4 

assuming F= 0.4 

PAEE WLM 

PAEC WLM 4.2490E-3 Potential alpha energy intake, I, 

of 222Rn or 220Rn progeny 

J 

*For 222Rn, Fdefault= 0.4; for 220Rn, Fdefault= 0.02 

3.18 Radium Monitoring by Breath Radon Analysis 

The measurement of 222Rn in breath as a bioassay for 226Ra was pioneered by Evans (Evans 1943) and by 

Harley of the AEC Health and Safety Laboratory (HASL) (Harley, Jetter, and Eisenbud 1951), and breath 

measurements were conducted by HASL personnel. Evans found that 1 pCi/L of 222Rn in exhaled breath 

indicated the presence of 0.1 μg (0.0989 μCi) of 226Ra in the body (Evans 1943). However, Stabin 

calculated that 1 pCi/L of 222Rn in exhaled breath indicated the presence of 0.252 μCi of 226Ra in the 

body (Stabin 2005). The latter value is more favorable to the claimant and will be used in this work. 

3.19 Determination of the Uncertainty Distribution for Annual Organ Doses 

Summed Over Multiple Intakes 

Assumptions favorable to the claimant are needed regarding the uncertainty distribution and uncertainty 

parameters for annual doses from intakes in all prior years. 
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When any uncertain variables described by distributions are added, their variances are added. For Normal 

distributions, this results in a straightforward means of determining the sum and its uncertainty, since the 

sum of Normally-distributed variables is Normally-distributed. For lognormal distributions, the product is 

lognormally-distributed, but the sum has no analytical distribution. This can be explored using Monte 

Carlo simulations. 

One defensible workaround would be to sum the arithmetic mean annual doses received in each year from 

intakes in all prior years, and sum their variances. Then the resulting sum could be deemed to be 

lognormally distributed with a mean equal to the sum of mean annual doses and a variance equal to the 

sum of the variances of the mean annual doses. This approximation may be good enough to be defensible, 

but it will need to be tested using Monte Carlo simulations. 

There is also the problem of uncertainty correlations across years4

. For external doses, uncertainties from 

one year to another will be somewhat correlated (for example, if there were systematic errors in dosimeter 

calibration) and somewhat uncorrelated due to random fluctuations. For annual doses from intakes of 

tenaciously-retained radionuclides, uncertainties are correlated with r

2

 ~ 1. 

Suppose an individual had an intake in 1950 for which we estimate a median and GSD, say x50 = 1 and 

GSD = 3. We use IMBA to calculate annual doses from the median value. Thus, if IREP picks, say, the 

80th %ile of the annual dose in 1950 for the intake in 1950, it should pick the 80th %ile of the annual 

doses for all subsequent years due to this intake. For the annual dose in 1951 due to the intake in 1950, 

the 80th %ile should be chosen. For the annual dose in 1951 due to an intake in 1951, IREP may pick, 

say, the 25th %ile. It should then pick the same percentile of each annual dose distribution due to the 

1951 intake. 

Using correlated uncertainties would result in broader dose distributions and larger PC values. Even 

external doses should have some degree of correlation because 1) systematic dosimetry calibration or 

response errors would correlate from year to year; and 2) an individual’s behavioral differences from 

some average would differ from year to year, e.g., his or her job may have required facing toward the 

source of radiation more than was assumed and this would be the same from one year to the next as long 

as the individual had the same job. 

Conclusions drawn from Monte Carlo simulations of the sum of random variables drawn from lognormal 

distributions with the same GSD: 

1. The arithmetic mean of the resultant sum is equal to the sum of the arithmetic means of the 

random variables. 

 

4

 While the correlated uncertainties here contain components of both bias and uncertainty, the bias component is an 

unknown bias. According to the ISO and NIST, such uncertainties, termed Type B uncertainties, are treated 

mathematically the same as random uncertainties (International Organization for Standardization (ISO) 1995; Taylor 

and Kuyatt 1994). The ISO and NIST chose to use the terminology “Type A” uncertainties (“those which are 

evaluated by statistical methods”, or “component of uncertainty arising from a random effect,”) and “Type B” 

uncertainties (“those which are evaluated by other means,” or “component of uncertainty arising from a systematic 

effect”). “There is not always a simple correspondence between the classification of uncertainty components into 

categories A and B and the commonly used classification of uncertainty components as “random” and “systematic.” 

The nature of an uncertainty component is conditioned by the use made of the corresponding quantity, that is, on 

how that quantity appears in the mathematical model that describes the measurement process. When the 

corresponding quantity is used in a different way, a “random” component may become a “systematic” component 

and vice versa. Thus the terms “random uncertainty” and “systematic uncertainty” can be misleading when generally 

applied. If the bias were known, it would be corrected for (with perhaps an uncertain correction factor). 
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2. The geometric mean of the resultant sum is greater than the sum of the geometric means of the 

random variables. 

3. The distribution of the sum is very skewed, not even remotely Normal, but not too different from 

a lognormal. 

4. The unbiased fit of a lognormal distribution5

 to the distribution of sums underestimates the upper 

95th and upper 99th %iles of the distribution of sums and thus is not favorable to the claimant. 

5. The unbiased estimate of the GSD of the resultant sum is less than the GSD of the random 

variables. 

6. Using the arithmetic mean of the distribution of sums and the largest GSD of the random 

variables results in substantial overestimates of the upper 95th and upper 99th %iles of the 

distribution of sums and thus is very favorable to the claimant. Table 3.8 shows results of 3 

separate Monte Carlo experiments that added random samples from 10 identical lognormal 

distributions. Clearly, the means add, on the average, to give the mean of the sum distribution. 

The median of the sum is significantly larger than the sum of the medians. Not surprisingly, the 

GSDs of the resulting sums are smaller than those of the distributions that were sampled. An 

unbiased lognormal fit u to the sums is done by the methods of Section 2.1.1. The unbiased fit’s 

u95 and u99 underestimate the upper 95th and 99th %iles y95 and y99, with the underestimate 

becoming more severe with increasing GSD. Finally, assuming that the sum has the same median 

as the simulation but has the same GSD as the sample distributions yields biased estimates b95 

and b99 as shown in Table 3.8. The assumption for the simulation with GSD = 4 that the resultant 

distribution has the same GSD (4) is shown as the dashed line in Figure 15. This assumption leads 

to a substantial overestimate of the upper 95th and 99th %iles, ranging from a factor of 3 for the 

upper 95th %ile with a GSD = 3 to a factor of 6 for the upper 99th %ile with a GSD of 5. 

Table 3.8. Results of adding random samples from 10 lognormal distributions with the same GSD. 

The results of the experiment for GSD = 4 are shown in Figure 15. Clearly, the farther one is from the 

median, the worse the simplistic lognormal approximation is. 

One conclusion from this Monte Carlo experiment is that it matters whether doses to a tissue or organ 

during a given year are summed prior to being put into IREP, or whether they are put into IREP 

individually. 

In the latter case, IREP will perform much as the Monte Carlo experiment above performed without 

fitting, i.e., the y values in Table 3.8. The former case can result from a workbook summing the doses 

within a calendar year for intakes over several calendar years, or from a dose reconstructor entering, say, 

10 intakes into IMBA and letting IMBA do the summation. 

 

5

 An unbiased estimate of a lognormal distribution of the sum is made by assigning the mean of the logs of the 

simulated sums to be the log of the geometric mean, and the standard deviation of the logs of the simulated sums to 

be the log of the GSD. 

Sample Distributions x Resultant Sum, y = ∑xi Unbiased Fit y50 & GSDx

x50 GSDx x y50 GSDy y y95 y99 u95 u99 b95 b99

1 3 1.828 16.6 1.53 18.28 33.8 49.7 33.5 44.7 101 214 

1 4 2.614 21.8 1.79 26.19 59.6 100 56.7 84.1 213 548 

1 5 3.652 27.4 2.04 36.99 91.8 189 88.4 143 387 1156 
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If the summation is done prior to inputting the sum into IREP, then the means of the various doses should 

be added (not the medians or modes) to get an unbiased estimate of the resultant mean. If one simply 

added the medians (= 1 in this case) of the initial distributions, one would use a median 10 × 1 = 10, while 

the medians of the Monte Carlo experiment shown in Table 3.8 were 16.6, 21.8, and 27.4, respectively, 

for GSDs of 3, 4, and 5. For this experiment, assuming a median of 10 is 60.2%, 45.9%, and 36.5%, 

respectively, of the actual medians for GSDs of 3, 4, and 5. 

The GSD of the resultant sum is significantly less than the GSDs of the distributions that were summed. It 

is not clear what the GSD would be if the initial distributions had varying medians and varying GSDs. If 

there is a positive correlation among the doses being summed, the assumption that the GSD of the sum is 

equal to the largest GSD of any of the input distributions may not result in much of an overestimate at the 

95th and 99th %iles of dose. 

Figure 15. Graph of 10,000 Monte Carlo trials adding 10 samples from lognormal distributions with x50 = 

1 and GSD = 4.

The geometric mean is needed as an input parameter to IREP. If the means of the input dose distributions 

were summed, the geometric mean would have to be determined from the correct mean and the 

overestimated GSD. For the first line in Table 3.8, the sum of the means (18.28) would be multiplied by 

0.5469 (using a GSD of 3) to get the median of 9.997. Doing the same for the other lines in Table 3.8 

gives a similar answer of 10. That is the same answer one gets if one simply adds the medians of the 

original distributions. Summing the medians and using the highest GSD of the individual distributions 

will produce a smaller overestimate than indicated by the Monte Carlo experiment, since the geometric 

mean is underestimated. 

A policy of simply adding the medians of the lognormally distributed doses and using the highest GSD

from the doses that were summed is chosen here because it is favorable to the claimant and technically 

feasible. 
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3.20 Representativeness of Air Samples 

General Area (GA) air samples, and even breathing zone (BZ) air samples, may not be representative of 

the concentration that the worker breathes (Caldwell, Potter, and Schnell 1967; Caldwell, Potter, and 

Schnell 1969; Cohen, Harley, and Lippmann 1984; Schultz and Becher 1963; West et al. 1995). Even if a 

constant airborne concentration were maintained where an air sampler is located, the concentration 

breathed by a worker may differ systematically from that concentration (U.S. Department of Energy 

(DOE) 1999). 

It is assumed that, on the average and in the absence of evidence to the contrary, an air sample 

distribution is unbiased. Thus the uncertainty distribution due to lack of representativeness must be 

unbiased, that is, have an arithmetic mean of 1. 

For the purposes of dose reconstruction, one of three assumptions concerning the lognormal uncertainty 

distribution due to lack of representativeness of air samples is to be made, as shown in Table 3.9. 

Table 3.9. Parameters of the lognormal uncertainty distribution due to lack of representativeness of 

an air sample distribution. 

Uncertainty Due to Lack of Representativeness of Air Sample 

Air Sample Measurement Type Mean Median, x50 GSD 

BZ 1 0.7864 = exp(−ln2

(2)/2) 2 

GA 1 0.2739 = exp(−ln2

(5)/2) 5 

Unknown air sample type, or 

combination of GA and BZ 1 0.2739 = exp(−ln2

(5)/2) 5 

3.20.1 Inferring Representativeness by Comparing BZ with GA Samples 

Many authors have studied the representativeness of air samples. For example, Caldwell (1972) compared 

breathing zone and general area air samples. The results, shown in Table 3.10 and Figure 16, show both 

bias and variability. For this small dataset for which early fecal clearance data were available, the average 

BZ sample was 118 times higher than the average GA sample, and the GSD was 3.35 (Strom, Watson, 

and Stansbury 2002). 
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Table 3.10. Breathing zone (BZ or Lapel) air sampling and general area (GA) air sampling (data digitized 

from Caldwell 1972; adapted from (Strom, Watson, and Stansbury 2002)). 

General Area 

(Bq m−3

) 

Lapel (Bq m−3

) Ratio 

Lapel/GA 

0.2 3 14.5 

0.22 4.7 21.1 

0.18 14 79.2 

1.62 28 17.5 

0.24 19 79.2 

0.35 161 458 

0.65 72 110 

0.45 72 160 

Average 118 

Standard Deviation 147 

Coefficient of Variation 1.25 

Geometric Mean 63.7 

Geometric Standard Deviation 3.35 

Minimum 14.5 

Maximum 458 

Number 8 

Figure 16. Graph from Caldwell (1972) digitized in Table 3.10. The straight line refers to 

predictions of the ICRP Publication 2 lung model 
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Figure 17. Comparision of lapel (BZ) air samples to general area air samples (Fig. 4 from Caldwell 

et al. 1967) 

As shown in Figure 17, work in the 1960s at NUMEC compared over 1,000 BZ and GA air samples for U 

and Pu (Caldwell, Potter, and Schnell 1967). These comparisons showed a bias on the average, and a very 

wide spread. Since the abscissa in Figure 17 is logarithmic, if the ratios were lognormally distributed, 

these curves would appear to be normally distributed. The text states, “Sixty-four per cent of Pu BZ's 

exceeded the GA concentration by a factor of 2 or more, 23% by more than a factor of ten.” This uniquely 

defines a lognormal distribution with parameters shown in Table 3.11. Furthermore, the text states, “The 

interval of general BZ - GA agreement (+ 100$, - 50%) covered 27% of the plutonium BZ samples and 

about 19% of uranium plant BZ samples.” The lognormal distribution described in Table 3.11 has 26.4% 

of its samples in the range of 0.5 to 2.0, indicating excellent agreement with the data. Unfortunately, there 

are not corresponding robust percentiles for the U data. 

Table 3.11. Summary statistics for the lognormal fit to the Pu distribution shown in Figure 17. 

Pu Lab (594 

samples) 

GeoMean 3.38 

GSD 4.33 

Calc. Mean 9.92 

Researchers as Los Alamos National Laboratory have investigated variability of air concentrations 

following glove box leaks (Gonzales et al. 1974). With small, point leaks, concentrations drop many 

orders of magnitude over short distances from the leak. They suggest a potential maximum ratio of BZ air 

concentration to GA concentration of 250, and their contour plots indicate measured spatial variations. It 
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is difficult to deduce the distribution of BZ to GA ratios, but it would appear that the LANL work is 

consistent with a GSD of 4.1 (~250(1/(2*1.96))) or more, based on the assumption that the range 1 to 250 

represents the 2.5 percentile to 97.5 percentile. 

If GA air samples cannot do better than a GSD of 4.33 in predicting BZ samples, then they should do 

even worse predicting intakes. But it is necessary to determine how well BZ air samples predict intakes to 

complete this line of reasoning. 

3.20.2 Inferring Representativeness by Comparing Excretion Rates Predicted from Air 

Samples with Measured Excretion Rates 

If bioassay is the gold standard for intakes, then representativeness of air samples can be deduced by 

comparing predicted urinary excretion rates with measured values. Without regard to bias (that is, 

systematic differences between biokinetic models and real workers), variability will be due to lack of 

representativeness of air samples as well as to other factors such as biological variability. However, such 

data are a start. 

Yu and Sherwood (1996) examined the relationships between urinary elimination, airborne concentration, 

and radioactive hand contamination for workers exposed to uranium. One cannot deduce the variability of 

an individual air sample and its subsequent urinary excretion value for individual workers, but one can 

understand average values. Airborne uranium concentration bore a strong, positive correlation with 

production index, with r

2

 = 0.73 at p<0.001. Urinary uranium excretion rate can be related to airborne 

concentration for 20 weekly average measurements of each variable, as shown in Figure 18. These GSDs, 

in the range of 1.5 to 1.6, are for weekly averages over many workers and many air samples, and thus 

represent a value much lower than an individual air sample and an individual person. Each of the 20 

weekly ratios represents a mean that is much more precisely known than any single ratio. 

1

10

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Standard Normal Deviate, z

Daily U Excretion / Average Airborne

Conc. [(μg/d)/(μg/m3)]

Sorted Data

ln(Data) Pred. GM= 2.73 GSD= 1.49

Uniform Pred. GM= 2.73 GSD= 1.55

Finney Pred. GM= 2.77 GSD= 1.58

Figure 18. Lognormal fit to the ratio of daily uranium excretion divided by the average airborne 

uranium concentration for maintenance workers, from Yu and Sherwood (1996). 
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Chase (1989) provides monthly individual exposure data for 2 workers. Again, as in the case of Yu and 

Sherwood (1996) these are monthly average air sample data, so variability has been smoothed out by 

averaging. Chase quotes the “[t]he reported LLD for the laboratory’s fluorimetric method of analysis was 

5 μg L−1

 U.” A value of 5/2 = 2.5 μg L−1 U was substituted for these analyses. The GSD for the mill 

operator (6 months’ data) was 1.6, while that for the mill mechanic (13 months’ data) ranged from 1.5 to 

1.7, depending on the fitting method chosen. 

From these analyses one concludes that breathing zone air samples are much more representative of what 

the worker breathes, but still bear a variable relationship to worker intakes. Since they are based on 

averages rather than individual measurements, using a GSD = 2 for BZ air samples is realistic. 

Combining the GSD for BZ representativeness of bioassay results (as a surrogate for intake) of ~3 and the 

GSD for GA representativeness of BZ results of ~4.3, one computes a GSD for GA representativeness of 

intakes of 

exp ln ( ) exp( ln (2.00) ln (4.33)) 5.06. 2 2

1/ 2

2 = + =

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ = ∑

i

GSDcombined GSDi 28

One obtains GSD = 4.03 if the value 3.35 is used in Eq. 28 in place of 4.33, and one obtains GSD = 4.81 

if the value 4.09 is used in Eq. 28 in place of 4.33. Based on this analysis, a GSD = 5 for GA 

representativeness will be used. 

3.21 Propagation of Medians and Uncertainties for Lognormal Distributions 

Products of lognormal distributions have simple analytical properties that permit inference of medians 

and GSDs directly from the parameters of the distributions being multiplied. These properties derive 

directly from the fact that multiplication of numbers is equivalent to addition of their natural logarithms 

and exponentiating, and that the variances of the logarithms of the distributions being multiplied are 

additive (Aitchison and Brown 1981). These properties are elaborated below. 

3.21.1 Propagation of Medians (not Means) for Products of Lognormal Distributions 

When lognormal distributions are multiplied, the resulting distribution is lognormal with a median x50

(geometric mean) that is the product of the medians of the underlying lognormals: 

= ∏

i

i x x , 50 50, 29

where the x50,i values are the medians of the contributing lognormals. Note that if the GSDs of the 

contributing lognormals are not the same, the arithmetic mean (average) of the product will not be equal 

to the product of the arithmetic means of the contributing lognormals: 

≠ ∏

i

i x x . 30

Annual point estimates of dose to a tissue or organ returned by IMBA will be the same type of statistic as 

the value put into IMBA. Thus if arithmetic mean or geometric mean values are put into IMBA, then the 

doses that IMBA computes will be the arithmetic or geometric mean, respectively. If a mean (also known 

as expectation value, arithmetic mean, and average) of a lognormal distribution is input, it must be 
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converted to a median (also known as the geometric mean) of the lognormal distribution for input into 

IREP. The median of a lognormal is always less than the mean, and is determined from the mean by 

exp( / 2), 2x50 = x −σ 31

where σ = ln(GSD). 

3.21.2 Propagation of Uncertainties for Lognormal Distributions 

Several uncertainties contribute to the uncertainty in an intake: 

• Air samples for a given operation or a given site show variability, typically characterized by a 

lognormal distribution. 

• General Area (GA) air samples, and even breathing zone (BZ) air samples, may not be 

representative of the concentration that the worker breathes. 

• There is uncertainty in biokinetic models and parameters, e.g., f1 (the uptake fraction from the 

gastrointestinal tract), and how a particular claimant may differ systematically and randomly from 

Reference Man used for the biokinetic models, including body weight, breathing rate, nose- or 

mouth breathing, smoking-induced changes in respiratory function, etc. 

If the intake is the product of lognormal distributions, then its GSD is combined with other uncertainties 

using 

exp , 2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = ∑

i

GSD σ i 32

where σi = ln(GSDi). For example, the GSD of a tissue or organ dose derived from an air monitoring 

distribution with GSD = 5, a representativeness with a GSD = 5, and a biokinetic modeling uncertainty 

GSD = 3 would be 

exp( ln (3) ln (5) ln (5)) 12.52. 2 2 2 GSD = + + = 33

3.22 Adding Doses with Differing Distributions 

When doses are added, the parameters that must be added are the arithmetic means (also known as the 

means, averages, or expectation values). As stated above, the median of a lognormal is always less than 

its mean, so adding the median of a lognormal to a mean of a normal distribution gives an answer that is 

biased low. The mode of a triangular distribution may be either greater or less than the mean. Summing 

means, medians, and modes creates a number whose meaning is unclear and which is likely, although not 

certainly, a low estimate of the true sum. 

In order to facilitate straightforward checking by the claimant, the Battelle Team will continue the 

practice of summing the IREP input parameters for the claimant’s lifetime dose report. The fact that this 

may be an underestimate of the true sum is of no consequence to the compensation decision, since IREP 

correctly uses the input distributions. 
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3.23 Adjusting Process-Specific Dose Rates or Air Concentrations for Time 

Trends over Periods of Years 

When average dose rates or air concentrations are available for a given span of years for processes or 

locations, and when separate information on time trends over periods of years are available, it is possible 

to adjust the average values for temporal changes. What follows is a specific example for uranium 

refining. 

Figure 19. Original Figure 1 from Christofano and Harris (1960) showing time-weighted average 

exposure trends over the years 1948-1956. 

Christofano and Harris (1960) showed that there was a large reduction over the years in time-weighted 

average air concentrations of uranium in various refining plants (Figure 19). 

The image in Figure 19 was digitized6

 and data analyzed for each year. The average value (mean) was 

computed from the digitized data for each year, and a lognormal was fit to the data for each year by 

computing natural logs, averaging them, and taking their standard deviation as described in BTIB-5000. 

The geometric mean and geometric standard deviation (GSD) was calculated for each year, and an upper 

 

6

 UnGraph V5, Biosoft, http://www.biosoft.com, PO Box 1013, Great Shelford, Cambridge, CB2 5WQ United 

Kingdom. tel: +44 1223 841700 fax: +44 1223 841802 
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95th percentile. These results are shown in Table 3.12 and Figure 20. Vertical bars are “times or divided 

by” one GSD. 

Table 3.12. Results of digitizing data from Figure 19. All concentrations (columns 3-7 and 9) are in 

dpm/m3

 

Year 

No. 

Data 

Points Mean 

Standard 

Deviation

Geometric 

Mean

Geometric 

Mean 

×GSD

Geometric 

Mean 

÷GSD GSD 

Upper 

95%ile

1948 17 7398 12417 2061 10792 393 5.24 31391

1949 9 964 1119 540 1902 153 3.52 4286

1950 25 349 951 101 434 24 4.29 1112

1951 8 521 569 219 1062 45 4.85 2940

1952 15 124 133 79 207 30 2.62 385

1953 31 71 121 40 102 16 2.53 186

1954 15 94 85 60 170 21 2.84 334

1955 8 63 27 57 91 36 1.58 122

1956 9 85 109 43 144 13 3.35 315

Total 137 
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10

100

1,000

10,000

100,000

1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957

Year

"Weighted Exposures" (Concentration [dpm/m3])

Mean

Geometric Mean

Upper 95%ile

Figure 20. Analysis of time-weighted average air concentrations from Christofano and Harris 

(1960; Fig. 1). Variability bars are the geometric mean times or divided by one GSD. 

Clearly, the mean concentration drops rapidly from 1948 through 1950, from 7,400 dpm/m3 to 350 

dpm/m3

 (a factor of over 20), as engineered workplace controls were installed at the dustiest locations. 

Alpha-emitting dust concentrations for the years 1953-1957 are roughly 100 times lower than they were 

in 1948. 

The first difficulty in interpreting Figure 19 is that it combines data from different processes, and we do 

not know which processes these are, that is, which are the dustiest or cleanest in a given year. 

The second difficulty in interpreting the data in Figure 19 and the analysis in Figure 20 is that data for the 

various processes may have been measured one or more times, and we do not know which times these 

were. So, while we know that workplace air became dramatically cleaner on the average, it is necessary to 

develop a defensible method to apply this knowledge to the tabulated values for the various operations 

that have been derived from the 1960 Christofano and Harris paper. 
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One approach to using these data would be to ignore process-specific information and use the upper 95th

percentile value in a given year for all process workers, and the median value for non-process workers. 

This assumption may be favorable to the claimant, but it ignores information that can be used. 

A preferred approach to using these data is to analyze all of the averages and create a year-specific scaling 

or correction factor for the tabulated values for the various operations that have been derived from the 

1960 Christofano and Harris paper. The time-dependent correction factors that result from this method are 

shown in Figure 21. The average of these factors is 1.00, so that a worker who was present for all 9 years 

would have a predicted intake that is unchanged by the application of this method. In this sense, the time 

correction method is unbiased. A worker who was present only for 1948 would have a nearly 7-fold 

higher intake, while workers who were present after 1953 would have dramatically lower intakes. The 

1948 value would be used for all years prior to 1948, and the 1956 value would be used for all years after 

1956. These factors are presented in Table 3.13. 

The annual intake is thus computed using 

 (process-specific concentration) (year -specific correction factor),

(1.2m /h) (actual or default hours per year) 3

annual

× ×

I = ×

34

which is a modification of Eq. 21 from Section 3.10. 

0.01

0.1

1

10

1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957

Year

Annual Mean/Overall Mean

Figure 21. Use of year-specific correction factors to adjust air concentrations for processes over 

time. 
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Table 3.13. Year-specific correction factors for tabulated process-specific air concentrations, derived 

from analysis of Christofano and Harris (1960) Fig. 1. 

Year 

YearSpecific 

Correction 

Factor 

1948 or before 6.89 

1949 0.897 

1950 0.325 

1951 0.485 

1952 0.116 

1953 0.0656 

1954 0.0875 

1955 0.0582 

1956 or after 0.0792 

Average 1 

The tabulated process-specific data reported by Christofano and Harris (1960) include measurements 

made over the duration of the survey. They state, “In substantially every case, the first survey conducted 

at any plant disclosed the highest exposures and the most recent measurements were lowest. In reporting 

occupational dust exposures, we have presented the range of exposures and also the numerical average of 

all such evaluations made during the 10 years from 1948 through 1957” (Christofano and Harris 1960, p. 

77)7

. The inclusion of measurements made at different times as ventilation and engineered controls 

improved means that some of the broader lognormal distributions include non-random variability due to 

temporal improvements in workplace controls, as well as a component of random variability due to other 

factors such as process differences between plants and air sample representativeness. Using the yearspecific correction factors to average air concentrations improves the precision of the estimates of air 

concentrations. 

 

7

 There are no data from 1957 in Figure 19, so this analysis goes only through 1956. 
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4.0 Glossary 

Note: Many of these definitions are borrowed from the DOE Standard on Internal Dosimetry (U.S. 

Department of Energy (DOE) 1999). 

activity median thermodynamic diameter (AMTD): “Fifty percent of the activity (thermodynamically 

classified) in the aerosol is associated with particles of thermodynamic diameter (dth) greater than the 

AMTD. A lognormal distribution of particle sizes is usually assumed” (ICRP 1994a). 

assigned protection factor (APF): The expected workplace level of respiratory protection that would be 

provided by a properly functioning respirator or a class of respirators to properly fitted and trained users 

(ANSI Z88.2-1992). 

bias: The deviation of a single measured value of a random variable from a corresponding expected 

value, or a fixed mean deviation from the expected value that remains constant over replicated 

measurements within the statistical precision of the measurement (synonymous with deterministic error, 

fixed error, and systematic error) (HPS N13.30-1996). 

bioassay: Another word for radiobioassay. 

biokinetic model: A series of often empirically determined mathematical relationships formulated to 

describe the intake, deposition in respiratory tract (if applicable), uptakes by the transfer compartment 

from intake compartment(s), uptakes by tissues or organs from the transfer compartment, translocation, 

retention, and elimination of a radionuclide from the body. 

censored data: Data that have been recorded as “less than” values rather than the observed numerical 

values (whether positive, zero, or negative). 

Class SR-0 gases: Insoluble and nonreactive gases and vapors (ICRP Publication 68, p. x, 1994a). 

Class SR-1 gases: Soluble or reactive gases and vapors (ICRP Publication 68, p. x, 1994a). 

Class SR-1 gases: Highly soluble or reactive gases and vapors (ICRP Publication 68, p. x 1994a). 

compartment: The smallest element in a biokinetic model for which a mathematical representation of a 

retained quantity is given. Compartments may be organs (e.g., lung, liver), tissues (e.g., bone marrow), or 

systemic (e.g., the transfer compartment). 

critical level: Same as decision level. 

decision level: The amount of a count (LC) or a count rate (LC) or the final instrument measurement of a 

quantity of analyte (DC or DC ) at or above which a decision is made that the analyte is definitely present 

(HPS N13.30-1996). 

deposition fraction: The fraction of the amount of a material inhaled that is deposited in a particular 

region of the respiratory tract. For an aerosol, this fraction is a function of the aerodynamic or 

thermodynamic diameter. 
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detection level (LD): This concept has been replaced by minimum detectable amount (MDA). 

diagnostic measurements: Measurements performed to estimate the amount of radionuclide deposited in 

a person when an intake is known or is suspected to have occurred (HPS N13.30-1996). 

direct radiobioassay: The measurements of radioactive material in the human body utilizing 

instrumentation that detects radiation emitted from the radioactive material in the body (synonymous with 

in vivo measurement.) (HPS N13.30-1996). 

equilibrium factor (F): The equilibrium factor F with respect to potential alpha energy is the ratio of the 

equilibrium equivalent concentration (EEC) to the actual activity concentration of radon in air. 

equilibrium equivalent concentration (EEC): The EEC of a non-equilibrium mixture of short-lived 

radon progeny is that activity concentration of radon in radioactive equilibrium with its short-lived 

progeny that has the same potential alpha energy concentration as the non-equilibrium mixture to which 

the EEC refers. 

exposure: (1) The general condition of being subjected to radiation, such as by exposure to radiation 

from external sources or to radiation sources inside the body. In this document, exposure does not refer to 

the radiological physics concept of charge liberated per unit mass of air. 

 (2) The product of exposure time to a radioactive aerosol and the average concentration 

during exposure, divided by the value of the DAC for the radioactive material in question (expressed in 

DAC-h). 

 (3) Exposure (of an individual to radon progeny) is the time integral of the potential alpha 

energy concentration in air over a given period (expressed in WLM) (adapted from ICRP Publication 65, 

p.4). 

gastrointestinal (GI) tract model: A mathematical representation of the behavior of radionuclides in the 

contents of the human gastrointestinal tract. 

indirect radiobioassay: Measurements to determine the presence of or to estimate the amount of 

radioactive material in the excreta or in other biological materials removed from the body (synonymous 

with in vitro measurement) (HPS N13.30-1996). 

intake compartment: One of four compartments from which systemic uptake can occur: the respiratory 

tract; the GI tract; a wound; or intact skin. 

intake route: A pathway by which radioactive material enters the body. The main intake routes are 

inhalation, ingestion, absorption through the skin, and entry through injection or a cut or wound in the 

skin. 

in vitro measurement: Synonymous with indirect bioassay. 

in vivo measurement: Synonymous with direct bioassay. 

lower limit of detection (LLD): Synonymous with minimum detectable amount (MDA). 

minimum detectable amount (MDA): The smallest amount (activity or mass) of an analyte in a sample 

that will be detected with a probability ß of non-detection (Type II error) while accepting a probability of 
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erroneously deciding that a positive (non-zero) quantity of analyte is present in an appropriate blank 

sample (Type I error). 

minimum detectable concentration (MDC): The minimum detectable amount (MDA) expressed in units 

of concentration (HPS N13.30-1996). 

potential alpha energy concentration (PAEC): The kinetic energy potentially released in a unit volume 

of air by alpha particles emitted by the short-lived radioactive progeny of 222Rn (i.e., 218Po and 214Po) or 

220Rn (i.e., 216Po, 212Bi, and 212Po). PAEC is expressed in working levels (WL). 

potential alpha energy exposure (PAEE): The average potential alpha energy concentration (PAEC) to 

which a worker is exposed, multiplied by the time of exposure in working months of 170 hours: that is, 

PAEE = PAEC × time. PAEE is expressed in working level months (WLM). 

quality assurance: All those planned and systematic actions necessary to provide adequate confidence 

that an analysis, measurement, or surveillance program will perform satisfactorily in service (HPS 

N13.30-1996). 

quality control: Those actions that control the attributes of the analytical process, standards, reagents, 

measurement equipment, components, system, or facility according to predetermined quality 

requirements (HPS N13.30-1996). 

radiobioassay: Measurement of amount or concentration of radioactive material in the body or in 

biological material excreted or removed from the body and analyzed for purposes of estimating the 

quantity of radioactive material in the body (HPS N13.30-1996). 

radon: Unless otherwise specified, the isotope 222Rn. 

respiratory tract model: A mathematical representation of the behavior of particles and gases in the 

human respiratory tract. 

retained quantity: The amount of material which, after being taken into the body by inhalation, 

ingestion, entry through an open wound, or absorption through the skin, exists in the whole body, a 

compartment, an organ, or a tissue at a specified time. 

screening measurements: Measurements made to detect radioactive material under routine conditions, 

but not used to quantify the amount of a given radionuclide (HPS N13.30-1996). 

thermodynamic particle diameter (dth): Diameter (in μm) of a spherical particle that has the same 

diffusion coefficient in air as the particle of interest (ICRP 1994a). 

thoron: The isotope 220Rn, also symbolized by Tn. Thoron is a “trivial name” like tritium. 

translocation: Movement within the body of a radioactive material after uptake, such as from bone to 

kidney. 

working level (WL): is any combination of the short-lived radioactive progeny in one liter of air, 

without regard to the degree of equilibrium, that will result in the ultimate emission of 130,000 MeV of 

alpha energy (1 WL = 2.083 E-5 J/m3

). 

Note: WL is the unit of potential alpha energy concentration (PAEC). 
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working level month (WLM): The unit of potential alpha energy exposure (PAEE), defined as exposure 

for 1 working month (of 170 hours) to an airborne concentration of 1 WL. (1 WLM = 1 WL × 170 hours 

= 0.00354 J.h/m3

). 

wound compartment: The compartment in a biokinetic model whose retained quantity is the amount of 

radioactive material in a wound that has not moved to the transfer compartment. 
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