Methods of Computation 

for Estimating 

Geochemical Abundance 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 574-B 
Methods of Computation 

for Estimating 

Geochemical Abundance 

By A. T. MIESCH 

STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 574-B 

A review of statistically efficient procedures 

for estimating the population arithmetic mean 

where the data are censored at the lower limit 

of analytical sensitivity or positively skewed 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1967 
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary 

GEOLOGICAL S-URVEY -

William T. Pecora, Director 

For sale by the Superintendent of Documents, U.S. Government Printing Office 

\Vashington, D.C. 20402- Price 20 cents (paper cover) 
CONTENTS 

Abstract-------------------------------------------

Introduction---------------------------------------

Problems in estimation of geochemical abundance ______ _ 

Analytical sensitivity _____ - ________ ---- _________ _ 

Smn.ll groups of data __ ----_- ______ ----_-_- ___ - __ 

Geometric clnsses_-------- _- _-------------- _---- AnnJyticnl discrimination _______________________ _ 

Recommended techniques _____ -- ____________________ _ 

Transformations __ -_------_- __ -- __ ----_--- ___ - __ 

Cohen's method for censored distributions ________ _ 

t and ta. estimators of Sichel and Krige ___________ _ 

Page 

Bl 

1 

2 

3 

3 

3 

4 

4 

5 

5 

7 

Page 

Recommended techniques-Continued 

Confidence intervals_____________________________ B8 

Summary of methods________________________________ 8 

Examples------------------------------------------ 10 

Molybdenum sulfide in drill core ----------------- 10 

Iron in sandstone_______________________________ 11 

Uranium in granite_____________________________ 12 

Arsenic in basalts and diabases.__________________ 13 

Conclusions________________________________________ 14 

Literature cited __________________ -----______________ 14 

ILLUSTRATIONS 

Page 

FIGURE 1. Histograms showing frequency distributions _____________ --_-_-- __ -------_~_. ______ --- ____ -- ___________ -__ B6 

2. Curves for estimating A for equations 5 and 6-----------------------------·----..:------------------------- 7 

3. Graphs of.,. as a function of the number of analyses, n, and the antilog of s or~----------------------------- 8 

4. Flow chart showing methods of computation for estimating geochemical abundance-------------------------- 9 

5. Probability graphs of frequency distributions A-Gin figure 1-------------------------------·--------------- 11 

TABLE 

Page 

TABJ,E 1. Summary of computations for estimating geochemical abundance from data represented in histograms in figure L.. B12 

III 
STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

METHODS OF COMPUTATION FOR ESTIMATING GEOCHEMICAL ABUNDANCE 

By A. T. MIESOH 

ABSTRACT 

Geochemical abundance of an element i·s regarded as the proportion of a total rock body, or group of rock 1

bodies, that is 

made up of the element and is equivalent to the population 

arithmetic mean of sample analyses, generally in units of percentage or parts per million. Computational problems encountered in estimation of abundances can arise from having limited ranges of analytical sensitivity, from having only small 

groups of data, and from reporting of data in broad geometric 

classes. 'l'hese problems can be partly .resolved by use of a combination of techniques described by Sichel (1952), Cohen (1959, 

1961), and Krige ( 1960). The combination of techniques allows 

efficient estimation of geochemical a·bundances from a wide 

variety of frequency distributi'On types if the analytical 

discrimination is sufficient to allow effective use of data 

transformations. 

Examples of data ·from the literature were used to demonstrate the techniques; up to 76 percent of the da·ta values were 

nrbitrarlly placed in the "not detected" class, and estimated 

arithmetic means agreed with those estimated from the complete data set to two significant figures. The types of frequency 

distributions displayed by the data ·examples are generally 

representative of those commonly encountered in geochemical 

problems. 

INTRODUCTION 

The esthnation of abundances of constituents in rock 

bodies, or in groups of rock bodies, is necessary in a 

broad range of geologic problems. In mining and ore 

reserve estimation, for example, unbiased and. precise 

estirnates of abundance of ore constituents are prerequ~site to efficient operational plannin.g, and become mandatory as the grade of the ore declines toward the 

minimum grade that can be mined profitably. As lower 

grade deposits of broad extent are being appraised as 

potential ores of the future, the need for accurate and 

precise abundance estimates will increase. Aside :from 

these economic problems, the estimation of element abundances is necessary in geochemical problems related to 

the origin of rock bodies and in both small- and largescale assessments of geochemical balance among 

different rock units and other material at the earth's 

surface. 

Estimates of ~bundance are generally expressed in 

terms of weight' percent or parts per million and ~re 

regarded as estimates of the proportion of the roc~ body 

that is made up of the constituent of concern. An 

abundance of 2 percent iron in a 500-million-ton rock 

body, :for example, implies that 10 million tons of iron 

are present. 

The estimation of reliable geochemical abundances is 

met with serious difficulties in both sampling and in 

computation. Errors due to sampling are undoubtedly 

the more serious in a majority of problems, but where 

sampling has :been well planned ~and successfully exe-

: cuted, correct computational procedures are increasingly 

·desirable. Where sampling has been inadequate (often.: 

unavoidably so owing to limited outcrop, :for example), 

· reliable abundance estimates are difficult to obtain. 

· Where the sampling has been unbiased but of limited 

· extent, the problem faced is to obtain the best estimate 

: of abundance possible; computational methods are 

; important :for this purpose. 

Many types of averages (such as medians, modes, 

geometric means, mean logs, and arithmetic means) are 

: suitable for specific geochemical problems, but the 

ari,thmetic mean is the only one of these that is a correct 

expression of abundance-or an unbiased estimate of 

abundance-under all circumstances. This was discussed and verified by Sichel (1947, 1952). I:f the 

. frequency distribution of values used in computing the 

' average is unimodal and symmetrical, the median, mode, 

'and arithmetic mean will be the same; but if the distribution is skewed (asymmetrical), they may differ 

widely. The geometric mean will always be less than 

the arithmetic mean regardless of the type of frequency 

distribution, except where they are equal owing to zero 

variance in the data. Any type of average may be 

appropriate :for a specific geochemical problem, depending on the purpose for which the average is estimated. 

The geometric mean, for example, is useful :for representing the typical concentration of an element in a 

Bl 
B2 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

group of rock specimens where the concentrations exhibit a symmetrical frequency distribution on a log 

scale. The median is a useful expression of average 

concentration among specimens regardless of the frequency distribution and will be the same. as the geometric mean where the distribution is symmetrical on a 

log scale. 

The equivalence of the arithmetic mean of the total 

population of samples in an entire rock body to geochemical abundance follows from the definition of 

abundance, A (in weight percent), as: 

where Wis the mass of the constituent in the rock body, 

and T -is the mass of the rock. body. If there are N 

potential discrete samples in the total rock body, the 

mass, W, of the constitue~t is given by: 

. . 1 .. N . 

w 100 ft;t XjSj, 

where x1 is the concentration (in weight percent) of 

the constituent in the jth sample and ·s1 is. the weight 

of the sample. If all samples ~re of the same weight, then 

and 

. T N" 

w lOON ft;t Xj, 

N 

~Xi 

A _j=l 

.-fr· 

The right side of this equation is the population arithmetic mean which can be estimated from analytical 

data by a number of different techniques. 

In some problems involving estimation of geochemical abundance, the weighted arithmetic mean is 

necessary to estimate the population arithmetic mean 

and has been known to provide accurate abundance 

estimates. This is well illustrated ·in many papers 

on, for example, polygonal.methods of ore-reserve estimation. In some other types· of problems, weighted 

arithmetic means are required to correct for highly 

uneven (biased) sampling, such as in estimation of 

the abundance of elements in large parts of the. earth's 

crust. Some of the problems met in this work were 

reviewed by Fleischer and Chao ( 1960). The methods 

described in this paper .. are not intended . for problems 

where weighted arithmetic means are inor~ suitably 

used. 

I am grateful to N. C. Matalas and L. B. Riley of the 

U.S. Geological Survey for technical criticism and much 

helpful discussion. 

PROBLEMS IN ESTIMATION OF GEOCHEMICAL 

ABUNDANCE 

Problems in the estimation of abundance (population 

arithmetic means) arise in both sampling and computation~ but only· the computational problems are considered il}.. this paper. The more common circumstances 

leading to ~omp:utational problems result from (1) limited ranges of analytical sensitivity,· (2) small groups 

of data (small n), and ( 3) data reported in geometric 

classes. vVhere none of these circumstances cause diffi-

. culty, ·abundance estimates derived from the ordinary 

expression for the arithmetic mean, 

- 1 

X=-~X n , (1) 

· will tend to be correct and will be at least relatively 

efficient. An efficient estimate, in the statistical sense, 

is one that has a small variance (Fisher, 1950, p. 12), or 

one that would be expected· to change little with the 

addition of new data. Concern for the efficiency of statistical estimates gave rise to the "maximum-likelihood 

method'' used in devising estimation methods (Fisher, 

1950, p. 14). Estimators derived in this manner are 

said· to yield statistics with minimum variance (maximum efficiency). The expression for the arithmetic 

mean in equation 1 is a maximum-likelihood ~stimator 

where the data are derived from a normally distributed 

. population; it 'is not a maximum-likelihood estimator 

for many of the problems in geochemical studies: 

In problems where data from a normally distributed 

population. are censored (some concentration values fall 

beyond . ~he range of sensitivity for the analytical 

·method), a maximum-likelihood method described by 

·Cohen (1959, 1961) may be used. Where the data are 

from a lognormal population, a maximum-likelihood 

. method given by Sichel (1952) will be applicable. A 

modification of the method given by Sichel will be useful where the· data indicate certain kinds of departure 

from the. lognormal (I{rige, 1960). 

Some estimation methods derived by the method of 

· maximum likelihood ·are slightly biased for small n. 

Unbiased likelihood estimators are those that have been 

corrected for the bias, where such correction is needed. 

Abundance estimates derived by maximum.,likelihood 

techniques will not differ greatly, in many instances, 

from estimates derived by other reasonable though less 

precise methods. In much geochemical work, especially 

·that c6nducted on a large scale, errors due to sampling 

·and aiuilysis are overwhelmingly dominant in determination of the total estimation error; computational 

errors are relatively small. Nevertheless, the possibility 

of large errors from other sources cannot serve to justify 

additional errors where they can be easily avoided. 
METHODS OF COMPUTATION FOR ESTIMATING GEOCHEMICAL ABUNDANCE B3 

ANALYTICAL SENSITIVITY . 

Every analytical method has limits of sensitivity beyond which it is ineffective· for determination of concentration valves. These limits may occur at both .the 

lower arid the upper bounds· for a concentration r3:nge. 

~he spectrographic method described by Myers, 

I-Iavens, and Dunton (1961), .for example, may be used 

to determine silicon within the range from 0.002 to 

10 perce~t. Concentrations judged. to be lower or 

hiO'her than this range are reported as < 0.002 percent 

or

0

>10 ·percent, respectively. Thus, the spectrographic 

data may be either left or right censored. Th~ term 

"censored" is applicable here because the number of 

values beyond each sensitivity limit is known for any 

set· ·of analyzed rock samples. In other types of problems, where the number of values beyond certain limits 

is unknown, the data are said to be truncated (Cohen, 

1959,p.217). . . 

A number of methods have been used by indivi.dual 

geologists for computing means from censored data. 

(See, for example, Miesch, 1963, p. 21-23). Other 

methods that are available, however, are a great deal 

more satisfactory (Cohen, 1959, 1961; Hald, 1952, p. 

30 · I-Iubaux and Smiriga-Snoeck, 1964). The maxinn~m-likelihood method described by Cohen will be 

applied to abundance· estimation problems in later 

sections. . 

Abundance estimates can be obtained directly from 

censored normal distributions, using Cohen's method, 

but not from other types of censored distributions. 

Where ·the analytical data ~o not indicate a norm~l distl:ibution, some transformation o~ the da•ta can be used, 

as will be demonstrated in a following section. The 

mean and varl"ance of the transformed values, in many 

problems, can then be used ~o derive abundance 

estimates. 

SMALL GROUPS OF DATA 

The precision, or reproducibility, of any statistical 

estim:a.te is larcrely dependent on the number of values 

on which 'the e;timate is based. Those estimates der~ved · 

from large data sets will be relatively precise, even 

w hei·e the statistical irtethod is not the most ·efficient one 

that could he u'sed (where the statistical method has .not 

been derived by the method of maximum likelihood). 

Where the data set. is small, however, an unbiased max.:. 

imum-likelihood technique should be used whereve~ possible if the ·precis! on· of the statistical estimate · is 

important. It is not possible here to state a value .of n 

which will serve to distinguish-between large and··smal~ 

data. sets because •this varue will vary according 'to the 

variation in the data. A better approach may. be to 

accept the inaximum-likelihood estimate, if obtainable, 

wherever it differs from an estimate derived by other 

techniques. . · . · .. 

Where the druta are derived from a normal d1str1bution, an estimate of abundance derived from ·equation 1 

is based on the method of maximum likelihood and is 

the most efficient estimate of abundance possible for a 

given number of samples .. However, most underlying 

frequency distributions in geochemical studies are _not 

normal; more commonly they· are ( 1) :asymmetrical 

wi-th a long tail toward high values (positive skewness), 

(2) asymmetrical with ·a long tail toward low values 

(negative skewness), or (3) multimodal with more than 

one peak in the frequency distribution curve. Where 

skewness or the presence of more than one mode is suffi-

' cient to indicate a departure of the underlying frequency .distribution from the normal form,· abundance 

estimates derived from equation 1 will not be as .efficient 

as maximum-likelihood estim·ates, if the latter ~r.e available. The difference in efficiency may be large if·n is 

sm~l · 

The most common departure of geochemical data 

from the normal distribution is a positive skei.vness. In· 

·many frequency distributions the ·skewness, ·along with 

· other properties of the distribution, may reflect an 

· underlying lognormal distribution. ·Where ·this is true~ 

an unbiased maximum-likelihood method for deriving· 

efficient estimates of abundance (the population arithmetic mean) can be applied (Sichel, 1952). : Where the 

. positive skewness in a particular data set is less or 

· greater than that ·which can be ascribed to a lognormal 

distribution, a modification of this method may be used 

(l(rige, 1960). The ·methods of Sichel arid' Krige 

:involve data transformations. · 

Unbiased maximum-likelihood methods of abundance 

. estimation applicable to 'frequency distributions which 

. are negatively skewed or multimodal are unknown to 

the writer. · · 

GEOMETRIC CLASSES 

Any quantitative analytical determination may be 

regarded as a reported range or class; a concentration 

reported as (8.62 percent, for. example, signifies the 

range fro~ 78.615 to 7.8.625 percent as the analyst's best 

· estimate of the true value. The classes are broader 

· where fewer significant figures ·are reported. ~ere 

the values are reported to an equal number of .deQimal 

. places, the class widths are equal; the class boundaries' 

are arithmetic, because they increase by a _constant increment. . . . . . . 

In much spectrographic· and colorimetric w9r~ t~e 

: a~alyst reports concentration yalues in broad cla~ses,. 

: without specifying ~ingl_e val)leS (except where t~ey are· 

meant only to identify . a cla"ss). Classes are used for· 
B4 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

reporting because of relatively poor discriminatory 

capacity of the· analytical methods. Generally, the· 

boundaries of the classes increase geometrically, each 

being higher than the previous boundary by a constant 

multiplier. Geometric classes are used because the 

error variance is at least approximately proportional 

to the amount of the constituent present. 

.An example of the geometric classes used in spectrographic work was given by Myers, Havens, and Dunton 

(1961, p. 217), who formerly reported the concentrations of 68 elements in classes having the boundaries0.00010, 0.00022, 0.00046, 0.0010, 0.0022, . . .-increasing by a factor equal to the cube root of 10, up to 10 

percent. Myers and other U.S. Geological Survey 

spectrographers currently report values in classes with 

boundaries increasing by a factor equal to the 6th root 

of 10 ( 0.00012, 0.00018, 0.00026, 0.00038, 0.00056, 0.00083, 

0.0012, . . . ) . Other similar systems of reporting, 

using various other factors for the generation of class 

boundaries, were described by Barnett ( 1961, p. 184). 

Although the reporting of spectrographic analyses 

in geometric classes has allowed spectrographers to produce a vast amount of data useful in geochemical problems, the practice has presented some difficulties in statistical analysis that have not, so far, been satisfactorily 

resolved. Because the class boundaries form a geometric progression, the class widths are unequal in size, 

and conventional methods of treating grouped data are 

awkward and less efficient than other methods that 

might be used. There is also the problem of choosing 

the best midpoint to represent a frequency class in the 

grouped-data computations. 'Where a number of concentration values are reported to occur within the range 

from 0.0046 to 0.010 percent, for example, it would seem 

proper to use the arithmetic midpoint of 0.0073, rather 

than the geometric midpoint of 0.0068, for computing 

the sample arithmetic mean for the whole data set. The 

use of the geometric midpoint, however, nearly always 

gives better answers. (An example is given later in this 

paper.) The reason for this is that the use of the lower 

value partly compensates for a positive bias caused by 

the grouping technique, but the compensation is without any sound theoretical basis and should not be relied on. 

'When a logarithmic transformation of the class 

boundaries is made, the class widths (in log units) become equal, thus allowing straightforward use of 

grouped-data computational methods for deriving the 

mean logarithm and standard deviation of the logs. By 

using these values, the abundance (population arithmetic mean) can be estimated according to the maximum-likelihood method of Sichel (1952), if the form 

of the underlying frequency distribution is at least approximately lognormal. 

ANALYTICAL DISCRIMINATION 

For many types of spectrographic or colorimetric 

analytical techniques the analyst is unable to estimate 

concentrations to more than one significant figure. If 

a number of concentrations of the constituent sought 

are similar among the samples analyzed, the suitability 

of the technique for discriminating among the samples 

is poor. Commonly, a large proportion of the determinations are reported as the same value. For example, 

Huff (1955, p. 111) gave copper determinations from 

a chromatographic field technique on 23 samples of 

Tapeats Sandstone (Cambrian) from near Jerome, 

Ariz. Seventeen of the 23 determinations are given as 

. 10 ppm (parts per million); the remaining 6 are either 

50, 100, or 150 ppm. Many other examples of poor 

analytical discrimination could be given involving data 

reported in geometric classes. In many instances only 

three or four adjacent classes are used in reporting, and 

commonly 20-50 percent of the concentrations occur 

within. one class. Discrimination among values in the 

same class, of course, is impossible. The proportion of 

values reported in a single class is dependent on the 

class widths, the variation in the concentrations, and 

. the form of the frequency distribution. 

As has been pointed out, some of the computational 

problems encountered in abundance estimation can be 

wholly or partly resolved by means of data transforma-

. tions. Where analytical discrimination is poor, however, some of the more useful transformations are 

impossible or ineffective. No transformation would be 

• useful, for example, in normalizing Huff's data referred 

·to ·a:bove, and transformations other than the logarith-

. mic transformation would be ineffective in treating 

· most data reported in broad geometric classes. 

RECOMMENDED TECHNIQUES 

Where large sets of uncensored data are a vail able, 

abundances may be estimated directly by using the con-

: ventional formula for arithmetic means in equation 1. 

. If the data are part of an underlying normal distribution of values, the abundance estimate will be the most 

efficient possible. If the underlying distribution is not· 

normal, it may be possible to obtain a more efficient 

estimate by other techniques. However, if n, the number of values, is large, the increase in efficiency may be 

. small. 

'Where sm'all sets of uncensored data are used, a bundance estimates derived from the conventional formula 

for the arithmetic mean will be as efficient as possible 

if the underlying frequency distribution is normal. 
METHODS OF COMPUTATION FOR ESTIMATING GEOCHEMICAL ABUNDANCE B5 

Where the distribution is positively skewed, abundance 

estimates that are significantly more efficient may be 

obtained by other methods. These methods begin with 

data transformations. 

Where the data are censored, abundances may be estiInated using the teclmiques available for estimating the 

population arithmetic mean, if the total underlying distribution is normal. Where it is not normal, data 

transformations may be employed. The mean and 

standard deviation of the transformed values may then 

be used to derive abundance estimates. 

T~NSFOR~TIONS 

1\{any types of data transformations have been used 

in geologic and geochemical problems to normalize observed frequency distributions, but the types that appear more useful in problems of estimating geochemical 

abundance are the log y and log ( y +a) transformations, where y is the concentration of the element, in 

percent or in parts per million, and a is a constant. 

(All logarithms used in this paper are to the base 10.) 

Data which can be transformed to the normal form by 

these functions are referred to as 2- and 3-parameter 

lognormal, respectively (Aitchison and Brown, 1957, 

p. 7, 14). . 

These two transformations generally appear to be 

effective where the distribution of the original analytical data is unimodal and posi~tively skewed. The simple log transformation of some positively skewed data 

will lead to a log distribution with negative skewness. 

For other data (fig. 1E) the log transformation will 

lead to a distribution with some degree of positive skewness remaining (fig. 1F). In either case the constant, 

a, can be estimated using techniques described by Krige 

(1960, p. 236) and used in the transformation log 

(y+a). Where a negative skewness of logs is to be 

corrected, a will be a positive value; where a positive 

skewness in the logs is to be corrected, a will be negative. 

In some cases the absolute quantity of the derived 

negative value of a will equal or exceed some of the 

concentration values, y, so that the quantity (y+a) is 

zero or negative and log (y+a) is undefined for some 

analytical values. Transformed distributions that result are, in effect, censored (fig. 1G), and abundance 

estimates may be obtained by using techniques for 

censored distributions. 

The choice of a transformation that may be effective 

in producing a normal, or at least symmetrical, frequency distribution can be governed by examination and 

testing of the data available or by previous experience 

with similar data. Selections based on the data avail240-308--67----2 

able are generally preferred, though this is not always 

possible when a group of data is small or when a large 

proportion of the data is censored. Many data pertaining to the concentration of minor elements in rock samples collected over l·arge areas correspond more closely 

to the lognormal form than to the normal, and the log 

transformation is appropriate in a large number of 

problems. The log (y+a) transformation can frequently he used to improve on the simple log transformation if a satisfactory estimate of a can be obtained. 

When the data have been transformed by a:= logy or 

w=log (y+a), the mean of w is estimated from equation 1 and the standard deviation is estimated from 

equation 2. The mean and standard deviation of 

these transformed values are of little value themselves 

in problems of abundance estimation, but they may be 

used to derive estimates of arithmetic means by techniques developed and described by Sichel ( 1952) and 

l(rige (1960). 

s=[~x2~n,X2Jtt2 (2) 

Equation 2 is a biased estimator of the population standard deviation but is used in this form in computations 

described 'later in the paper. Where unbiased estimates 

of standard deviation are needed, n in the denominator 

of equation 2 is replaced by n-1. 

COHEN'S METHOD FOR CENSORED DISTRIBUTIONS 

Cohen (1959, 1961) presented maximum-likelihood 

techniques for estimation of the mean and standard 

deviation from either censored or truncated normal distributions. These techniques may be used whether the 

unknown part of the distribution is in the low- or 

the high-value region (left- or right-censored, respectively). Because left-censored distributions are by far 

the more common in geochemical problems, only the 

part of Cohen's techniques applicable to these distributions will be discussed here. Although Cohen's methods 

are strictly applicable to normal distributions only, 

it can easily be demonstrated that the method for leftcensored distributions provides satisfactory estimates of 

arithmetic means wherever the total distributions are 

symmetrical about one mode. The method given by 

Cohen has not been corrected for a small bias, and other 

methods for treating censored distributions may be 

more accurate when n is less than about 10 (Cohen, 

1959, p. 218). 

The methods given by Cohen. (1959, 1961) are preferred to others given by Hald (1952) and Hubaux and 

Smiriga-Sn<;>eck ( 1964) only because of the simplicity 

of computatiOn. 
B6 

30 

t 20 z 

LLJ 

:::> 

0 

LLJ 

ff: 10 

STATISTICAL STUDIES IN FIELD GEOCHE:MISTRY 

0.2 

A 

0.3 0.4 0.5 

MoS2, I~ PERCENT 

0.6 0.7 

URANIUM, IN PARTS PER MILLION 

E 

n =58 

4 .6 8 10 

ARSENIC, IN PARTS PER IV)ILLION 

-0.4 

40 

30 

20 

10 

-11/3 -2/3 0 2/3 

LOG IR,ON, IN PERCENT 

D 

n =185 

-0.2 0 0.2 0.4 0.6 0.8 1.0 

LOG URANIUM, IN PARTS PER MILLION 

0 0.4 0.8 

LOG ARSENIC, IN PARTS 

PER MILLION 

~ 

f,l 

C: 

1J 

~ ·0 

(/) 

c 

Q) 

(.) 

G 

n =58 

L£..:-UL..~c..£..£..~.£.L.L~ 

-1.o· ...:...0.2 o.6 

LOG (ARSENIC, IN PARTS 

PER MILLION,-0.6> 

1.2 

FIGURE I.-Frequency distributions. A., MoS2 values for 300-degree samples at 2-foot intervals in drill core from 

Climax, Colo. (from Hazen and Berkenhotter, 1962, p . .S4-86). B, Iron in sandstones (from Miesch, 1963, pl. 3). 

0 and D, "Uranium in granite a deux micas de Pontivy" (from Hubaux and Smiriga-Snoeck, 1964, p. 1207). 

E, F, and G, Arsenic in basalts and diabases (from Onishi and Sandell, 1955, tables 5, 10). 
METHODS OF COMPUTATION FOR ESTIMATING GEOCHEMICAL ABUNDANCE B7 

In Cohen's technique X0 is taken as either the transformed value of the lower limit of analytical sensitivity 

or as the limit of sensitivity (in percent or in parts per 

million) if the data transformation is not required. 

The quantity n' is the number of concentration values 

that are below the limit of sensitivity; such concentrations are generally reported by the analyst as "not 

detected" or zero. The ratio h=n' fn is the fraction of 

the total number of analyzed specimens in which the 

element was not detected. The mean and standard 

deviation of the analytical values above the limit of 

sensitivity are computed as: 

and 

-

X'=-k....l ""'x __ 

n-n' 

~x2 ]112 s'= ---(x')2 • n-n' 

(3) 

(4) 

The mean and standard deviation of the entire distribution are then estimated from: 

and 

1.9 

1.8 

1.7 

1.6 

·1.5 

1.4 

1.3 

1.2 

1.1 

1.0 

0.9 

0.8 

" - (- ) J..I.=X'-X X'-.X0 

u=[ (s')2+X(x' -Xo)2]112. 

(5) 

(6) 

The value X in equations 5 and 6 is a function of hand 

the quantity (s')2/(x' -x0 ) 2• Values of X for 0.01::; h 

::; 0.90 and 0.00::; (s')2/(x' -x0 ) 2::; 1.00 were tabulated 

by Cohen (1961, p. 538), and graphs of X were given in 

an earlier paper (Cohen, 1959, p. 231). The graphs 

are reproduced here in figure 2, with Dr. Cohen's 

permission. 

t AND ta. ESTIMATORS OF SICHEL AND KRIGE 

'¥here the original analytical data have been used in 

computation without transformations, the derived values of x ( eq 1) or J'j. ( eq 5) may be taken directly as 

estimates of geochemical abundance. However, when ro 

or P.· are derived for w=log y or m=log (y+a, abundances can be estimated using methods described by 

Sichel ( 1952) and Krige ( 1960). The method by Sichel 

is an unbiased maximum-likelihood technique; that by 

1\::rige is a modification of Sichel's method, to be used 

where the log (y+a) transformation is required. The 

estimators, where the x=log y and x=log (y+a) 

transformations are used, are referred to as t and ta, 

respectively, and should not be confused with Student's 

t, which has wide application in statistical procedures. 

1.9 

1.8 

1.7 

1.5 

1.4 

1.3 

1.2 

1.1 

( s') 2;(J(! xo>2 

FIGURE 2.-Curves for estimating X for equations 5 and 6 (from Cohen, 1959, p. 231; reproduced with 

author's permission). 
B8 STATISTICAL STUDIES IN FIELD GEQCHEMISTRY 

The t and ta statistics provide useful estimates o:f the 

ar~thmetic mean and geochemical abundance when the 

distribution o:f log y or log (y +a), respectively, approximates the normal distribution :form. 

Where log y is normally distributed: 

or ] ' (7) 

where x and ~ are the arithmetic means of log y estimated from complete (eq 1) and censored (eq 5) 

distributions, respectively. The factor , is a function 

of n, the total number of specimens analyzed and either 

the antilog of 8 ( eq 2) or of ~ ( eq 6), depending on 

whether the distribution is complete or censored. 

Values of , may be read from graphs in figure 3; for 

more exact work the reader is referred to the· original 

equation and tables (Sichel, 1952, p. 275, 284-288) 

from which equation 7 and the graphs in figure 3 were 

derived. Simplified equations and tables were given 

recently by Sichel (1966). Other equations and tables 

that are mathematically equivalent to those of Sichel 

were given by Aitchison and Brown (1957, p. 45, 

156-158) and were based on the work of Finney (1941). 

Where log (y+ a) is normally distributed: 

or 

6 

5 

T4 

3 

2 

ta=(-rX10'Z)-a 

n= number of analyses 

s =standard deviation (equation 2) 

a-= standard deviation (equation 6) 

} ' (8) 

1 ~~~~~~~~~~~~~~~~~~~~ 

1 2 4 5 6 7 

10 5 or 10b 

FIGURE 3.-Grapbs ofT as a function of the number of analyses, 

n, and the antilog of s or cJ (based on tables by Sichel, 1952). 

where x and ~ are the arithmetic means of log (y+a) 

estimated from complete (eq 1) and censored (eq 5) 

distributions, respectively. The equations in 8 are 

modified forms of one given by Krige (1960, p. 239). 

The factor , is a function of n, the total number of 

specimens analyzed, and of the standard deviation 

of the transformed values, log (y+a). If complete 

distributions are used, the standard deviation is estimated by 8 (eq 2); if censored distributions are used, 

it is estimated by~ (eq 6). The value of, can be read 

from figure 3 by using n and the antilog of either 8 

or ~ for the transformed values. For more exact 

computation the reader is referred to Krige (1960, 

p. 239) and tables given by Sichel (1952, p. 284-288), 

or to Sichel (1966). 

CONFIDENCE INTERVALS 

Estimates of geochemical abundance should be accompanied by estimates o:f the confidence intervals wherever 

possible. The method for estimating this interval about 

the arithmetic mean derived from equation 1 is well 

known (see Dixon and Massey, 1957, p. 128) and leads 

to approximately correct intervals even where the distributions are not normal. Cohen (1959, p. 231-232) 

gave methods for estimating confidence intervals of p. 

derived from censored distributions. Appr:oximate 

confidence intervals for the t and ta, may be estimated 

by using equations given by Aitchison and Brown 

(1957, p. 46) ; more exact confidence intervals can be 

obtained by using methods given recently by Sichel 

(1966). 

SUMMARY OF METHODS 

A flow chart indicating recommended computational 

procedures :for estimating geochemical abundances is 

given in figure 4. With large groups of uncensored 

data, abundances may be estimated as arithmetic means 

in the conventional manner, using equation 1. However, where means obtained from equation 1 differ :from 

those obtained from procedures indicated in figure 4, 

the means obtained from the latter procedures should 

be preferred. The higher precision of the t and ta estimators have been adequately demonstrated by Sichel 

( 1952, p. 276-278) and Krige ( 1960, p. 242-244), respectively, and the advantages of treating censored distributions by the statistical method of Cohen (1959, 1961) 

over the approximate methods used in the past will be 

obvious. 

The methods of Sichel ( 1952), Krige ( 1960), and 

Cohen ( 1959, 1961) are individually inadequate :for a 

large number of computational problems met in analyzing geochemical data. Sichel's t estimator is useful 

where the data correspond to the lognormal frequency 

distribution and are uncensored. Krige's ta estimator 
[1] START J 

Data are in 

arithmetic classes 

[2) Examination of 

frequency distribution of t---

metal val ues<y) 

Frequency distribution is symmetrical 

~ I Let x =y [3] 

~ Frequency distribution ~lUI"' 

of xis nonnormal 

r Data are in geo- ~ [7] Plot frequency 

distribution on Censored J--

1 metric classes distribution log scale (x) 

L.._ (5] Frequency .distribution of x is approximately nor- I---

r---+ 

f---..+ 

r---

y 

Frequency distribution mal is unimodal and H [4] Transform 

positively skewed; x =logy Frequency distribution 

analytical discrimina- of x is nonnormal 

tion is satisfactory ~ 

J [6] Transform J 

Frequency distribution 1 x=log <y+a) 

is unimodal and positively skewed; ana- f----. STOP Frequency distribution of lytic a I discrimination xis approximately is poor normal, but x is un-

-- defined for some values of Y 

Frequency distribution 

multi modal and Frequency distribution of I---STOP is 

positively skewed x is approximately 

f-- normal, and x is defined for all known 

values of y 

Freqency distribution STOP Frequency distribution is negatively skewed 

of·x is nonnormal 

-r Frequency distribution I STOP 

is nonnormal r 

Censored ~ A • ) • 

distribution y J.& (eq~at1on 5 .•s an 

Frequency distribution 1-- unboased .estomate 

Complete 

distribution 

is approximately normal of abundance 

x·(equation 1) IS an 1-------------------J unbiased estimate 

of abundance 

Complete }-

distribution 

~ distribution Censored ~ 

c-y distribution 

Complete ~ 

STOP 

FIGURE 4.-Methods of computation for estimating geochemical abundance from small groups of data. 

Compute ~ and b 

of logy from 

e· q u at i o n s 5 

and 6. 

Compute x and s of 

Jog y from equations 1 and 2 

t (equation 7) is 

an unbiased 

estimate of 

abundance 

Compute A and fT 

of log ( y +a) 

from equations 

5 and 6 

Compute x and s of log ( y + a: ) 

from equations 

1 and 2 

t

4 (equation 8)is 

an unbiased 

estimate of 

abundance 

tr-

...._: 

...._ 

f. 

a:: 

trJ 

8 

s 

t::1 

UJ 

0 

'%J 

n 

0 

~ Cl 

0 

~ z 

'%J 

0 

~ 

trJ 

~ 

~ 

~ 0 

0 

trJ 

0 n· 

t:Q 

trJ 

a:: 

1-4 

n 

~ 

~ Cl 

z 

t::1 

~ trJ 

t::d 

<:0 
BlO ,. STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

is applicable to a wider variety of distribution types, 

but, as with the t estimator, the data must be uncensored. · 

Moreover, with highly skewed frequency distributions, 

the log (y+a) transforn~ation may impose censoring 

of data which are completely above the lower sensitivity 

limit of the analytical method. Cohen's method is 

directly applicable to data which approximate a normal 

distribution, but data of this type are not common in 

minor-element studies pertaining to areas larger than a 

single mine or small outcrop. Generally the required 

normal distribution can be achieved only through a 

data transformation; estimates of the mean of the transformed variate· are not valid estimates of abundance 

but may be transformed to valid abundance estimates 

using the meth6.ds of Sichel or Krige, if the necessary 

data transfonnation is accomplished by 'log y or log 

(y +a). The ti•ansformations are, indeed, sufficient in 

a wide variety of geochemical studies. 

Some properties. of small data sets prohibit the estimation of precise arithmetic means, or abundances. 

These are indicated by STOP. signs on· the flow chart 

in figure 4, and are as foll<;>ws: 

1. A censored frequency distribution which, though 

believed to be part of a symmetrical distribution, 

departs widely from .the norma] form (for example, some censored multimodal :distributions). . 

2. A frequencj,.distribution:whic,h is markedly asymmetrical and cannot be transformed to an approximate normal distribution by log y or log ( y +a). 

This particularly includes multimodal skewed dis.: 

tributions ~nd all negatively skewed distributions. 

3. A frequency distribution that is unimodal and positively skew'ed but is based on data that cannot be 

normalized: by the logy or log (y+a) transformations owing to poor analyticaldiscrimination or 

other factors. · 

4. Data reported in broad geometric cla.sses, but not 

approximately normal on a log scale. 

Where these properties are present, the conventional 

method of estimating the population arithmetic mean 

( eq 1) may be the only way .readily available to the 

geologist for estimating geochemical abundance. 

The use of the flow chart, figure 4, is demonstrated 

in the following section ·by employing ex·amples of data 

from the literature. 

EXAMPLES 

Four data sets were selected from the· literature to 

illustrate use of the recomm.ended methods for widely 

differing types of data. ·One data set is approximately 

normally distributed, two are approximately lognormal 

(one of these is reported in geometric classes) , and a 

fourth set is approximat¢ly l9gnormal after adjustment 

by a constant, a. However, except for the data in geometric classes, none of the data sets indicate a close 

correspondence to the normal or lognormal form; this 

selection has been intentional to demonstrate that the 

di,stribution requirement is not rigid .. 

·As shown by these and other examples, the principal 

requirement is that the distributions be unimodal and 

approximately symmetrical on any of the three scalesthe scale of original measurement, y, or one of the two 

transformed scales, logy or log (y+a). 

The number of values in each of the four data sets 

used as examples (fig. 1) is large compared with the 

number available in many geochemical problems. Most 

of the estimated abundances, therefore, agree fairly well 

with the abundances derived using the conventional 

method for estimating the population arithmetic mean. 

Large data sets were used so that this comparison could 

be made to verify the accuracy of the techniques for use 

in actual abundance estimation problems where n is 

small or where the data are censored. 

MOLYBDENUM SULFIDE .IN DRILL CORE 

The first set of data is from Hazen and Berkenhotter 

( 1962, p. 84--86). The assays, of percent MoS2, were 

made on samples of drill core ( 300-degree segments) 

from the Climax Molybdenum mine, Lake County, Colo. 

We shall assume, for purposes of illustration, that the 

assays are an objective and unbiased sample of those 

that might have been obtained from the block of ore 

penetrated by the drill hole (that is, there is no sampling problem). The frequency distribution of the 

original assays is represented by histogram A in figure 

1 and by the probability graph for distribution A. in 

figure 5. Both illustrations indicate that the frequency 

distribution of assays (at least the central part) is approximately symmetrical. The distribution was arbitrarily censored successively at three points indicated 

by the small arrows (fig. 1A.), and four abundance estimates were made using the complete and the partial 

data sets. The estimates, with intermediate computational values, are given in table 1. 

In reference to figure 4 : 

[1] The MoS2 assays are in arithmetic classes. Proceed to [2]. 

[2] The frequency distribution of MoS2 assays, y 

is approximately symmetrical (figs. 1A., '5A.). 

Proceed to [3]. 

[3] If the complete distribution is used, the abundance 

. of MoS2 in the ore block is estimated, lby the conventional method in equation 1, to be 0.359 percent. 

If only assays equal to or greater than the arbitrary 

(or hypothetical) analytical · cutofl:'s-0.25, 0.35, 
METHODS OF COMPUTATION FOR ESTIMATING GEOCHEMICAL ABUNDANCE Bll 

and 0.40 percent-are used (table 1, col. 2), the 

quantities indicated in columns 6-14 (table 1) 

are derived from the indicated equation~, and 

the successive abundance estimates are derived 

as shown in column 17. The abundance estimates are in 'agreement to two figures, even 

though as much as 63 percent of the data was 

arbitrarily censored (table 1, col. 8). 

99~-----------------------------------*--------------------~ 

98r---------------------------~~-----7~~~~ 

~ 95~----------------------7=----~~-r~--------~ w 

~ 90~--------------~~~----~------------~ w 

Q. 

~ 80~------------~~--------~----------------~ 

~ 70~-----------+~~-----&------------------~ 

~ 60~------~~-+------~----------------------~ :J 

·~50~--------~--~------~~------------------------~ 

~ 40~-------+--------7-----+-------------------------~ 

~ ~~----~~----~--~~--------------------~ 

~ 20~--~----~r---~----------------------~ :J 

~ 10~~-------------+----~----------------------------~ 

0 

A, MoS2. IN PERCENT 0 1 2 3 4 5 6 7 8 9 10 11. 12 13 

C, URANIUM, 'AND E. ARSENIC, 

IN PARTS PER MilLION 

99~r-r-~~~,-~~-r-r~~~~~~r-r-r 

98~--------------------------~--------------~ 

~95~------------------------~--~--------------~~ 

z 

~90~----------------~~-+------------~--~ 

0:: 

w 

Q. 80~----------------~~-+------------~~----~ 

z 

~70~------------------+---r-----------~-7~----~ 

~ 60~----------~~-r---------=~----~=-------~ 

~ 50~---------------r--r---------~~-,~~--------~ 

~ 40~--------------~-r--------~r----+--------~ 0:: 

~ 30~------~_,~- ~--~----~--------~ w 

~ 20~------~~-r---- ·~~----~----------~ s i 10~--~--~----- +-------L----------~ 

:J 

0 5~~~---------

2~~------------------------------------~ 

0.2 0.4 0.6 0.8 1.0 1.2 

D, LOG URANIUM, IN PARTS PER MILLION 

:-1 -2/3 -1/3 0 1/3 

'B, LOG IRON, IN PERCENT 

-1.0 -0.6 -0.2 0.2 0.6 1.0 

F, LOG ARSENIC, IN PARTS PER MILLION, 

AND G, LOG (ARSENIC, IN PARTS 

PER MILLION,-0.6) 

ll'IOURE 5.-Probabiiity graphs of frequency distributions .A.-G 

in figure 1. 

Had the data, in fact, been censored a,t the arbitrary 

cutoff values, it would have been necessary to judge the 

nature of the total frequency distribution from as little 

as 37 percent of the data oc.curring above the cutoffs. 

In this extreme example of data censoring, the judgment regarding the form of the frequency distribution 

could be made only on the basis of prior experience with 

similar data. Where the censored part of the distribution is minor (less than about one-third of the total distribution), ~the probability graphs may still be useful 

for this purpose. 

IRON IN SANDSTONE 

The iron concentration in 85 drill-core samples of 

quartzose sandstone from the Salt Wash Member of the 

1\{orrison Formation (Jurassic) in San Miguel County, 

Colo., are represented by l}istogram B in figure 1. The 

data are from Miesch ( 1963, pl. 3) and were obtained 

by means of a spectrographic method (Myers and others, 1961) wherein the results are reported in classes 

having the boundaries 0.046, 0.10, 0.22, 0.46, ... percent. The corresponding logarithms of the boundaries 

are -11fg, -1, -%, -,lh, . . . . The boundaries, in 

percentage concentration values, are at geometric intervals and increa:se by a factor of 2.15. The boundaries, 

in log values, increase by an increment of one-third. 

vVe shall -assume that the analyzed samples are an 

unbiased representation of the body of rock for which 

the geochemical abundance is to be estimated. 

The estimate of iron abundance in the sandstone, 

derived from the grouped data form of equation 1, is 

0.38 percent. The grouped-data form of equation 1 is: 

(9) 

where /i is the number of values in the ith class and w, 

is the class midpoint. The midpoints used were the geometric centers of the classes-the values 0.07, 0.15, 0.32, 

0.68, and 1.46 percent. There is little justification for 

accepting the value of 0.38 percent as a valid estimate of 

abundance other than the fact that it agrees closely wi,th 

the abundance of 0.37 percent derived with the theoretically justified t estimator. It is not expected .that equation 9, used in the manner described here (with 

geometric midpoints), will consistently lead to unbiased 

and efficient abundance estimates. The use of arithmetic 

midpoints (0.07, 0.16, 0.34, 0.73, and 1.58) leads to an 

abundance estimate of 0.40 percent. Although the estimate of 0.40 percent is not great]y different from that 

of 0.37, it does demonstrate the positive bias in the 

technique by which it was obtained. The positive bias 

is commonly much larger. 
B12 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

TABLE 1.-Summary of computations tor estimating geochemical abundance from data represented in histograms in ftgttre 1 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 

--- -----------x---------------- Hypo- n' (eq1) 8 (eq2) (8') 2 

~ (eq 5) ; (eq 6) 

EstlDistribution 1 thetical Transformation a Xo n' n h=-.- or x' or 8' -- >.(fig. 2) t (eq 7) ta (eqS) mated 

analytical n (eq 3) (eq 4) (x'-x.)' a buncutoff dance 

Molybdenum sul~de (percent) 

A ________________ ------------ None.----------------- -------- -------- 0 101 

A---------------- 0. 25 _____ do _________________ -------- 0. 25 20 101 

A----------------- . 35 _____ dO----------------- -------- . 35 48 101 

A________________ . 40 .•••• dO------------------------- • 40. 64 101 

0 0.359 

.20 .396 

.48 .445 

.63 .475 

0

: &~~ ---o~39- ---o~27- --o~35ii- --o~iis- ======== ======== . 072 • 58 . 94 . 356 .117 -------- -------- • 066 • 77 1. 55 . 359 .114 -------- --------

0.359 

.356 

.356 

.359 

Iron (percent) 

B----------------- ------------ x=log 71--------------- -------- -------- 0 85 B.---------------- .10 ..... dO----------------- -------- -1. 000 5 85 B.---------------- , 22 ....• dO----------------- -------- -. 667 30 85 B----------------- . 46 _____ dO----------------- -------- -. 333 65 85 

0 -.543 • 06 -.504 .35 -.355 • 76 -.100 

. 314 -------- -------- -------- -------- .281 .34 .08 -.544 .313 

.209 .45 .59 -.539 .318 • 133 • 33 ""2. 04 -. 576 • 359 

• 37 -------- .37 

• 38 -------- .37 --------

.37 

.37 

.38 

.37 

Uranium (ppm) 

0----------------- ------------ None __________________ -------- -------- ('_________________ 2. 6 ....• dO----------------- -------- 2. 6 c_________________ 4. 0 _____ do _________________ -------- 4. 0 

g================= --------2:6- -~~~~~~---~============= ======== ---:415- c_________________ 4. o _____ do_________________ ________ • 602 

0----------------- 5.1 _____ do _________________ -------- • 708 

0 185 23 185 

80 185 

0 185 

23 185 

80 '185 

135 185 

0 

.12 

.43 

0 

.12 

.43 

• 73 

4. 53 

4.89 

5. 78 

. 612 

.660 

. 742 

. 845 

~: ~ ----:so- ---·:~s· --4:48-- --2.-26-- ======== ======== 2. 03 1. 30 • 94 4.11 2. 66 -------- --------

: ~~~ ----:39- ----:iii- ---:ii2i- ---:isi- !: g: -------- .125 . 77 . 86 • 620 . 181 4. 54 -------- .105 . 58 2. 00 • 571 . 220 4. 24 --------

4.53 

4.48 

4.11 

4.54 

4.55 

4.54 

4.24 

Arsenic (ppm) 

E---------------- ------------ None __________________ ----------------

~================= ---------:7- -~~~~0~---~============= ======== ·:::155" F----------------- 1. o _____ do _________________ -------- • 000 

F----------------- 1. 6 _____ dO----------------- -------- , 204 

0----------------- . 7 x=log <v+a). --------- -0.6 -1.000 o_________________ 1. 0 _____ do_________________ -. 6 -. 398 

o_________________ 1. 6 _____ do_________________ -. 6 . 000 

0 

0 

8 

22 

42 

9 

22 

42 

·58 

58 

58 

58 

58 

58 

58 

58 

1 The letters in this column refer to frequency distributions shown in figs. 1 and 5. 

In reference to figure 4 : 

[1] The data are grouped in geometric classes. Proceed to [7]. 

[7] The frequency distribution, on a log scale, is shown 

in figure 1B. The probability graph for this distribution is given in figure 5. No important 

departure from the normal forin is indicated, if 

the log scale is used. Proceed to T 5]. 

[5] The abundance of iron in the sandstone, using 

Sichel's t estimator, is estimated to be 0.37 percent. If the data are censored at the arbitrary 

points, 0.10, 0.22, and 0.46 percent, and 6, 35, and 

76 percent of the data is effectively lost, the 

abundance estimates are 0.37, 0.38, and 0.37 percent, respectively (table 1, col. 17). 

If only druta values above the 0.46-percent class boundary (20 of the 85 values) are used, a large positive 

skewness is apparent from the fact that the median is 

between 0 and 0.46 percent-far below the central part 

of the total range of concentrations ( 0-2.2 percent) . 

Therefore, a log transformation could be judged to be 

an appropriate step toward achieving a distribution 

closer to the normal form. 

If the data had shown a significant departure from 

the lognormal form, a "STOP" sign would have been 

0 

0 

.14 

.38 

.72 

.14 

.38 

• 72 

1. 69 

. 096 

.152 

.257 

.488 

-.208 

• 032 

.375 

1. 78 

.299 

.283 

.267 

.244 

.510 

.377 

.296 

-------- -------- -------- -------- 1. 58 -------- . 85 . 22 . 085 • 318 1. 58 -------- 1. 07 . 75 . 064 . 348 1. 59 -------- . 74 1. 96 -. 069 . 467 1. 51 -------- .41 • 20 -. 367 • 621 -------- 1. 77 • 76 . 72 -. 279 . 526 -------- 1. 68 • 62 1. 92 -. 345 • 598 -------- 1. 75 

1. 69 1. 58 1. 58 1. 59 1. 51 1. 77 l.li8 

1. 75 

encountered in figure 4. Because the data are in broad 

geometric classes, the log (y +a) transformation would 

have been awkward, 'and no method is available that· 

could have been used to obtain a precise estimate of 

abundance. 

URANIUM IN GRANITE 

The data represented in histogram 0 (fig. 1) were 

originally from Coulomb (1959), but were reproduced by Hubaux and Smiriga-Snoeck (1964, p. 1207) 

and used in testing a digital-computer method for d.eriving m·eans and standard deviations of censored dis-

• tributions. Hubaux and Smiriga-Snoeck derived mean 

iogarithms rather than arithmetic means or abundance 

estimates. The data represent uranium concentrations 

in samples from a homogeneous granitic massif; we 

shall assume that the sampling was unbiased. 

The frequency distribution of the original data (fig. 

10) exhibits a clear positive skewness, and the data, 

therefore, were transformed logarithmically (fig. lD) . 

The frequency distribution, as noted by Hubaux and 

Smiriga-Snoeck ( 1964, p. 1206), is only imperfectly 

log11ormal. 'fhe e~e~tiveness of the log transformation 

in bringing the data closer to the normal form can be 
METHODS OF COMPUTATION FOR ESTIMATING GEOCHEMICAL ABUNDANCE 

seen by comparing the probability graph for distribution 0 with that of distribution D (fig. 5). 

If altl. the data ( n= 185) are used, the geochen1ical 

abundance of uranium in the granitic massif is estimated, as the arithmetic mean ( eq. 1), to be 4.53 ppm 

(table 1, col. 17). 

In reference to figure 4 : 

[1] The uranium data are in aritlunetic classes. Proceed to [2]. · 

[2] t"rhe frequency distribution of uranium" analyses 

is unimodal and positively skewed; analytical 

discrilnination is satisfactory (figs. 10 and 50). 

Proceed to [ 4]. 

[4] The frequency distribution of the logarithms of 

the uranium analyses is approximately normal 

(distribution D; in figs. 1 and 5). Proceed to 

[5]. . 

[5] If the complete distribution is used (n=185), the 

abundance, derived with the t esti·mator of Sichel 

( 1952), is 4.54 ppm, virtually the same as the 

value, 4.53, derived by the conventional 

procedure. 

After censoring the data at the arbitrary analytical 

cutoffs 2.6, 4.0, and 5.1 ppm and effectively discarding 12, 43, and 73 percent of the data, respectively, the abundance estimates, as derived with 

the t estimator, are 4.55, 4.54, and 4.24 ppm 

(table 1, col. 17) ·. 

I-Iad the decision been made to use the :original data 

rather than to transform them logarithmically (that 

is, had the distribution represented by histogram 0 in 

figure 1 been judged approximately symmetrical), the 

computation method would have proceeded as in the 

previous example for MoS2 assays. Censoring the data, 

then, at 2.6 and 4.0 ppm would result in abundance estiJnates of 4.48 and 4.11 ppm, respectively (table 1, col. 

17). These are somewhat poorer than the corresponding values of 4.55 and 4.54 obtained by way of the t 

estimator, and the importance of the log transformation 

is apparent. 

The problem of judging, from censored data, whether 

or not the total concentration values (fig. 10) exhibit a 

symmetrical frequency distribution is not as difficult 

as for MoS2 assays in the previous example. Because 

only 27 percent of the uranium values are above 5.1 

ppm, for example, the median of all the data is known 

to occur between 0 and 5.1 ppm. As the data extend 

to more than 14 ppm some positive skewness is evident. 

However, judging the degree to which the log transformation corrects the skewness is more difficult. Where 

a large proportion of the data is censored but the remainder is sufficient to indicate definite positive skewness in the original data, the log transformation might 

be used as an approximation. Where a lesser proportion of the data is censored, the transformation using 

log (y+a) might be used if needed, and thereby offer 

a means for improving the accuracy of abundance 

estimates. 

ARSENIC IN BASALTS AND DIABASES 

The data with extreme positive skewness, represented 

by histogram E in figure 1, are from Onishi and Sandell 

(1955, tables 5, 10) and were used by Ahrens (1957, 

p. 207-209) in a discussion of frequency distributions 

of minor elements in igneous rocks. The data-arsenic 

determinations, in parts per million-were obtained on 

58 samples of basalt and diabase from widely separated 

localities in the United States and from localities in 

Japan and Sicily. For purposes of illustrating the 

computational techniques we shall assume that the 

samples adequately represent the rock bodies from 

which they were taken. 

The geochemical abundance of arsenic in the basalts 

and diabases, estimated by the conventional method 

(eq 1), is 1.69 ppm (table 1, col. 17). 

In reference to figure 4 : 

[1] The arsenic data are in arithmetic classes. Proceed to [2]. 

[2] The frequency distribution is unimodal and positively skewed; analytical discrirnination is satisfactory. Proceed to [4]. 

[4] The frequency distribution of the logarithms of 

the arsenic analyses (distribution F in figs. 1, 5) 

retains a definite positive skewness. Proceed to 

[6]. 

[6] If the methods described by 1\::rige (1960, p. 236) 

are used, the appropriate constant, a, is estimated 

to be -0.6. (I-Iad the skewness of distribution 

F in fig. 1 been negative, the estimated constant 

would have been a positive value.) Eight of 

the anaiytical values are equal to or less than 0.6, 

and as a is negative, th~ quantity log (y +a) for 

these values is undefined; the transforn1ed data 

(distribution G in fig. 1) are censored with 

h=%8=0.14. The probability graph for the 

newly transformed data, drawn using log 

(y-0.6), is shown in figure 5 (distribution G). 

Neither the histogram nor the probability graph 

indicates a large departure of the transformed 

data from the censored normal form. 

If Cohen's technique for censored distributions 

( eq 5, 6) and Krige's ta. estimator are used, the 

abundance estimate for arsenic in the basalts and 

diabases is 1.77 ppm (table 1, col. 17). Because 

the number of analyses in this example is small 

(n='=58) and the data are highly skewed, it may 
B14 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY 

be argued that the estimate of 1.77 ppm is better 

than the estimate of 1.69 derived from the conventional procedure. However, the two estimates are in at least fair agreement. 

When censoring the data at the arbitrary cutoffs, 

1.0 and 1.6 ppm, and effectively losing 38 and 

72 percent of the data, respectively (table 1, col. 

8), the abundance estimates derived with the ta 

estimator are 1.68 and 1.75 ppm (table 1, col. 17). 

I-Iad the data actually been censored below 1.6 ppm, 

estimation of the constant a may have been difficult, 

but some useful estimate might have been made unless 

the point of censoring was lower than about 1 ppm. 

Whether the point of censoring had been at either 0.7, 

1.0, or 1.6 ppm, a high positive skewness would have 

been apparent from the :fact that the median lies well 

below the central part of the known range of the data0-10 ppm. If the log transformation is used (without 

adjusting the data by the constant a) and an the data 

are used, the abundance estimate is 1.58 ppm. If only 

data equal to or greater than 0.7, 1.0, and 1.6 ppm are 

used, the abundance estimates are 1.58, 1.59, and 1.51 

ppm, respectively (table 1, col. 17). These estimates 

are notably poorer than those derived using the ta 

(rather than t) , estimator, but they are, nevertheless, 

sufficiently good :for many types o:f geochemical studies. 

CONCLUSIONS 

Computational methods given by Cohen (1959, 1961), 

Sichel ( 1952, 196.6), and Krige ( 1960) are useful in 

providing accurate and efficient estimates o:f geochemical abundance in many problems where the conventional 

method :for estimating population arithmetic means is 

not applicable (owing to censored data) or is inefficient (because of smrull data sets from nonnormal distributions). A combination o:f the methods may be 

useful where censol'ed data are :from an underlying 

frequency distribution that is nonnormal. Each of the 

methods requires that the data, or transformations o:f 

the data, be normally distributed. Two transformations that have proven satisfactory in much geochemical 

work are log y and log (y +a). Neither transformation 

is effective, however, where analytical discrimination 

has been poor. Moreover, selection o:f the appropriate 

transformation may be difficult where a large proportion of the distribution has been censored. 

The methods discussed are recommended :for esti-:-

mating geochemical abundance primarily because they 

are the most efficient methods available. Indeed, the 

methods developed by Sichel and Cohen are the most 

efficient possible where the :frequency distribution requirements are satisfied. Those developed by Sichel 

and Krige, moreover, have been corrected :for a small 

bias that exists where n is small. The method o:f 

Cohen does contain some bias where n is small, but the 

bias is probably much less than that introduced by 

most arbitrary methods used in handling the censoreddata problem. · 

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METHODS OF COMPUTATION FOR ESTIMATING GEOCHEMICAL ABUNDANCE B15 

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0 

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