Theory of Error in 

Geochemical Data

GEOLOGICAL SURVEY PROFESSIONAL PAPER 574-A
Theory of Error in 

Geochemical Data

By A. T. MIESCH

STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 574-A

A classification of geo chemical errors 

and discussion of their effects in 

data interpretation

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

WALTER J. HIGKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

First printing 1967 

Second printing 1970
CONTENTS

Abstract______..._._______._

Introduction. _________________

Basic concepts......______________

The population_____._____._ 

Restrictions due to limited access.

Other basic concepts. ___....__ 

Mathematical models and sampling error.

Face 

Al

1

2

2

2

4

5

Page

Special models for geochemical sampling-... A9

....-__ 10

.._._._. 13

____ 14

.____ 14

........ 15

.____ 16

Analytical error.... .

Effects of error types in statistical analysis.. 

Analysis of variance. 

Trend analysis........ 

Summary-. __ _ 

Literature cited ____ ...

ILLUSTRATIONS

Page 

FIGURE 1. Diagram showing directions of stepwise inferences from specimens to the target population.... _ .. A3

2. Diagram showing directions of stepwise inferences from the specimens to the target population where sampling

localities comprise subpopulations_____________._________... 6

3. Graphs of analytical determinations on prepared standards of detergent in water.. - - 12

STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

THEORY OF ERROR IN GEOCHEMICAL DATA

By A. T. MIESCH

ABSTRACT

All geochemical data accumulated in field studies directed at 

the determination of spatial variation in rock bodies contain 

error due to both sampling and analysis. Sampling error associated with a rock specimen is the compositional difference 

between the specimen and the part of the rock body that the 

specimen is intended to represent. Analytical error is the 

difference between the analysis and the concentration of the 

constituent in the total specimen. Error that is unbiased and 

of about equal variance from one sampling locality to another 

may not greatly hinder attempts at data interpretation; additional data will always serve to reduce the error effects. However, either sampling or analytical error may contain an overall 

bias (when the mean error does not tend toward zero), a variable 

bias (when bias is variable from one sampling locality to another), 

or variable precision (when the variance of the error differs from 

one locality to another). Errors of these types may significantly 

affect interpretation, and their relative importance is about the 

same whether the methods of interpretation are based on formal 

statistical procedures or on intuitive judgment.

Some types of error can commonly be avoided or controlled by 

taking suitable precautions in the field and laboratory. But 

where errors result from conditions of the rock body and its 

outcrop or from an inherent shortcoming of an analytical technique that must be used, they are unavoidable. Detrimental 

effects of some error types may be overcome by suitable data 

transformation prior to interpretation. Other types of error, if 

of sufficient magnitude, may invalidate any attempt at interpretation.

INTRODUCTION

Geochemical field studies are highly varied in scope 

and objectives, but they are most commonly designed 

to describe and interpret the compositional variation 

within bodies of rock, soil, or alluvium, or the compositional variation among different rock bodies. The 

distinction between these two types of problems is not 

necessary to a discussion of error, and here we need 

consider only the variation within a single rock body. 

Depending on how it is defined, this rock body may or 

may not contain sharp contacts or discontinuities that 

would allow its division into multiple rock bodies.

Any geochemical field study of this kind involves the 

problems of obtaining specimens to represent larger 

bodies of rock and of obtaining analyses to represent 

the specimens. Thus, two main kinds of error that

due to sampling and that due to analysis are always 

present, even though the importance of each may vary 

greatly from one study to another. Both the type 

and the magnitude of error should be considered; some 

types of error need not seriously affect data interpretation even if they are large. Other types can never be 

overcome and, if of sufficient magnitude, have serious 

effects on data interpretation.

A classification of types of error important in geochemical prospecting was described briefly in a previous 

paper (Miesch, 1964). The same classification as it 

applies to a broader class of geochemical field problems 

is presented here, along with some basic concepts that 

may be essential to understanding the nature of error. 

Geologists have always employed most of the concepts, 

but some geologists express them in terms of a rigid 

statistical framework within which experiments in field 

geochemistry are carried out, whereas other geologists 

view the statistical framework informally and conduct 

their experiments according to intuitive judgment. 

Actually, the concepts employed by the two groups are 

virtually the same. Moreover, there are few requirements demanded of data, and of data errors, for rigid 

statistical analysis that are not demanded otherwise, 

and it can be shown that sampling and analytical errors 

similarly affect statistical and nonstatistical interpretations of the data. The concepts and mathematical 

frameworks used in statistical experiments, and the 

requirements and assumptions regarding data errors, 

are not so different from those most geologists have been 

employing for many years without formal statistical 

analysis.

The principal theme of this report is in support of 

the notions presented in the preceding paragraph. 

The report aims to show that adoption of rigid experimental design techniques in field geochemistry 

does not necessarily require radical departure from 

methods and concepts now being used, but, instead, 

requires mainly a greater awareness of things we already 

know. Statistical principles are commonly stated for

Al
A2 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

the general case, and their application to individual 

problems is not always obvious. It is hoped that this 

discussion will be of some help in interpreting a few 

basic statistical principles in terms of the particular 

situations in field geochemistry. To avoid making a 

subject that is already seemingly complex even more 

so, the geologic and analytical problems used as 

examples will be kept simple, perhaps even trivial. 

The examples, however, will be ones that are familiar 

to nearly every field geochemist or petrologist.

Geochemical errors in general are departures of 

determined values from true or expected values or 

from values one is attempting to estimate; but such a 

departure does not necessarily imply that a mistake 

has been made in sampling or analysis. According to 

definition (Webster's Seventh New Collegiate Dictionary, 1963, p. 282), the term "error" "* * * may 

suggest an inaccuracy where accuracy is impossible." 

Error may result from mistakes or blunders in relatively 

rare cases. Thus, the degree or error in a particular 

geochemical study may be controlled by the nature of 

the study itself, and not necessarily by the care with 

which the study is conducted. Two geochemical 

studies that are identical in objectives and are conducted with identical degree of precaution in sampling 

and analysis may produce different types and magnitudes of error, depending on the natures of the two 

areas or rock bodies involved or on the methods of 

analysis used. In studies directed at rock bodies that 

are highly variable in composition, more sampling 

error must be expected than would be met in studies of 

relatively homogeneous rocks. Where rapid lowcost analytical methods are used, the errors from 

analysis are commonly greater than where more 

elaborate methods are used, but the high precision of 

the elaborate methods may be unnecessary, and of 

little advantage, in studies where sampling error is 

large.

Acknowledgments. I am grateful to Dr. W. C. 

Krumbein, Department of Geology, Northwestern 

University, and to Dr. R. F. Link, Department of 

Mathematics, Princeton University, for valuable criticism of an early draft of the manuscript.

BASIC CONCEPTS 

THE POPULATION

Geochemical sampling is normally directed at rock 

bodies, or parts of them, whether they are granitic 

plutons, selected stratified layers, covers of soil or 

alluvium, or whatever is of geologic interest. The 

boundaries of some rock bodies are distinct and well 

defined. The boundaries of others are arbitrary and 

may or may not be well defined. Nevertheless, the 

rock body is a single unit and cannot, while considered

in this sense, be regarded as a geologic population; 

moreover, a rock body in this sense has a bulk composition but has no mean composition or compositional variance until it is conceptually divided into 

parts. In studying a rock body it is convenient for 

the geologist to collect specimens or samples, and the 

population is frequently regarded as all the potential 

specimens that the rock body contains. The fact that 

some potential specimens cannot be obtained has led 

to the concepts of target and sampled populations 

introduced by Cochran, Mosteller, and Tukey (1954) 

and used by Krumbein (1960, p. 351).

Characteristics of the population of specimens, such 

as the variance, may largely depend on specimen size. 

In a study of uranium ores from the Colorado Plateau, 

Shoemaker, Miesch, Newman, and Riley (1959) used 

samples split from crushed and homogenized ore shipments aggregating at least several tons. The compositional variation among these samples was less than 

might have been found in samples cut from only a few 

pounds. If the samples were so small that each included only a few mineral grains, the amount and 

nature of the variation from one sample to another, 

again, would be entirely different. Laffitte (1957, 

p. 101-105) discussed efficiency of sample size in relation to size of constituent mineral grains.

The geologic population in geochemical sampling, 

then, is determined by the geologist when he decides 

on the target of his investigation. Commonly the 

population consists of all possible specimens of about a 

few pounds' size that occur within a rock body. Because most rock bodies under study are large, and 

because possible specimens can overlap, the population 

is, for all practical purposes, infinite in size.

RESTRICTIONS DUE TO LIMITED ACCESS

Although the geologic population of interest in most 

geochemical field studies comprises all possible specimens occurring within a rock body, sampling is generally 

limited to parts of the rock body that are exposed at 

the surface. In some work, mostly that bearing on 

resource appraisal, deep drilling can be conducted in 

such a way that the entire rock body is theoretically 

accessible for sampling; in other work, shallow drilling 

or trenching may furnish samples from within a few 

feet of the surface. Except where drilling or trenching 

makes the entire rock body accessible for sampling, 

there may be a large difference between the composition 

of the target population (the population of interest) 

and the sampled population (the population that can 

be sampled). The difference is due to both original 

compositional variation in the rock body and secondary 

variation caused by weathering and other related 

processes.
THEORY OF ERROR IN GEOCHEMICAL DATA A3

Sampling limitations frequently necessitate that one 

or a series of indirect inferences about the target population be made from the collected specimens. The 

inferences may proceed in stepwise fashion, as illustrated in figure 1. The correctness of the inferences 

may partly depend on the degree to which the sampled 

population corresponds to the target population; the 

sampled population, in turn, may be determined by 

the facilities available for sampling.

If facilities such as deep-drilling equipment are available, making the entire rock body accessible for sampling, the inference from the specimens to the target 

population may be direct and straightforward. Objective schemes of drilling and of sampling the drill core 

or cuttings would be especially helpful in validating 

the inference.

Where shallow-drilling devices or trenching equipment is available, the sampled population may consist 

of all possible specimens from within a few feet of the 

surface (A, fig. 1). Inferences from the specimens to 

this sampled population may be direct and straightforward, but the additional inference from the sampled 

population to the target population is more difficult. 

The validity of this second inference depends on the 

nature of the rock body being studied, especially on 

such factors as compositional zoning and depth of 

weathering.

If the sampling devices are used only to obtain specimens from the outcropping parts of the rock body (such 

devices are necessary, for example, in sampling a smooth 

face of dolomite or ganite that cannot be sampled satisfactorily with a hammer and chisel; see Baird and 

others, 1964, p. 258), an additional indirect inference 

may be required to interpret the nature of the rock 

body from the nature of the specimens (B, fig. 1). 

One must infer that the outcropping part of the rock 

body adequately represents the part of the rock body

near the surface, and that this part, in turn, adequately 

represents the rock body as a whole.

In most geochemical field problems the only available 

sampling devices are a hammer and chisel or similar 

tools. (Most rock specimens are collected by means 

of only a geologic pick and hammer. However, the 

portion of an outcrop that can be sampled with an 

ordinary pick and hammer is commonly greatly limited 

in comparison with that which can be sampled using 

a 2-pound sledge hammer and a heavy steel cold chisel. 

Thus, bias in sampling may be significantly reduced by 

using a chisel for sampling many types of rock.) 

Consequently, depending on the type of rock being 

studied, not all parts of the outcrop can be sampled. 

Specimens are generally taken only where the rock 

protrudes, as along bedding planes or fractures. This 

is especially true where the rock is hard and dense, as 

are many quartzites, dolomites, and granites. Where 

a hammer and chisel are used as sampling devices, the 

sampled population consists only of specimens that can 

be collected with this equipment (C, fig. 1), and the 

inferences to be made about the target population 

from the collected specimens, with few exceptions, 

are subject to severe limitations.

These few examples illustrate the kinds of problems 

met in inferring the composition and compositional 

variation of an entire rock body from specimens 

collected with limited sampling access, but the 

problems are similar to those commonly encountered. 

The difficulty of inferring from the sampled populations 

(A, B, and (7, fig. 1) to the target population, as defined 

in figure 1, may discourage one from such attempts.

In some specialized geochemical problems the target 

population is not the potential specimens from the 

entire rock body but only those near or at the surface. 

This is true, for example, in studies directed at environmental health problems, because it is primarily the

Target population: Sampled populations:

B

All possible specimens 

from entire rock 

body, or from selected part of the 

rock body

Total access for sampling (by deep-drilling equipment)

All possible specimens 

from within several 

feet of the surface

Limited access for 

sampling(by shallowdrilling or trenching 

equipment)

All possible specimens 

from outcropping parts

Limited access for 

sampling (by portable drilling equipment)

All possible specimens 

that are obtainable 

with a hammer and 

chisel

Very limited access 

for sampling (hammer and chisel 

only)

SPECIMENS

FIGURE 1. Directions of stepwise inferences from specimens to the target population.
A4 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

near-surface part of the rock body that affects this 

environment. This may also be true in geochemical 

exploration problems where the target population comprises only the upper few inches of soil or alluvium; 

here the target population corresponds to the sampled 

populations designated by A or B in figure 1, and the 

inference from the specimens to the target population 

may be direct and straightforward.

In other geochemical problems the inference from 

sampled to target population may be so difficult that 

the geologist confines his assessments of geochemical 

properties to the sampled population, labeling any 

further inference as speculation only.

Where the target population corresponds to the part 

of the rock body that is accessible for sampling, or 

where inferences are confined to the sampled population, sampling bias can generally be eliminated. 

Specimens, in these studies, may always be collected 

by some objective operational procedure (Krumbein, 

1960, p. 351).

In many geochemical field problems it is convenient 

to employ sampling localities, each locality being defined as some restricted part of the total rock body. 

Each part of the rock body within a sampling locality, 

then, contains a target subpopulation, and the portion 

of the target subpopulation that is accessible for 

sampling is a sampled subpopulation. Further discussion of sampling localities and subpopulations is 

given in a following section.

Krumbein (1960) presented a comprehensive discussion of the concepts of target and sampled populations 

as they may be applied in geology.

OTHER BASIC CONCEPTS

Other concepts basic to the theory of error in geochemistry are accuracy, bias, precision, and variance. 

A few additional remarks concerning the experimental 

nature of geochemical field investigations may also be 

in order.

Accuracy, as used in this discussion, implies close 

agreement between the average of a large number of 

estimates and the correct value. The difference 

between the average of a large number of estimates and 

the correct value is called bias. Differences between 

the average of a small number of estimates and the 

correct value may or may not reflect bias; if the difference tends to diminish toward zero as the number 

of estimates is increased, bias is absent and the method 

of estimation is said to be accurate. In practice, of 

course, the correct value is never known and is taken 

as a value derived by some technique considered 

inherently more likely to be correct than the technique 

used for arriving at the estimate.

In theory of statistical estimation, usage of the term 

"accuracy" differs somewhat from that given in the 

preceding paragraph, and the term "consistency" is 

frequently used in much the same manner that "accuracy" is used here. The distinction is necessary 

only in discussing relative merits of various formulas 

used in estimation. As we are concerned here only 

with errors arising from sampling and analysis, the 

term "consistency" is of little use. The estimation 

formulas referred to are both accurate (unbiased) and 

consistent, even though they may not be the most 

efficient in certain problems. The terms "accuracy," 

"consistency," and "efficiency" in statistical estimation 

are discussed in numerous texts, including those by 

Ostle (1963, p. 87) and Bennett and Franklin (1954, 

p. 134-136, 209).

Precision implies reproducibility, or close agreement 

among replicate estimates, and is independent of accuracy. Methods of estimation may be both precise 

and accurate, neither precise nor accurate, precise but 

not accurate, or accurate but not precise (Ostle, 1963, 

p. 105).

The accuracy and precision of individual estimates, 

whether they are chemical determinations, statistical 

averages, or other types, depend on the accuracy and 

precision of the method by which they are derived.

Imprecision, or the lack of perfect reproducibility in 

measurements, is the type of error that statistical 

methods are designed to overcome and is the least 

serious type of error which can occur. Moreover, imprecision can always be effectively overcome by the 

collection of more data (by additional sampling or by 

replicate analyses of specimens).

Variance is a statistical device for describing the 

degree of variation among a set of variables or measurements. Considering X} as the jih measurement in a 

group of n(l <j </&)> the variance of X is estimated by:

where x is the arithmetic mean of the n values. Among 

the reasons for the widespread utility of variance is its 

property of additivity (Tippett, 1952, p. 167). When 

two or more sets of independent variables are added, the 

sum of their variances, each based on n measurements, 

is equal to the variance of the n sums. This can be 

illustrated with a small group of numbers if the numbers 

are adjusted to be perfectly independent, by eliminating 

spurious effects, as these effects tend to be eliminated 

in large sets of data. Using the columns of numbers 

below, suppose that duplicate analyses (j=l and j=2) 

have been made of two specimens (i=l and i=2) so 

that four analytical values, Xij} are available. Suppose
THEORY OF ERROR IN GEOCHEMICAL DATA A5

also that each value is determined by the sum of an 

unknown correct value for the specimen, T{, and an 

unknown analytical error, ai}.

+

15

5

15

9

12

12

14

14

+3

-7

+ 1

-5

Adjustment of the numbers in this example consisted 

of fixing the mean of an and «12 (+3 and 7) to equal 

the mean of az\ and a22 (+1 and 5), so that the 

correlation between T and a is exactly zero. The 

variances of Xv, Tit and a^ are 24, 1%, and 22%, 

respectively, as derived from equation 1. The variance 

of Tt is the between-sample variance component, and 

that of aij is the within-sample, or error, variance. 

Wherever these two quantities in real data are additive, 

analysis of variance procedures can be used to estimate 

each of them. They will not be additive unless the 

errors are independent a property insured in this 

artificial example by fixing the mean errors for each 

specimen to be equal. In actual data sets the mean errors 

will not be precisely equal, but satisfactory estimates of 

the variance components can be obtained if the mean 

errors tend to be equal as the number of values in the 

experiment is increased.

Because of the additive character of the variance, 

data from suitably designed experiments can be used 

to apportion the total variability among geologic or 

other factors considered in an experiment through 

analysis of variance techniques. These techniques, 

as well as other techniques of statistical analysis, are 

based on formal mathematical models discussed in 

following sections.

Geochemical field investigations may be regarded as 

"experiments," even though geologists rarely use this 

term in oral or written discussions. Use of the term, 

however, helps to clarify the fact that each geochemical 

field investigation is a trial that is subject to repetition 

and confirmation. If the experiment is conducted 

according to some definite operational procedure, it can 

be replicated and confirmed; differences among replicate 

experiments can be used to measure the precision 

(reproducibility) of the experimental results.

MATHEMATICAL MODELS AND SAMPLING ERROR

Mathematical models are used either formally or 

informally in all geochemical sampling. Informal 

models are used when the geologist is attempting to 

estimate amounts or differences but does not formalize 

the problem in a mathematical equation. The nature 

of the model depends on the purpose of the specific 

study. The purpose may be, for example, to estimate

the composition of the total rock body, to estimate its 

variation in composition, or, commonly, to determine 

the spatial variation with respect to geologic factors 

such as sedimentary source areas, the margins of igneous 

plutons, ore deposits, and structural features. In 

this section some simple models for geochemical field 

problems will be developed and used to define sampling 

error. It is assumed here that no analytical error is 

present. In the following section the basic model 

will be extended to account for variation resulting from 

this source.

If a rock body were perfectly homogeneous on a 

specimen scale, so that every potential specimen had 

the same composition, the model might be expressed as:

*,=* (2)

where Xt is the concentration of some constituent in 

the jth specimen, and /* is the mean concentration for 

all specimens in the population being studied. That 

is, each specimen would have the same composition 

or the composition of each specimen would be equal 

to the mean composition of all specimens. Clearly, 

sampling error would be impossible if such an idealized 

rock body existed.

A more realistic model might be written as:

"V I /O\ <X.j=fJ.-]-Vj. (o)

Here each specimen is regarded as having a composition determined by M, the arithmetic mean for all 

specimens in the population, plus a positive or negative 

deviation, vi} from the mean. The purpose of much 

geochemical sampling is to estimate /*, to estimate the 

expected magnitude, or range, of u/, or to examine the 

magnitude of vf with respect to geologic or spatial 

factors.

If the geologist is actually interested in the variation 

among potential specimens over the entire rock body, 

and no one part of the rock body is of any greater 

interest than another part, it seems appropriate to 

collect specimens randomly, or at about even intervals 

ovar the entire rock body, or over the part that is 

accessible for sampling. Sampling need not be either 

random or on a grid, but the sampled portions of the 

rock body should not be purposely clustered.

If the rock body is relatively homogeneous on a local 

scale, the collection of a small number of specimens 

broadly distributed over the rock body may be effective. However, if the rock is locally heterogeneous, 

the local variation will complicate detection of regional 

differences. The effects of the local variation may be 

at least partly overcome by employing sampling 

localities each locality defined as a part of the rock 

body larger than a single specimen and collecting 

a number of samples at each locality. The local
A6 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

variation is smoothed (or "averaged out") in this 

manner.

Employing sampling localities requires a revised 

approach to the field problem. The objective is to 

determine not the variation among specimens but 

the variation among parts of the rock body larger than 

a specimen. According to terminology of Krumbein 

and Slack (1956, p. 744), these larger parts may be 

referred to as "master sampling units," or according 

to more common terminology, as "major sampling 

localities." Each specimen, or group of specimens, 

collected within a major sampling locality, then, is 

intended to represent the locality a target subpopulation. A locality may be defined as a point on 

a sampling grid, a cell within a grid, an outcrop, or a 

stream valley, or in any manner convenient and meaningful for the particular field problem. The purpose 

is to determine the amount and nature of variation 

among localities (among locality means); variation 

among specimens collected within a locality is of no 

particular interest, except that it determines what 

may be called the residual variation (or the withinlocality \rariation) and controls the number of specimens that must be collected in each locality to represent 

the locality adequately. (See Miesch and Connor, 

1964, p. D84.)

When sampling localities are employed as described 

here, the stepwise inferences from the specimens to the 

target population (fig. 1) must also be reframed. (See 

example in fig. 2.) If that part of the rock body that 

constitutes a locality is completely accessible for sampling, inferences from the specimens to the locality may 

be direct and straightforward. If it is not completely 

accessible for sampling, direct inferences may be made 

about the part that was sampled (the sampled subpopulation), but further inferences concerning the total 

locality (the target subpopulation) are necessary. 

Finally, inferences may be drawn about the total rock 

body the prime target population from the nature of 

the localities. The validity of these inferences depends 

partly on the spacing of sampling localities, which, in 

turn, may depend on the purpose of the experiment. 

These matters, beyond the scope of this paper, occur 

within the realm of sampling design problems.

Where major sampling localities are employed in a 

geochemical field investigation, the model in equation 3 

is extended to:

Here X^ is the quantity of some constituent in the jth 

specimen from the ith locality. The term n, again, is 

the grand mean of the constituent of all possible speciTarget population: 

All possible specimens 

from all possible 

sampling localities

Target subpopulation 1

All possible specimens from

sampling locality 1

Sampled subpopulation 1 

All specimens that are accessible within locality 1

Limited access 

for sampling

Specimens from locality 1

Target subpopulation 2

All possible specimens from

sampling locality 2

Sampled subpopulation 2 

All specimens that are accessible within locality 2

Limited access 

for sampling

Specimens from locality 2

FIGURE 2. Directions of stepwise inferences from the specimens to the target population where sampling localities comprise

subpopulations.
THEORY OP ERROR EST GEOCHEMICAL DATA A7

mens in the total rock body. The vt component in 

equation 3 is now represented by two components, & 

and o)tj) where fit is the deviation of the ith locality 

mean from the grand mean, n, and o>w is the deviation 

of the jth sample from the ith locality mean.

The value M+& is the true mean concentration of a 

constituent in the ith locality (the ith subpopulation), 

and the quantity ji+/ff<+«« is the true concentration 

in the jth specimen from the ith locality. The u> i} 

component is the difference between the concentration 

in the specimen (**+&+««) and the true mean for the 

locality (ju+fr).

As the purpose of the jth specimen from the iih 

locality is to represent the mean of the ith locality 

(M+&)J *he component u>i} may be regarded as the 

sampling error associated with the specimen, and the 

mean of all the wi} components associated with specimens from the ith locality is the sampling error associated with the locality. Thus, sampling error is 

caused by local variation within the rock body. Relatively homogeneous rock bodies can be sparsely sampled 

with little chance of large sampling error, whereas 

highly variable rock bodies require taking a large 

number of specimens to represent them adequately.

The «w component associated with the specimen or 

the mean u{} component associated with a sampling 

locality is never known in practice. Its computation 

would require prior knowledge of M+&, the true mean 

for the locality, and this, in turn, would require complete sampling of the locality and analysis of the 

specimens (determination of Xi}) by a perfectly accurate 

and precise technique. Nevertheless, we must make 

judgments about «<, prior to any sort of interpretation 

of geochemical data.

Four aspects of the «w components are discussed next. 

Three are of particular significance, regardless of 

whether the interpretation is statistical or intuitive. 

The fourth is important mainly where the statistical 

methods used involve certain kinds of probability theory.

First aspect

The grand mean of the sampling error components 

(«<j) over all specimens and localities should tend toward zero as the number of localities, m, and of specimens per locality, n, is increased. That is,

as m,'(5)

If equation 5 is not true, an overall bias is present 

in the data. Such bias may occur when the total rock 

body (the target population), or the total sampling 

locality (the target subpopulation), is not accessible 

for sampling, or when sampling is not done according 

to some objective operational procedure (Krumbein,

1960, p. 351). In the latter instance the selection of 

samples may involve personal bias on the part of the 

sampler. Personal bias may be easily avoided, but 

bias caused by limited accessibility of the rock body 

such as lack of outcrop is more difficult to overcome.

For example, parts of the rock body rich or deficient 

in some constituent may tend to be either covered or to 

crop out as smooth surfaces that are difficult to sample 

with a hammer and chisel, as previously discussed. In 

this circumstance the geologist may resort to another 

sampling device, such as a portable drill. If drilling or 

other sampling devices are not practical, and the outcropping parts of the rock body are unlike the total 

sampling locality, the inference from the sampled subpopulation to the target subpopulation (fig. 2) may be 

difficult. Nevertheless, this is the common situation, 

and consideration of possible differences between the 

sampled and target populations is required in most 

field studies.

Where the target subpopulation, however defined, is 

completely accessible for sampling, objective sampling 

procedures (commonly involving some randomization 

in sample selection) can reduce or eliminate the possibility of personal bias.

Overall bias in sampling, if of sufficient magnitude, 

may seriously affect the estimation of means, but it 

will not hamper attempts to describe variation among 

sampling localities. Unless the bias differs from one 

locality to another, as discussed under aspect 2, the 

errors in locality means tend to be constant, and estimates of differences among means tend to be correct.

Second aspect

The mean of the sampling-error components should 

tend toward the same value for each locality as the 

number of specimens per locality is increased. That is,

as HI, . . . Wi-XX),

(6)

where n{ is the number of specimens from the ith locality. 

Where equation 6 is not true, variable bias is present. 

If each term in equation 6 tends toward zero, neither 

variable bias nor overall bias is present. Variable bias 

in sampling, or bias that varies from one sampling 

locality to another, may result from differences in the 

circumstances under which the sampling is done. This 

can be caused, for example, when localities are sampled 

by different individuals having different personal biases, 

or when the outcrop characteristics vary among localities. 

The problem of personal bias is easy to resolve, but the 

problem caused by differences in outcrop characteristics 

is more difficult. For example, all localities within the 

rock body may actually have the same total composi-
A8 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

tion, but parts rich in, say, calcium carbonate may tend 

to crop out at one locality and to be covered at another. 

Even objective sampling of the exposures at each locality, then, may result in maps showing regional variation caused perhaps by climatic differences. Variable 

bias is often unavoidable, and it is therefore necessary to 

consider all the factors (even those which are nongeological) that may be important in the interpretation of 

the experimental results. Variable sampling bias, 

however, can never be completely overcome in data 

interpretation, and if it is of sufficient magnitude it may 

seriously hamper attempts to describe regional variation in the rock body. Variable bias, however caused, 

may be the most serious type of error that can be 

present in geochemical data.

Third aspect

The variation, or precision, of the sampling error, 

conveniently expressed as the variance, should tend to 

be the same for each locality as the number of specimens 

per locality is increased. That is,

rai

Z (<"u-

 w<, as Tii, (7)

where w* is the mean sampling error component of 

the ith locality. Using s£ 4 for the variance of ww at 

the ith locality, equation 7 may be written as:

as HI, (8)

If equation 8 is not true, variable precision in sampling 

is present. The conditions defined by equations 7 and 

8, unlike those defined by equations 5 and 6, may be 

tested from the sampling results if analytical errors 

are sufficiently small to insure that sampling precision, 

rather than analytical precision, is being observed. 

The ability to test for the relation in equations 7 and 

8 results from the fact that the quantity w^~&>* for 

the ith locality is equal to X^ Xi. Therefore, equations 7 and 8 are identical with:

, n2, (9)

where s^t is the variance of a constituent in samples 

from the ith locality.

If all samples from a particular locality have about 

the same composition, those locality means estimated 

from repeated sampling experiments will also be closely 

similar, or reproducible. If, on the other hand, the 

samples differ widely in composition, the means from 

repeated experiments, each involving a small number of 

samples, will also differ widely. The expected variance

of means may be estimated for a single set of samples 

by use of the statistical relation:

(10)

That is, the expected variance of means for the ith 

locality (the degree to which a locality mean may be 

expected to vary in repeated experiments) is l/nt of 

the variance among the samples from the locality.

If objective sampling is employed at all the sampling localities, the variance among collected specimens 

is controlled by the nature of the rock body (its local 

variability) and its outcrop characteristics. Variable precision, then, is caused by variation of these 

factors from one locality to another. For example, 

samples from one locality may be widely different 

in composition because the rock body there is highly 

variable and completely exposed for sampling, whereas 

samples from another locality may be virtually identical because the rock body is relatively homogeneous 

or only a relatively homogeneous part of it is exposed. 

Variable sampling precision may commonly occur in 

minor-element data because the variation of minorelement concentrations in rocks tends to be proportional to their mean concentration. Where the means 

vary widely among localities, the variation, and therefore the sampling precision, may also vary widely.

The significance of deviations from the mean is 

difficult to evaluate where sampling precision is variable. Locality means are estimated with varying 

degrees of confidence or reliability, and a minor deviation of one locality mean from the grand mean for the 

rock body may be geologically significant, whereas 

a major deviation for another locality mean may 

result only from sampling error. The relative statistical significance and possible geologic implications of 

deviations are not easily determined from their 

magnitude.

Variable precision may be overcome, in some experiments, by using suitable data transformations. For 

example, if variation within localities tends to be 

proportional to the locality means, the variation of 

the logarithms of the concentrations tends to be 

constant among the localities. Transformation of 

the data to logarithms, in this circumstance, would 

be a valid and effective means of overcoming variable 

precision. Transformations must be used cautiously, 

however, as complicating factors may thereby be 

introduced into the experiment. Link, Koch, and 

Gladfelter (1964, p. 35) referred to the difficulties that 

had to be overcome by Sichel (1952) and Krige (1960) 

in estimation problems where the data required logarithmic transformation.
THEORY OF ERROR IN GEOCHEMICAL DATA A9

Variable precision of means among localities can 

be overcome by collecting different numbers of specimens at the several localities. This is equivalent to 

adjusting the values of nt in equation 10 so that the 

value of s|i tends to be the same across localities. 

The adjustment of nf values would have to be made 

on the basis of preliminary estimates of sjf, obtained 

from a pilot sampling survey (Miesch and Connor, 

1964).

Fourth aspect

The sampling errors (the co f^ components of equation 

4) for any particular sampling locality should tend to 

be normally distributed; that is, the frequency distribution of UK should approximate the bell-shaped 

normal, or Gaussian, curve. This requirement, as 

that in aspect 3, can be tested by using the observed 

data, Xtj, if analytical error is relatively small. The 

frequency distribution of the data, Xii} for the iih 

locality will have the same form as that of the theoretical component, ui}.

Frequency distributions that are highly asymmetrical are less easily interpreted than symmetrical distributions, regardless of the method of interpretation; 

but only in methods based on certain probability 

theory must the frequency distributions be normal 

(Gaussian). Most of these methods, however, allow 

at least a moderate, and commonly a large, departure 

from normalcy.

SPECIAL MODELS FOR GEOCHEMICAL SAMPLING

The geochemical sampling model given in equation 4 

is applicable to the general case where the objective 

of the field study is to detect and describe the variation 

among parts of the rock body larger than a single 

sample in other words, among sampling localities. 

These localities are commonly about evenly spaced 

over the rock body, or over the part of the rock body 

that is of interest, and are referred to as the major 

sampling localities.

More specialized models have been employed in 

geochemical sampling by Krumbein and Slack (1956) 

and by Baird, Mclntyre, Welday, and Madlem (1964), 

and models similar to theirs are currently being used 

in several geochemical investigations by the U.S. 

Geological Survey. The purpose of each such investigation is to determine the type of variation present 

in a rock body and to devise an efficient sampling 

plan based on the type of variation found. Krumbein 

and Slack, for example, measured radioactivity in a 

shale bed associated with a cyclothem deposit over 

a large part of southwestern Illinois. In the regional 

phase of their study they collected and grouped samples 

by source as within mines, selected mines within

townships, and townships within supertownships. 

Their experimental model (Krumbein and Slack, 1956, 

p. 754) is given as:

(ID

where Xiikm is the radioactivity of the mth sample 

(l<m<d) from the kih mine (l<k<c) in the jth 

township (1 <j<b) in the iih supertownship (1 <i<a). 

The term "p" is the grand mean radioactivity for all 

potential samples in the shale bed (the true mean), 

is the true mean for the iih supertownship, 

is the true mean for the jih township in 

the ith supertownship, and /*+«<+ j8«4-7i#; is the true 

mean for the kih mine in the jih township in the iih 

supertownship. If a sampling design based on this 

model is used, the total sampling error is spread over several 

levels. As the purpose of each sample is to allow 

estimation of a mean for a mine (the quantity

, the mean of the Sijkm components ( -j^ 8t}km )

V*m=l /

may be regarded as the mean sampling error associated 

with the mine. As the purpose in sampling mines is 

to allow estimation of a township mean (M+ai+jS^), the

/ 1 e d 

mean of the yijlt, and 8 ijkm components I ; 22S

\ca *=i m=i

may be regarded as the mean sampling error

for the township. Similarly, the mean of the fti}, yi}k)

/ i b e d \

and 8tjkm components ( r- i S S S Pti+Tftflt+8tt*m } \oca j**i k~i m=i /

is the mean sampling error for the supertownship. 

In Krumbein and Slack's experiment, all supertownships in the area of interest were sampled, so no error 

was involved in selection of supertownships the 

master sampling units, or major sampling localities.

Overall bias, variable bias, and variable precision, 

discussed in the previous section, and defined in terms 

of the model in equation 4, are equally applicable to 

more specialized models such as that given in equation 

11. These several kinds of error, however, will now 

occur at the several levels of sampling. For example, 

the selection of mines may involve an overall bias, but 

selection of townships may be unbiased yet cause 

variable precision from one supertownship to another.

The purpose here is to show how the concept of 

sampling error used in the previous section can be 

extended to more complex sampling plans. The concept of sampling error will be applicable whenever a 

collected specimen, or group of specimens, is intended 

to represent a sampling locality, the locality consisting 

of some part of the rock body larger than an individual 

specimen.
A10 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

ANALYTICAL ERROR

The preceding discussion has been based on the 

assumption that no analytical error is present in the 

experiment. In studies where the effects of analytical 

error are evaluated, the model must be extended to:

(12)

where aijk is the difference between the &th analytical 

determination (Xijk) on the jfth specimen from the ith 

locality and the correct concentration of the constituent 

in the specimen (/*+&+««) The term "aijlc" is the 

error associated with an individual analysis. The 

model in equation 12 is applicable in experiments 

designed to evaluate analytical error and to determine 

the variance of sampling error, w<j, separate from that 

of the analytical error. The model may be extended 

further for experiments designed to separately evaluate 

parts of the analytical procedure or methods of sample 

preparation (such as crushing and sieving) (Shaw, 1961).

Like the special model described in the previous 

section, the model in equation 12 is useful in preliminary 

studies; here the studies are designed, in part, to determine whether an analytical method is sufficiently 

precise for a particular field problem. If the variance 

of aijjc is not small in comparison to the variance of ww 

or 0<, the method of analysis may be inadequate for 

describing variance among specimens within localities, 

or among localities within the rock body.

In final sampling programs we are more concerned 

with the mean error associated with a specunen (the 

error of the mean of all analyses on the specimen) than 

with the error of an individual analysis, and it will be 

convenient to designate this mean error by the symbol 

OH, where,

1 ^ (13)

The quantity Otj is the difference between the true 

concentration in the specimen and the mean of the p 

analyses of that specimen. The error in the mean 

analysis of a specimen is the important factor in field 

geochemical problems, even though the error in individual analyses determines the nature of the mean error. 

Of course, where only one analysis is made per specimen, 0<j=aui.

If Otj is part of the model, the analytical error is 

placed on a level with sampling error, and, even though 

the model in equation 12 is used in evaluating analytical 

error, the role of the error in the type of geochemical 

field problems with which we are concerned here is best 

described in terms of equation 14:

(14)

In considering total experimental error that due to 

both sampling and to analysis it will be convenient 

to use:

*«,=M+A+ («+«)<* (15) 

or

tu, (16)

where c=w+0, the total sampling and analytical error 

associated with a specimen. Actually, it is the component of total experimental error, c w, that is of major 

concern in data interpretation, even though the character of the total error is determined by the character 

and magnitude of sampling and analytical errors 

present. It is possible, and even likely in some studies, 

that either sampling or analytical error can be overwhelmingly dominant and thus control the nature of 

the total error. The advantages of high analytical 

precision, for example, might be almost lost where 

the sampling variance is large, or a small sampling 

bias may be insignificant compared with a large analytical bias. Similarly, skewed analytical error with 

small variance might be almost completely obscured 

by normally distributed sampling error with large 

variance; the deviation of the total error from a normal 

distribution might be undetectable and unimportant. 

However, characteristics of both the sampling errors 

and the analytical errors are reflected in the total error 

to some degree, and the individual components «w and 

QH in equation 15 are still of basic importance. It 

would be rare indeed for undesirable features of sampling and analytical errors to compensate in such a 

way that the features were not present in the total 

error. Unbiased normally distributed sampling and 

analytical errors, of variance slt and «§,, respectively, 

will give rise to unbiased normally distributed total 

error with variance sli-{-s^ i (if the sampling and analytical errors are independent).

Determination of the absolute value of 6ti for any 

set of analyses is not generally possible because the 

correct concentration of any constituent in a rock 

specimen is never known. The value of 0tj, or of «<#, 

can be closely estimated for analyses of artificially 

prepared standards, however, and this is common 

laboratory practice in the evaluation of a new analytical 

technique. In spite of the fact that Oiit the mean 

analytical error for a specimen, is never known in an 

actual field experiment, we are faced with the necessity 

of making judgments about this value prior to data 

interpretation, just as we must judge the character 

of the sampling error, ww.

Three aspects of analytical error, 0W, as for sampling 

error, are important in efforts to detect and describe 

variation among sampling localities: overall bias, variable bias, and variable precision. Also, the frequency
THEORY OF ERROR IN GEOCHEMICAL DATA All

distribution of the analytical error may be important, 

especially when normal probability theory is employed 

in data interpretation. 

Overall bias

Overall bias is absent when the mean analytical 

error for all specimens and all localities tends toward 

zero as the number of localities and specimens per 

locality is increased.

, « »,*-»-. (17)

If equation 17 is not true, overall analytical bias is 

present. This may result from a large number of 

causes (including sample contamination) that may occur 

at any time from when the sample was collected to 

when the analytical determination is received by the 

geologist. (This all-inclusive notion of analytical error 

is not fair to the analyst, but errors due to sample 

handling and preparation must be accounted for somewhere. The part of the analytical error that is the 

responsibility of the analyst may be referred to as 

laboratory error when such subdivision is necessary.)

There are numerous ways in which overall bias and 

other types of error could occur in analysis. Many 

are well known to chemists and other analysts, and 

examples could be given with reference to any particular analytical method. However, it may be useful 

here to refer to the well-known method of estimating 

uranium content by measurement of radioactivity. 

In the commonly used laboratory method a small powdered rock specimen is placed in a lead chamber containing radioactivity counters of various types; the 

lead chamber eliminates nearly all radioactivity other 

than that from the specimen itself. Results of the 

radioactivity measurements are commonly expressed in 

units of equivalent uranium (percentage eU), where 

equivalent uranium is defined as the ratio of the net 

counting rate of a specimen to the net counting rate 

per percentage of a uranium standard in equilibrium 

with all of its disintegration products, both measured 

under similar geometry (Rosholt, 1954, p. 1307). 

Viewed as a measurement of radioactivity, the eU 

analysis is generally unbiased. As a method of uranium 

analysis, its accuracy depends on two main assumptions:

1. That all radioactivity in the specimen is from uranium and its radioactive decay products.

2. That uranium and all its radioactive decay products are in equilibrium within the specimen; that

is, each radioactive member of the uranium series

is disintegrating within the specimen at the same

rate at which it is being formed.

The first assumption may be invalid when, for

example, part of the radioactivity from the specimen

arises from thorium or potassium, rather than from the

uranium series. The second assumption may be invalid when radioactive decay products of the uranium 

series have been removed from the specimen or where 

they are present in excessive amounts. Either situation can give rise to overall bias in uranium assays by 

the eU method. The mean of the assay error may tend 

toward some positive or negative value, rather than 

toward zero. Rosholt (1959) discussed this subject 

thoroughly.

Overall bias in low-level radiometric uranium determinations was illustrated by Newman (1962, table 

33). The data consist of fluorimetric uranium and eU 

analyses of 95 mudstone specimens. The fluorimetric 

analyses indicate that most of the specimens contain 

less than 10 parts per million uranium; the radiometric 

analyses are somewhat higher and, if the fluorimetric 

analyses are accepted as correct, have an average error 

of plus 13 parts per million. The bias is probably due 

to the presence of potassium (containing K40) in the 

analyzed specimens (Newman, 1962, table 33).

Variable bias

Variable bias is absent when the mean analytical 

error tends to be the same from one locality to another 

as the number of specimens per locality is increased. 

That is,

(18)

If equation 18 is not true, variable analytical bias is 

present. Analytical bias that is variable from one 

sampling locality to another may result when the specimens from different localities are treated differently 

or when the localities' characteristics that affect the 

analytical procedure vary. For example, if the analytical 

work extends over a long period of time, improvements 

in technique or analysts are likely to occur. If the 

specimens are treated locality by locality radier than 

in randomized sequence, the analytical bias associated 

with localities can easily vary. Variation in the chemical or physical characteristics of sampling localities 

can also cause variable bias. Where the rock at some 

sampling localities contains large amounts of one constituent that interferes with the determination of 

another constituent, variable bias could occur if suitable precautions were not taken. In some spectrographic procedures, for example, high concentrations 

of manganese may interfere with the determination of 

silver (Bastron and others, 1960, p. 179). If steps had 

not been taken to allow for the interference, biased 

determinations of silver would result only on specimens 

from localities where manganese concentrations are 

high.
A12 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

Finnell, Franks, and Hubbard (1963, p. 46-49) 

discussed an excellent example of variable bias in ell 

analyses among sampling localities within uraniumcopper deposits in San Juan County, Utah. Because 

of differential uranium leaching, specimens from 

weathered parts of the outcrop contain less uranium 

than the etl analysis would indicate, whereas unweathered specimens from underground contain about 

the same amounts of uranium as indicated by the etl 

analyses. If the ell analyses had been viewed as uranium determinations rather than as measurements of 

radioactivity, the bias would have ranged from some 

high positive value at localities where weathered outcrop was present to some value near zero at unweathered 

localities.

Variable bias from one sampling locality to another 

may also result when the amount of bias is related to 

the amount of the constituent present. In some types 

of colorimetric methods an analyst may tend to overestimate low concentrations and underestimate high 

concentrations. As the amount of the constituent 

present can be expected to vary from one sampling 

locality to another, the bias related to the amount 

present also varies among localities. Bias related to 

the amount present may also result from incorrect 

standards or from the chemistry involved in the analysis 

itself.

Examples of both overall bias and bias that varies 

with amount of constituent present were given by 

Wayman and Miesch (1965), with regard to field 

methods for determining the amounts of detergent in

I.Or

0.8 -

0.6 -

0.4 -

0.2 -

/ 

water. Analyses of prepared standards containing 0-1 

part per million detergent (alkylbenzenesulfonate) were 

made by two analysts using three methods. Some of 

the results are given graphically in figure 3. Example 

A is from a standard laboratory technique; the mean of 

the deviations from the accepted values of the prepared 

standards is about zero at each level of concentration. 

Neither overall nor variable bias can be demonstrated. 

In example B, based on a rapid field method, the bias 

present is clearly related to concentration; it is small or 

absent at low and high concentrations but large (and 

negative) at intermediate concentrations. As concentrations at different sampling localities may certainly 

vary when the technique is applied in a field problem, 

variable bias among the localities is certain to result. 

As the biases are not compensating, an overall negative 

bias will also be present.

Variable precision

Variable precision in analysis is absent when the 

variance of the analytical error associated with specimens, Oij, tends to be the same from one sampling 

locality to another as the number of specimens per 

locality is increased. That is,

as (19)

If equation 19 is not true, variable analytical precision 

is present in the data. This type of error may be 

expected when the samples from various localities are 

treated differently or when characteristics of the 

localities that affect the analytical precision vary.

1.0 r

0.8 -

0.6 -

0.4 -

A

/ '

4

/

0.2 h >

0.2 0.4 0.6 1.0 0.2

I 

0.6 1.0

A. LABORATORY METHOD B. FIELD METHOD

FIGURE 3. Analytical determinations on prepared standards of detergent in water. X= concentration in standard; 

Y= measured concentration. Concentrations are given in parts per million. From Wayman and Miescb 

(1965).
THEORY OP ERROR IN GEOCHEMICAL DATA A13

The conditions under which variable precision is likely 

to occur are about the same as those that might lead to 

variable analytical bias. It seems, however, that 

variable analytical precision is much more common 

than variable bias, especially in studies directed at 

minor elements. The precision of minor-element 

determinations, as is well known, nearly always varies 

with the concentration. Fortunately, however, variable analytical precision is an easily recognized type 

of error and, because it is frequently systematic, can 

often be corrected by suitable data transformations 

(Bennett and Franklin, 1954, p. 356). Variable precision may also be corrected by making a number of 

replicate analyses of each sample proportional to 

their variance by adjusting p in equation 13 proportional to the variance of ai)k.

Variable precision of the analytical error, aijk in 

equation 12, from one specimen to another may be 

detected by computing the variance of analytical values, 

XM> f°r eacn specimen and applying several statistical 

tests for variance homogeneity (see Bennett and 

Franklin, 1954, p. 196-200). The same tests may be 

used to detect variable precision of the total sampling 

and analytical error, e fy in equation 16, among sampling 

localities.

Frequency distribution.

The components of analytical error, Ql)} should tend 

to have a normal distribution. The frequency distribution of the total experimental error for sampling 

localities, e y in equation 16, may be examined by 

constructing the frequency distribution of analytical 

values, XtJ, representing the locality. If the distribution departs widely from the normal, it can commonly 

be corrected by suitable data transformations (Bennett 

and Franklin, 1954, p. 91). The requirement of a 

normal distribution is necessary only when probability 

theory is to be employed in data interpretation, 

although symmetrical frequency distributions are easier 

to interpret byk any method.

The frequency distribution of the analytical error 

alone is best determined from a large number of 

replicate analyses of the same specimen, or from a large 

number of estimates of aiile obtained by comparing 

analytical values with corresponding values known to 

be more correct.

EFFECTS OF ERROR TYPES IN STATISTICAL ANALYSIS

In the preceding discussion several types of sampling 

and analytical error have been described, and the effects 

of each error type in data interpretation are briefly 

discussed. The effects are the same whether the error 

is due to sampling or to analysis.

Overall bias is present when the mean of the error 

components does not tend toward zero, and its effect is

merely to distort the estimate of the mean composition 

of the rock body. Unless the bias is variable from one 

sampling locality to another, the estimated locality 

means tend to be wrong by a constant amount, and 

estimates of differences among locality means tend to 

be correct.

Bias that is variable from one sampling locality to 

another may have disastrous effects on data interpretation. As the bias for each locality varies and is 

unknown, determining the differences among localities 

the primary objective of most geochemical field 

sampling may be impossible.

Variable precision of sampling and analysis from one 

locality to another presents difficulties in judging both 

the statistical and geologic significance of differences 

but does not present as severe difficulties as does 

variable bias.

The type of frequency distribution of errors is of even 

less concern in data interpretation, though symmetrically distributed error is easier to evaluate by both 

formal statistical or other methods. When some types 

of probability theory are used in data interpretation, 

however, the type of frequency distribution displayed 

by the error components should be considered.

In formal statistical methods of data analysis, certain 

assumptions pertaining to the kinds of sampling and 

analytical error are required to make valid use of 

statistical models as given in previous parts of this 

paper. These models are the bases for formal statistical 

analysis of the data. However, it is a reasonable 

circumstance that the required assumptions regarding 

data errors are much the same for formal statistical 

analysis as for other methods of data analysis. Moreover, the relative importance of error assumptions 

required for formal statistical methods is about the 

same as for other methods. The lone exception here 

is the importance of the normal distribution in statistical 

methods based on normal probability theory.

The required assumptions regarding data errors are 

about the same for all statistical methods, even though 

the requirements of the data themselves, and the manner in which the data are collected, may differ. Multivariate statistical methods involve some assumptions 

regarding data errors, such as uncorrelated errors 

among variables, that are not considered here; but the 

requirements of most statistical methods that deal 

with one estimated variable at a time can be illustrated 

by reviewing two methods that have been shown to be 

useful in geochemical problems. These methods are 

analysis of variance and trend-surface analysis (polynomial multiple regression). Applications of these 

methods to geochemical and petrologic problems are 

so numerous that citation of individual papers is 

unnecessary.
A14 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

ANALYSIS OF VARIANCE

Analysis of variance has been applied in geochemical 

problems for testing compositional differences for 

statistical significance and for partitioning the total 

variability of a constituent among several factors. 

Applications of the second type are efforts to ascribe 

variability to geologic or spatial factors in the correct 

proportions. The assumptions that are required for 

analysis of variance techniques in general were summarized by Eisenhart (1947). These assumptions, as 

stated for the general case, appear somewhat formidable; viewed in terms of a particular problem, they 

are more easily understood. In a companion paper to 

Eisenhart's, Cochran (1947) reviewed the effects of 

failure of the data to meet the requirements set forth by 

Eisenhart. Insofar as experimental error is concerned, 

Cochran assumed that all the errors have zero means, 

and he tabulated the other assumptions required as 

follows:

1. Errors must be independent.

2. Errors must have a common variance.

3. Errors should be normally distributed.

If the assumption of zero means among the errors 

fails (overall bias is present), the analysis of variance 

may still be valid so long as the errors are independent, 

have a common variance, and are normally distributed. 

The grand mean and individual means for sampling 

localities will be biased, but the analysis of variance 

method is still valid for testing the significance of 

differences among them.

The three error requirements will be discussed as 

applicable to analysis of variance based on the model 

given in equation 16.

Errors lack independence when the amount of one 

error is related to that of another. This may occur 

in geochemical field sampling when the errors associated 

with specimens from one locality tend to be distributed 

about one value and those associated with specimens 

from another locality tend to be distributed about a 

different value. The errors, then, are related to the 

localities and lack independence. The errors associated 

with at least one of the localities, moreover, are biased, 

and the bias is variable among localities. The effect 

of variable bias, or dependent errors, in analysis of 

variance is that the additivity of variances is destroyed, 

as shown below in an example analogous to one given 

previously:

-1

+2 

+2

 3

Here the mean of the c w values ( 1 and +2) added

to /*+&=2 is different from that of the «w values 

(+2 and 3) added to /t+&=4. The variance of 

M-f£< (= 1,J0 plus the variance of tj (= 6) is equal to 

7%, whereas that of Xij is 6. Because the additive 

character of variance is destroyed by variable bias, 

the analysis of variance method, based on models 

given in this paper, is invalid for estimating variance 

components and testing the significance of differences 

among sampling locality means. Mean square esimates derived in the analysis of variance procedure 

tend to be incorrect.

Common variance among the errors is absent when 

the errors in one category of the experimental design 

have a different variance than those in another category. 

This occurs in geochemical field problems when the 

precision of sampling or analysis, measured by the 

variance, varies from one sampling locality to another. 

Variable precision prohibits valid estimation of mean 

squares in analysis of variance because the true component of error variance is not a single value, but a 

range of values. Data transformations that may be 

suitable for obtaining a common, or homogeneous, 

variance were reviewed by Bartlett (1947).

The requirement of a normal distribution for experimental errors does not affect the validity of the 

analysis of variance method for estimation of variance 

components, but it does affect tests for significance 

of differences among sampling locality means. The 

effects, however, are not severe, and non-normality 

is not regarded as a rigid requirement when JP-tests 

are applied (Cochran, 1947, p. 24). The F-tests, 

because of this lack of high dependence on a normal 

distribution, are said to be "robust." When the 

normal distribution assumption cannot be satisfied, 

nonparametric methods of statistical analysis may be 

useful (Siegel, 1956).

Two-tailed <-tests for estimating the significance of 

difference between two means a special application 

of analysis of variance and methods for estimating 

the confidence interval of a mean based on the tdistribution are also "robust"; the normal-distribution 

requirement is not severe. On the other hand, methods 

based on the one-tailed ^-distribution, as for comparing 

a mean with some critical value, lack "robustness," 

and the normal-distribution requirement is more 

critical (Cochran, 1947, p. 24-25). The lack of "robustness" of Bartiett's test for common, or homogeneous, 

variance is well known (Box, 1953).

TREND ANALYSIS

The method of trend analysis, as currently applied to 

geologic problems, is a form of multiple regression 

wherein polynomial or other mathematical surfaces are 

fitted to map data by least-squares techniques. Several
THEORY OF ERROR IN GEOCHEMICAL DATA A15

objectives can be achieved by this technique, but chief 

among these is the separation of total map variability 

into regional and local components. The essential features of the method, as applied in geology, were summarized by Krumbein (1959). Trend analysis may be 

viewed as an empirical tool or as a descriptive technique 

in geologic data analysis; as such, the data errors involved are not subject to unusual requirements. Overall bias in the data merely affects the estimation of the 

constant term in the computed regression equation and 

the absolute values on the contours of the regression 

surface. Variable bias affects the regression coefficients and the form of the fitted surface. Variable 

precision results in unequal reliability of the locality 

means described by the fitted surface. These effects 

are the same as would be present if the data were merely 

contoured. The frequency distribution of the error 

has no unique effect on either trend analysis or contouring procedures.

Testing the statistical significance of fitted trend 

surfaces, by F-tests as outlined in Bennett and Franklin 

(1954, p. 431-436), requires the same assumptions 

regarding data errors as listed previously. Other assumptions regarding the data and the mathematical 

function fitted to the data in trend analysis are beyond 

the scope of this paper. (See Link and others, 1964.)

From the above examples it is apparent that data 

containing errors causing the data to be unsuitable for 

statistical analysis are probably unsuitable for any 

other rigid type of interpretation. When neither overall bias, variable bias, nor variable precision of the 

errors is of large magnitude, the data, insofar as the 

error is concerned, are suitable for interpretation on 

either statistical or substantive grounds, although statistical methods are more likely to overcome the effects 

of unbiased error. However, statistical methods have 

no greater power to overcome effects of overall and 

variable bias than have other methods.

SUMMARY

The entire purpose of statistical analysis in geochemical field problems is to evaluate and allow for the effects 

of error in the data in drawing geochemical conclusions. 

However, neither statistical nor other types of data 

analysis can be effective when certain kinds or error are 

present. Some kinds of error can be effectively reduced 

by collecting more data or by making data transformations; but other kinds, if of sufficient magnitude, may 

persist and invalidate any type of data interpretation.

The individual components of sampling and analytical error are never known in a real field problem, 

though the variance of total experimental error and 

the form of its frequency distribution can be determined. In spite of the fact that the error components

are unknown, we are faced with the necessity of making 

judgments about them prior to any sort of data interpretation. These judgments are necessarily based on 

knowledge of the sampling and analytical procedures 

and the circumstances under which they were carried 

out. No sampling or analytical procedure gives 

results that are perfectly precise, or reproducible; some 

imprecision is present in all observational data. Overall bias, variable bias, and variable precision, however, 

may or may not be present in important amounts, and 

the presence of one is not necessarily related to the 

presence of another. In some studies the occurrence 

of these error types can be reduced or controlled by 

suitable precautions in the field and laboratory. In 

other studies, owing to the nature of the rock body 

and its outcrop characteristics or the nature of an 

analytical method that must be used, the occurrence 

of these error types in important amounts is beyond 

control.

Overall bias may be controlled when it is due to 

misjudgment on the part of the sampler or analyst. 

The sampler may be persuaded to use an objective 

sampling technique, and the analyst may change his 

analytical methods. Overall bias may also result, 

however, under circumstances beyond the sampler's 

or analyst's control, as when the sampler is restricted 

to outcropping parts of the rock body and the outcrop 

is not representative of the rock body as a whole. 

Overall bias in analysis may be beyond control, for 

practical purposes, when a method that must be used, 

owing to cost or other factors, involves such shortcomings as inadequate sample digestion or inadequate 

complexing or precipitation. Overall bias in analysis 

is subject to tests by analysis of prepared standards 

or by comparison with other methods accepted as 

more precise and accurate, but bias in sampling can only 

be inferred from knowledge of the field procedures and 

the nature of the rock body.

Variable bias in sampling and analysis, where it is 

due to variation in the field and laboratory procedures, 

may be reduced by exercising precautions to keep the 

procedures uniform. However, variable bias may 

be caused by circumstances that are beyond control, 

as when outcrop characteristics differ from one sampling 

locality to another and when attributes of the rock 

that affect analytical bias vary among localities. 

Variable sampling bias may only be inferred from 

knowledge of the field conditions and procedures, but 

variable analytical bias sometimes may be detected by 

analysis of prepared standards or by comparison with 

methods known to be more precise and accurate.

Variable precision in sampling, like overall and variable sampling bias, will rarely be introduced by the 

sampler who uses some objective method of sample
A16 STATISTICAL STUDIES IN FIELD GEOCHEMISTRY

selection. Variable precision resulting, however, from 

natural differences in the amount of local variation 

over different parts of the rock body is unavoidable. 

Variable sampling precision is also unavoidable when 

caused by the variable nature of the outcrop from one 

sampling locality to another, if only outcrops can be 

sampled. Analytical precision that varies with the 

amount of the constituent present is an inherent feature 

of most analytical methods for minor-element determinations and is generally unavoidable. Variable 

precision in total experimental error, that due to both 

sampling and analysis, is comparatively easy to detect 

in the analytical data.

Experimental results which are affected by bias in 

sampling cannot be corrected; effects of analytical bias 

may be corrected in certain restricted instances where 

it is either constant or systematically variable (Youden, 

1962). The effects of variable precision commonly may 

be overcome by suitable data transformations; these 

same transformations may also bring the frequency 

distribution closer to normal (Bartlett, 1947, p. 40). 

Variable precision among sampling locality means may 

be overcome by collecting additional specimens at the 

more variable localities.

The effects of overall bias in geochemical errors are 

usually not severe, unless the purpose of the investigation is to estimate absolute mean concentrations of 

constituents to several significant figures. Variable 

bias, however, may seriously affect both statistical or 

intuitive interpretation of the data. Variable precision or a non-normal frequency distribution of error 

components renders judgments of the relative significance of differences difficult and may prevent use of 

some useful statistical methods based on probability 

theory.

Despite the difficulties that can and do arise in the 

field and in the laboratory, sound geological judgment 

in designing a study and use of objective sampling plans 

can do much to reduce their effects. The many valid 

inferences made in geochemical studies attest to the 

fact that in many instances the errors are not great 

enough to invalidate the findings of the study. Nevertheless, we cannot afford to ignore the implications of 

error. The different types of error that may be present 

must be recognized and the kinds of effects each may 

have on data interpretation should be known. Only 

through a clear understanding of these factors can we 

arrive at sampling designs and methods of data interpretation that are both consistent with the geologic 

problem under consideration and efficient with respect 

to the available field and laboratory resources.

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