Sample Size and 

Meaningful Gold Analysis 

By H. EDWARD CLIFTON, RALPH E. HUNTER, FREDERICK J. SWANSON, 

and R. LAWRENCE PHILLIPS 

SHORTER CONTRIBUTIONS TO ECONOMIC GEOLOGY 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 625-C 

A technique for estimating the size of 

sanzple required to provide a representative 

sample for analysis for gold and other 

particulate heavy metals and minerals 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON 1969 
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary 

GEOLOGICAL SURVEY 

William T. Pecora, Director 

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

Washington, D.C. 20402 - Price 35 cents (paper cover) 
CONTENTS 

Page 

Abstract___________________________________________________________________________________ C1 

Introduction_______________________________________________________________________________ 1 

Adequate sample size________________________________________________________________________ 2 

Preconcentration requirements________________________________________________________________ 6 

Particle-size variability______________________________________________________________________ 9 

Applicability to natural samples __ ----------------____________________________________________ 10 

Mathematical analysis_______________________________________________________________________ 12 

Conclusions-------------------------------------------------------------------------------- 15 References cited ___________________________________________________________________________ . 17 

ILLUSTRATIONS 

[All figures are graphs] 

FIGURE 1. Relative errors and relative standard deviation in relation to number of gold particles in sample ________ -- ___ -_ 

2. Relationship between number of particles per 1-kg sample, particle mass, and grade or tenor of the sample in 

parts per million ________________________________________________________________________________ _ 

3. Size of sample required to contain an expected 20 particles of gold as a function of the combination of gold particle size and grade, assuming all gold particles to be of uniform size and randomly distributed in the 

deposit __ --- _______ ---- __ ------------------------------------------------------------------------------

4. Scales to convert sample size as indicated by figure 3 for 20 particles per sample to sample sizes for other particle- per-sample ratios _______________________________________________________________________________ _ 

5. Percent of all possible samples in which the relative error (E) is less than or equal to -100, -50, 0, +50, and 

+ 100 percent, plotted in relation to the expected number of gold particles (>.) in the sample _____________ _ 

6. Relation of gold grain size and grade to preconcentration requirements for some common sizes of analytical 

portions--------------------------------------------------------------------------------------- 7. Cumulative curve of gold grain-size distribution in a sample of beach sand ________________________________ _ 

8. Distribution of analytical values for three sample sizes ___ -------- ________________________ -------- ______ _ 

9. Relationship between particle niass and grain size (diameters of spheres) of different particulate heavy metals 

and minerals as a function of specific gravitY-------------------------------------------------------

TABLES 

Page 

C3 

4 

5 

6 

7 

8 

11 

12 

16 

Page 

TABL1D 1. Estimate of effective number of gold particles from distribution of analytical values__________________________ C10 

2. Occurrence of gold in size fractions of a 5,150-g beach sand sample_________________________________________ 10 

3. Measures of gold particle size for splits of a large sample, adequate sample sizes predicted from these data, and 

statistical parameters by which these were calculated_________________________________________________ 12 

HI 
SHORTER CONTRIBUTIONS TO ECONOMIC GEOLOGY 

SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS 

By I-I. EDWARD CIJIF'l'ON, RALPH E. HuNTER, FREDERICK J. SwANSON, and R. LAWRENCE PHILLIPS 

ABSTRACT 

A sample of geologic material is of adequate size for gold 

nnalysis if it is large enough to insure tbllit the analysis will 

have n specified degree of precisio·n. It can be shown ma.themntically tha:t, if (1) gold particle mass is uniform, (2) g'Old 

particles make up less than 0.1 11ercent of all the particles, (3) 

the snmple contains n:t least 1,000 particles of all kinds, (4) 

nnnlyticnl errors are absent, and (5) the gold particles are 

l'IIJHlomly distributed thi"'ugh 'the deposit being sampled, the 

precision is determined solely by the number of gold particles 

in l:he snmple. 'l'he precision afforded by 20 panticles per sample 

is deemed sufficient ror most purposes. 1:\. graph has been devised 'by which one can determine, for different combilllations 

o·f gold grade and particle size, the weight of 'sample expected 

to contain 20 particles. Modifications of this graph allow its 

use for determiniug smnple sizes necessm·y fo·r other degrees of 

precision, for determining to what extent field snrnples require preconcentration for gold ·analysis, and for determining 

sample sizes for heavy minerals other bhan gold. 

In addition to varia·bility in the number of gold particles per 

SfW11ple, wtriability in gold particle size is a major cause of 

imprecision in golcl analysis. Several methods may be used to 

determine sample size where particle size varies. Analysis of a 

nattwnl s·and sample demoustra1tes the use of the procedures 

for def:ermining adequate smnple size. 

INTRODUCTION 

The collection of an adequate san1ple constitutes the 

first step for a.ny a.nalysis of geologic material To be 

meaningfltl, the smnple must accurately represent a 

ln.rger entity-a deposit or some portion of a deposit. A 

sample too small to be representative of this entity is 

without value. The problem of sample size is palticularly acute in the analysis for gold, which occurs significantly in trace amounts, parts per million, commonly 

represented by only a fmv pa.rticles per sa.mple. The 

problem is further co~nplic:tted by the size of the lwboratory specimen ultimately :tm'\.lyzed; it mn.y be much 

smaller tlmn the initia.l field sa.mple. 

For exn..mple, a.tomic-a,bsorption techniques, currently used by the U.S. Geological Survey for routine 

n.nnJysis for gold (LaJdn and N a.kaga,wa, 1D65), utilize 

only 10 g (gr1un) of materiaL If the field smnple conta..ins reln,tively few gold particles, rrlJ1dom selection 

of the analytical portion may not provide a portion 

that is representative of the field sample. In such cases, 

the gold in the initial saJnple ·must be concentrated into 

the analytical portion (Clifton aJld others, 1967) or the 

sample must be reduced by grinding to produce a 

greater number of gold particles; otherwise, the most 

careful field sampling ma.y prove futile. 

The purpose of this paper is to provide simple workable means of establishing the minimun1 sample size 

~Ldequate and to indica.te the circun1staa1ces wherein concentration prior to annlysis is necessa.ry. C0nsidera,tion 

is given mainly to analysis for gold by a.tomic-a;bsorption techniques; the principles; ho\vever, a.pply to any 

kind of a.na.lysis for pttrticula.te trace elements.· 

The problems of determi1~ing saJnple siz~ for 1ninei·al 

or chemical analyses have beeii studied in i~ecent years 

by Becker ( 1964a, )964b, 1965a, 1965:b, 1966) and Gy 

( 1954, 1956, 1967). Their results a.re applicruble to 

sampling problems in geneeal, whereas the present approach ·is based 'on. assumption~ that seem appropriate 

f.o~· the special problem o·f saanpling for gold ana.lysis. 

Consequently, the mathematical derivations presented 

here .are less complex tha.n those of Becker and Gy. 

· Tl~e results given in this· paper can be shown to be 

essentia.lly the same as those of Gy if the r'estrictions 

made in the present study a.re introduced into his equatimi-rela.ting the va.riance of a.iut~ytieal results to facto·ts 

such as sa.mple size (Gy, 1967, p. 51, eq 4.D8; Ottley, 

1966, p. 42). The present res~1lts :tre aJs~ essentially the 

saane as those of de l\1agnee ( 1956) ~tnd Prigogine 

( 1961), who 1nade 1~estric.tions siri1ilar to those in this 

study. I-Im~ ever, the graphs presented here a.re perhaps· 

rnore easily grasped intuit.i\rely than are the equations 

of Gy, de Magnee, and Prigogine. They assume a for1n 

irery similar to those shown by Gy (1956, p. 95, 97), 

b~t g!ve sample siz~s con~aining the expected number 

of golc~ particles. . 

· The ~p.pro·ach of this paper also differs frmn the work 

of Gy, de Magnee, and Prigogine in its us~ of relative 

er~rors dete11nined from the binominal and Poisson distributions. This approach ·gives niore cmnplete infor01 
C2 SHORTER CONTRIBUTIONS TO ECONOMIC GEOLOGY 

mation about the distribution of analytical results than 

is given by the equations of Gy, de Magnee, and Prigogine for the variance of analytical results. The more 

complete information is especially useful for relatively 

small samples containing few gold particles. 

The writers have profited greatly from the work of 

Gy and Prigogine, who have treated the problems of 

sampling in much greater detail than we have here. 

The reader is referred tD their papers for additional 

information as well as to Krumbein and Graybill 

(1965) and Miesch (1967b) for discussions of the basic 

philosophy and methods of sampling geologic materials. 

·vve wish to acknowledge that the analytical work was 

carried out by Kam Leong and Arthur Hubert, assisted 

by Oliver Roman, and that T. R. Alpha assisted in the 

field and laboratory. 

ADEQUATE, SAMPLE SIZE 

In order to determine adequate sample size, the precision expected in the analysis of gold content must be 

quantitatively defined. The precision of a gold analysis 

can be defined by two numbers. First, the true gold content of the deposit is expected to be within a certain 

range of values surrounding the value obtained by 

chemical analysis of a sample. For example, we will 

require here that the true gold content be no more than 

approximately 50 percent larger or smaller than the 

gold content obtained by chemical analysis of a sample. 

Jiowever, because of random variation in gold content 

from sample to sample, it is impossible to be completely 

certain that the true gold content will be within a given 

range of values. Therefore, one must specify a second 

number, the confidence, or probability that the true gold 

content will be within the required range of values. For 

example, we will require here that it be 95 percent 

probable that the true gold content of the deposit be 

no more than approximately 50 percent larger or smaller 

than the gold content obtained by chemical analysis of 

a sample. The number, 50 percent, will be referred to as 

the relative error. 

In the following discussion, a basic assumption will 

be made concerning the deposit from which ~the sample 

is taken. It will be assumed that its variability in gold 

content from point to point is due entirely to random 

spacing of the individual gold particles and that there 

are no systematic variations in gold content in any 

direction across the deposit. Most actual geologic deposits do not approach this ideal, instead they contain 

gold preferentially concentrated· in layers, veins, or 

pockets. :However, the ideal may be approached by 

limited portions of geologic deposits, which can be considered separate entities or deposits themselves. 

If systematic trends in gold content do occur in a 

. deposit, the precision of the gold analysis may be less 

than that predicted for the ideal type of deposit' assumed here. The undesirable effeot of systematic trends 

in gold content can be reduced by a proper sampling 

plan, but such procedures are beyond the scope of this 

paper. Sampling plans are treated in detail by l(rumbein and Graybill (1965) and by Miesch (1967b). 

It can be shown mathematically (see the section 

"Mathematical analysis") that the nun1ber of gold 

particles in the sample is the only factor controlling the 

precision of the chemical analysis if, in addition to assuming that the gold particles are distributed randomly, 

it is also assumed that ( 1) gold particle n1ass is uniform, 

(2) gold particles make up less than 0.1 percent of all 

the particles, ( 3) the sample contains a:t least 1,000 particles of all kinds, and ( 4) analytical errors are absent. 

The relation between precision and number of gold 

particles in the sample is shown in figure 1, which indicates that the degree of precision specified in this 

study will be attained if the sample contains 20 particles 

of gold. A sample has the minimum adequate size then 

if it is large enough that it can be expected to contain 20 

particles of gold. For reconnaissance studies, a smaller 

sample containing fewer particles of gold may suffice. 

It is important, however, to note that as the expected 

number of particles per sample falls below five, the 

chance of having no gold particles in a given sample 

greatly increases (fig. 5). 

Unfortunately, it is seldom, if ever, possible to count 

the number of gold particles in order to determine if a 

sample is of adequate size. However, in any material, an 

interrelationship exists between the number of particles 

of a particular component per unit mass of sample, the 

1nasses of these individual particles, and the grade or 

concentration by weight of the component. If the particle masses and their concentration are known, the number of particles can be determined. If particle mass is 

assumed to be uniform, the general relationship between 

these factors can be shown on a simple graph (fig. 2 or 

Clifton, 1967) or chart (l\Eesch, 1967a). The density 

and particle shape of any specific component relate the 

particle mass to particle size. Figure 2 applies specifically to gold in the grain-size relationships shown on 

the right margin of the figure. Grain size is shown in 

terms of both spheres and flakes of the shape that gold 

particles commonly exhibit. The ranges of particle mass 

and grade shown in the figure are limited to the field 

occupied by most deposits of detrital gold; the relationships can readily be extrapolated to cover particle 

masses and grades beyond the scope of the figure. 
SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS C3 

z 

0 

~ 

~ 0::0 

0 0 1.1..0:: 

(/)<( 

a:: 0 oz 1 

ffi ~ (100 PERCENT) 

~~ i= 1- 0.5 

~ ;5 (50 PERCENT) ~~ 

~~ ~~ ffit::i 

z 0 

oz 

z <( 

<(~ UJ .._, 0.1 

~ ~ (10 PERCENT) - z 

Q_ 

~~ -

z 0.05 8 (5 PERCENT) 

z 

UJ 

(.) 

a:: 

UJ 

a. 

'-> 

EXPLANATION 

The precision of a chemical analysis for gold may be 

calculated by the formula 

P0 +E~P <P<P0 + EtPo· 

where P =proportion or grade of gold in deposit sam· 

pled, 

P0 =proportion or grade of gold in sample, 

E; =positive relative error at c percent con· 

fidence. 

Ec =negative relative error at c percent con. 

fidence · 

0.01 

(1 PERCENT) l~----~~~~~5~~-~10----~--~--~5~0~-L1~0LO-----L--~~-5~00-L~~l~OO-O----~--~-L5-0~0~0-L~10~.000 

NUMBER OF GOLD PARTICLES IN SAMPLE (X) 

FIGURE 1.-Relative errors and relative standard deviation in relation to number of gold particles in sample. The relative 

errors for 68 and 95 percenlt confidence are shown. Note that the negative relative errors have negative values. The d'ashed 

portion of the 111-oG curve was calculated on the basis: of the Poisson distribution. An example of how to use figure 1: .Suppose that a sample is found by analysis to have a goltl content (Po) of 1.00 ppm and that the sample size and gold particle 

size are such that the number of gold pa-rticles, X, is calculated to beJ 20. At 95-percent confidence, what limits can be 

pla<:ed on the gold content of the deposit from which the sample was ~taken? From the figure, E+os--0.54 and E-oG=-0.34. 

Therefore, one can be 95 percent confident that the gold content (P) is between (1.00-0.34) ppm and (1.00+0.54) ppm, 

or between 0.66 ppm and 1.54 ppm. 

The size of sample required to provide the desired 

particle-per-sample ratio can be determined from figure 

3, which is derived from figure 2. The ratio depends on 

both the mass and concentration of ~the particles. In figure 2, a 1-kg (kilogram) sample contains 20 particles at 

any of the mass-grade combinations intersected by the 

20-particle vertical line. Likewise, a 500-g sample will 

contain 20 particles at any of the combinations intersected by the vertical line in the figure at which 1 kg 

contains 40 particles. The grade to grain-size relationships which provide 20 particles for samples of other 

sizes can be similarly calculated. In this way, figure 3 is 

derived; it shows the sample sizes that contain 20 particles per sample at various combinations of grade and 

particle mass. 

Figure 2 also can readily be used to determine sample-size requirements to provide other particle-per-sample ratios. For example, to obtain 40 particles per sample 

for any particle mass-grade combination in figure 3, the 

indicated sample size should be doubled. To obtain 10 

particles per sample the indicated size should be halved. 

Figure 4 indicates the changes necessary to modify the 

base of figure 3 to apply to·specifically desired particleper-sample ratios and relative errors.· . . . 

Use of figure 3 requires that the sampler have some 

idea of the grade and particle mass likely to be encoun-
•) 

'.l 

.., 

J 

C4 SHOUTER CONTRIBUTIONS TO ECONOMIC GEOLOGY 

4.0 

2.0 

1.0 

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t-

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I0-2 

0.008 

I0-3 

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lQ-4 

I0-1 JOO JQI 10 2 J03 jQ4 JQ5 J06 

NUMBER OF GOLD PARTICLES PER !-Kg SAMPLE 

l!.,IGURE 2.-Relationship between number of particles per 1-kg sample, paJrticle mass (assuming all particles to be of uniform 

mass), and grade or tenor of the sample in parts per million. Scales· to right relate grain size of gold spheres and flake~ 

to particle mass. 
SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS C5 

.IQG 

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lQ-5 II I I I IIIII I I I . l1111 I I L liiii I I I IIIII I I I IIIII I I I IIIII I I I 1111 I I I IIIII 

lOOOkg 500kg lOOkg 50kg lOkg 5kg lkg 500g !DOg 50g lOg Sg lg SOOmg 100mg50mg 10mg5mg 

SIZE OF SAMPLE EXPECTED TO CONTAIN 20 GOLD PARTICLES 

l!..,IGURE 8.-Size of sample required to contain an expected 20 particles of gold as a function of the combination of gold particle 

size attd grade, assuming all gold particles to be of uniform size and randomly distributed in the deposit. 

328-455--69----2 
C6 SHORTER CONTRIBUTIONS TO ECONOMIC GEOLOGY 

l1111 I I I I I I II It I !11 I II II ! "IIIII! II 1111!111 I 111 II II I I 11111 I l l l1111r 1 r 

lO,OOOkg 1000kg 100kg lOkg lkg 100g lOg 1 g 100 mg 

400 PARTICLES PER SAMPLE (Ets= 10 percent, E9s= -9 percent) 

I I I I1!Jl! I I 11111 I I I 

lOOkg 

!11 II I I I I !11 I I I I I I !11 II 1 I I I '" I 1 ._,1 lc_l,___.__....ull_ul 1_._1 "--'1 [,__,_( _.___--'-'-!I w..ll .J...J! l-~.!__.1--L.-

lOOOkg JOkg 1kg 100 g 10 g 1 g 100 mg 

75 PARTICLES PER SAMPLE (E{s = 25 percent, E95 = - 20 percent) 

II I I 1111 1 1 r !111!1 r 1 

lOOkg lOkg 1kg 100 g 10 g 1 g 100 mg 10 mg 

7 PARTICLES PER SAMPLE (Eg's = 100 percent, E95 = -50 percent) 

lOOOkg IOOkg IOkg !kg 100 g 10 g 1 g 100 mg 10 mg 

20 PARTICLES PER SAMPLE (E9\=54 percent, E9s= -34 percent) 

FIGURE 4.-Scales to convert sample size as indicated by figure 3 for 20 particles per sample to sample sizes for other 

particle-per-sample ratios (7, 75, and 400). E+u5 and E-95 show the positive and negative relative errors at 

95 percent confidence for each of the ratios. For example, if the precision given by 400 particles per sample is 

required and the grain size-grade combination in figure 3 indicates a sample of 1 kg for 20 particles per sample, 

conversion shows that a sample of about 30 kg is needed. The conversion scale can also be applied to figure 6 

to establish preconcentration requirements consistent with the degrees of precision shown. 

tered. This may be estimated :from the expected range 

:for the type o:f deposit, established by preliminary analyses o:f selected size :fractions, or estimated by a procedure described in a :following section o:f this paper. 

PRECONCENTRATION REQUIREMENTS 

The :foregoing discussion describes how to select a sample large enough to adequately represent the deposit. 

Analysis o:f such samples, however, may be meaningless 

i:f made only on a randomly selected portion o:f the original sample. For example, assume that a 2-kg sample of 

gold containing 20 particles o:f gold averaging 0.1 mg 

(milligram) each is brought :from the field. The sample 

has a gold concentration o:f 1.0 ppm (parts per million). 

It is to be analyzed by an atomic-absorption process 

that utilizes a 10-g analytical portion. The hypothetical 

sample can be divided into 200 analytical portions but 

contains only 20 particles o:f gold; the chances are only 

about 1 in 10 that any randomly split analytical portion 

will contain 1 or more gold particles. I:f a particle did 

occur in the sample, the analysis would indicate a misleading 10.0 ppm, compared to the true concentration in 

the sample o:f 1.0 ppm. 

Such a problem results :from the particle-sparsity effect ( CH:fton and others, 1967), whereby the analysis :for 

a component such as gold, based on a split o:f unprocessed 

sample, depends more upon the chance occurrence o:f 

particles in the analytical portion than upon the actual 

concentration within the sample. Figure 5, based on calculations described in the section "Mathematical analysis" graphically illustrates the particle-sparsity effect. 

In the hypothetical sample described above, :for example, 

the expected . number o:f gold particles in each 10-g 

analytical portion is 20/200, or 0.1. Figure 5 shows that, 

:for this value o:f expected nu1nber o:f gold particles, 

slightly more than 90 percent o:f all the possible analytical portions would contain no gold particles. 

As the hypothetical sample has a gold value o:f economic interest, about $1.00 per ton, a 90-percent chance 

o:f finding no gold by analysis is intolerable. Even i:f the 

gold particles were only one-tenth as large, weighing 

0.01 mg apiece, a gold concentration o:f 1.0 ppm would 

lead to only 1.0 gold particles expected in a 10-g analytical portion. Figure 5 shows that, :for this value o:f expected number o:f gold particles, the chances o:f an 

analysis finding no gold would be 37 percent, still an intolerably high figure. 

The particle-sparsity effect pertains to some degree 

whenever the analytical portion contains :fewer than 

the number o:f particles required :for a given degree o:f 
SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS 

90 --------

----)----

80 

70 

a.. 

~ 

::;: 60 <;15 

...... 

-' 

CQ 

~ 0 

a.. 

=I 50 < 

..... 

0 

...... 

<.::1 

No gold in sample 

t5 

ffi 40 0 

ffi 

a.. 

30 

20 

10 

0 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 

EXPECTED NUMBER OF GOLD PARTICLES (A) IN SAMPLE 

FIGURE 5.-Percent of all possible samples in which the relative error (B) is less than or equal to -100, -50, 0, 

+50, and +100 percent, plotted in relation to the expected number of gold particles (A) in the sample. 

The curve for which B=-100 percent shows~, for a given value of A, the percentage of all possible samples 

that will contain no gold. The curve for which B~+100 percent shows the percentage of all possible 

samples in which the observed gold content will be no more than twice the gold content of the deposit. 

Similarly, the curves for which B~+50 percent, B~O. and B~-50 percent show the percentages of all. 

possible samples in which the observed gold content will be no more than 1.5, 1.0, or 0.5 times the gold 

content of the deposit. For example, if the grain size-grade combination of the deposit is such that a sample 

of given size is expected to contain 1 gold particle, then 37 percent of all possible samples of that size will 

contain no gold, 74 percent of the samples wiU have gold analyses equal to or less than the gra:de of the 

deposit, and 93 percent of the samples will have gold analyses no more than twice the grade of the deposit. 

C7 

precision. The effect can largely be overcome by concentrating all of the gold in a sample into the analytical 

portion or portions prior to analysis (a procedure referred to herein as "preconcentration"). Figure 6 shows 

those combinations of grade and grain size which 

necessitate preconcentration. It also distinguishes those 

samples that need to be preconcentrated, not to produce 

a representative analytical portion, but to increase the 

grade of gold to the level of detectability by atomicabsorption techniques. Distinction between these two 

purposes is important. In ~the small range below the 

level of detectability it is possible to analyze samples 

without preconcentration to determine whether or not 

gold is present at the limit of detec~tability. Elsewhere 

in the field of the graph, preconcentration is required 

for even this decision. 
CB 

J03 

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~ 

ii? 

<.:> ]02 0 

c:: 

(..) 

~ 

~ 

vi 

~ ~ 

u.J 

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(..) 

;:::: 

c:: 

cC 

a.. 

JOO 

J0-1 

10-2 

J0-3 

JOOkg 

SHORTER CONTRIBUTIONS TO ECONO~C GEOLOGY 

!Okg !kg 

Area of no preconcentration for 1000 g 

analytical portions 

100 g 

Area of no preconcentration for 100 g 

analytical portions 

Area of no preconcentration for 29.166 g 

(I assay ton) 

Area of no preconcentration for 10 g 

portions 

Area of no preconcentration for 2 g 

portions 

10 g 1 g 

SIZE OF ANALYTICAL PORTION EXPECTED TO CONTAIN 20 GOLD PARTICLES 

U') 

c:: 

u.J 

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:g 

3 

~ 

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f1 

5 

:I:: 

e; 

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0 

4.0 

2.0 

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0.016 

0.008 

0.004 

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0.062 

( .031 

FIGURE 6.-Re'lation of gold grain size and grade to preconcentration requi-rements for some common sizes of analytical 

portions. (For explanation see opposite page.) 
SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS C9 

PARTICLE-SIZE VARIABILITY 

The assumption has been made that the gold to be 

sampled is of uniform size. In practice, gold particles almost certainly range across a size spectrum, although in 

some deposits this range may be relatively restricted 

(Clifton and others, 1967). The variation in the sizes 

of gold particles is another source of error which must 

be considered when selecting adequate sample size. As 

the size range of the partiCles increases, the relative 

errors of the analyses will increase, even if the number 

of gold particles per sample remains the same. Therefore, the relative errors for samples in which the particle 

size varies will be greater than indicated in figure 1, 

and larger samples will have to be taken in order to 

maintain a specified precision. 

The effects of nonuniform particle size can be taken 

into account by assuming that the gold particles have 

some uniform mass larger than the average mass per 

gold particle in ·the sample. By average mass per gold 

particle, we mean the total mass of gold in the sample 

divided by the number of gold particles in the sample. 

If an adequate sample size is determined from figure 3, 

assuming that the mass per gold partide is larger than 

the average mass, the sample would actually contain 

more than 20 gdld particles. If .the proper mass per gold 

particle is chosen, the sample could be said to contain 

effectively 20 gold particles, in the sense that the precision of the gold analysis would be the same as that 

predicted by figure 1 for a sample· containing 20 gold 

pn.rticles of uniform size. This proper mass per gold 

particle will be termed the "effective" mass per gold 

partic1e. 

Gy (1967, p. 51, eq 4.96) and Prigogine (1961, p. 22, · 

eq 32b) have shown that, if the grain-size distribution 

of the particles is known, the measure of grain size that 

can be used to calculate the effective gold-particle mass, 

and which therefore may be called the effective grain 

size or diameter, de, is calculated by the equation 

where M1 is the mass of gold in size grade j, d1 is the 

midpoint diameter of size grade j, and M is the total 

mass of gold in all size grades. Gy (1967, p. 33, eq 3.6) 

shows that the proper value for d1

8 is one-half the sum 

of the cubes of the sieve openings bounding size grade j. 

The effective grain diameter, do, may be converted to 

the effective particle mass by use of the vertical scales 

in figure 2 or 3. 

Preliminary analysis of unsized splits provides another means of estimating effective particle size. For 

example, a 1-kg sample could be split randomly into 

a hundred 10-g portions, each of which is independently 

analyzed. The variability in the analytical values, as 

measured by the relative standard devia;tion or coefficient of variation (~table 1), gives an estimate of the 

effective number of gold particles in .the sample (fig. 

1 and table 1). Note that the value of the relative standard devia;tion is intermediate between the values of 

positive and negative relative errors at 68-percent confidence (fig. 1). If gold is not detected in some of the 

analyses, the percent of the analyses in which it is not 

detected may be compared with the percent of all possible analyses in which no gold particles are present (fig. 

5) to give another estimate of the number of gold particles (table 1) . 

. EXPLANATION OF FIGURE 6 

Preconcentration is required for any combination of grain size and grade to the left of the areas indicated. For example, 

a sample containing gold particles that are unifotrm.ly 10 mg in mass and 4 ppm in grade need not be concentrated prior 

to analysis in 100 or 1,000 g portions but woll'lid require concentration prior to being analyzed in one of ·the smaller sized 

portions shown. The preconcentration specified here maintains the degree of precision given by 20 particles per sample 

provided the particle size is assumed to be uniform. Vertical boundaries are located by grain size-grade combinations that 

will produce 20 particles in the analytical. portion indicated. Diagonal boundaries are located by the minimum detectability 

for each size of analytical portion. Below the diagonal boundaries, preconcentration is required, not to provide a statistically representative analytical portion but to increase the grade of the portton to the level of detectability, The detectable 

limits shown apply specifically to the atomic-absorption analytical methods descri'bed by VanSickle and Lakin (1968), as 

modified, for 1,000 g portions, by K. W. Leong. The limits of detectability may diffel'l slightly depending on the instrument 

used and the D1ature of the material to be analyzed (K. W. Leong, oral commun., 1968). For different requirements of 

precision, the conversion scales given by figuM 4 may be use!d. 
ClO SHORTER CONTRIBUTIONS TO ECONOMIC GEOLOGY 

TABLE 1.-Estimate of effective number of gold particles from 

distribution of analytical values 

Distribution of analytical values 

Gold detected in all analyses, very to fairly uniform values (8,<0.22) '--- ____________________________________ .- __ -- _ ----

Gold detected in all analyses, some variation (8,=0.22-0.50) __ Gold detected in most analyses, much variation (8,=0.50 

-1.0) ------------------------------------------------------ Gold detected in less than hall of the analyses, much variaGold tion (8r> 1.0) ----------------------------------------------- not detected in any of the analyses ____________________ _ 

Effective number of 

gold particles per analyzed subsample 

>20 

4-20 

1-4 

<1 

0 

' The relative standard deviation, or coefficient of variation, 8r, is the standard 

deviation of the gold values divided by the mean of all the values. The standard 

deviation, 8, may be calculated. by the equation 

( 

N )2 N 2; A; 2; A;2--'~-t-. ,_, N 8= . N-1 

where A, is the value of the gold analysis of the itb subsample and N is the number 

of analyzed subsamples (Dixon and Massey, 1957, p. 19). 

The estimate of effective number of particles per kilogram may be plotted in figure 2 against the average 

grade produced by the combination of the analyses o:f 

all the spHts. The intersection of these two parameters 

·gives an estimate o:f the effective size o:f gold particles 

in the sample. The ·accuracy of the estimate improves as 

the number of analyzed splits is increased, but the method can be used even with a small number of splits. 

If the size distribution of the gold particles is uricertain and analyses of splits are not available, adequate 

sample size can be determined on the assumption that 

all o:f the gold in the sample is uniformly as large as the 

maximum significant size. By maximum significant size 

is meant the size near the coarse end of the size frequency curve, a size ·beyond which coarser particles contribute only insignificantly to the total gold content. 

The coarsest gold particles, if by chance occurring in a 

sample, will produce a misleadingly high analytical 

value; however, their likelihood of occurrence is directly 

proportional to their significance within the deposit. 

Once the gold-particle-size distribution is roughly 

established or estimated, ;the maximum significant size 

can be defined in accordance with the requirements o:f 

the study. In our studies we ·have arbitrarily considered 

the coarsest 5 percent '(by weight) o:f the gold to be 

insignificant. Adequate sample size can then be determined by applying the estimated maximum significant gold size rather than the estimated effective gold 

size to figure 3. The assumption that all of the gold 

in a sample is uniformly as large as the coarsest significant gold provides an added safety :factor in determining adequate sample size and should help to ensure a 

representative sample. 

APPLICABILITY TO NATURAL SAMPLES 

Detailed analysis o:f a single sample demonstrates 

the.application o:f figures 2 ~and 3 to natural samples. A 

sample· weighing 80 pounds was taken from magnetiterich beach sand north o:f Gold Beach, Oreg. The sample 

was thoroughly mixed ·and then split into ·analytical 

fractions through a Jones-type sample splitter in order 

to avoid the possibility that ~arirubility in analytical results might be due to internal inhomogeneity of the gold 

distribution within the sample. The gold-particle-size 

range ·and grade in this sand were established by analysis 

o:f sieved fractions. The effective :and maximum significant gold parti'Cle size determined fro ~rt,e analyses 

o:f the sieved fractions were compared to the effective 

gold particle size estimated from the distribution of 

analytic data from a number of unsieved splits. Adequate sample size was determined from figure 3 by 

using the various estimates of effective and maximum 

significant gold particle size, and the different sample 

sizes thus indicated were evaluated by comparing the 

results o:f analysis of samples of different sizes. 

·Initially, a split weighing a:bout 5,000 g was sieved 

and the size fractions analyzed for gold, using a combination of wet chemical and atomic-absorption techniques (Lakin and Nakagawa, 1965). Table 2 and 

figure 7 summarize ther;:;e analytic data, which show 

the gold to range in size from less than 0.044 to 0.354 mm 

(millimeter). The size frequency curve of this distribution indicates a maximum significant gold particle size 

of about 0.27 mm (fig. 7) ·; particles larger than this 

constitute less than 5 percent of the total gold. Gold 

grains from the deposit were examined under the microscope and found to be :flakes whose thickness is about 

one-tenth of the diameter. The average gold content of 

all analyzed splits of this sample is 0.38 ppm. The inter-

. section of this grade in figure 3 with the maximum significant diameter indicates that about 1,500 g should 

consti·tute an adequately representative sample. In conTABLE 2.-0ccurrence of gold in size .fractions of a 5,150-g beach 

sand sample 

Size class 

(mm) Weight of gold in size class 

~g) 

o.354-o.5oo_ ____ __________ _________ <o. 4 

0.250-0.354_----------------------- 138 0.177-0.250________________________ 195 

0.124-0.177------------------------ 327 

0.088-0.124_-- --------------------- 680 0.062-0.088________________________ 340 

0.044-0.062_ ----------------------- 126 <0.044________________________ 8. 6 

------'---'-- TotaL ____ --------___________ 1815 

Percent of 

total gold 

0. 0 

7. 6 

10.7 

18.0 

37. 5 

18. 7 

6. 9 

. 5 

99. 9 
SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS Cll 

90 

~ 70 (.) 

i= 

~ 

~ 

9 60 0 

(!) 

LL 

0 

LJ.J 

(!) 

~ 50 

z 

LJ.J 

(.) 

~ 

LJ.J 

a. 

1-

::r: 

(!) 

L;j 

~ 

40 -:-

LJ.J 

> 30 

~ ::> 

:::?:: 

::> 

(.) 20 

10 

5 

0.354 

Vl 

c: 

E 

0 

c. 

01) 

0 

§ 

"C 

Q) 

> Q) 

'iii 

c: 

:::1 

c: 

0 

"C 

~ ro 

.0 

Q) 

Q) N 

N 'iii 

'iii Q) 

Q) 

] ~ ro 

~ c. c. "C 

"C 0 

0 01) 

01) 

c 

ro 

~ ·c: 

01) 

'iii 

0.250 

Vl 

c: 

:e 

0 

c. 

01) Vl 

'iii 0 >. s -m 

c: "C ro c: Q) ro > 

s Q) 'iii 

"C c: Q) 0 > Q) "C 

'iii ~ c: ro :::1 .0 

c: Q) 

0 N 

"C 'iii Q) Q) Vl 

~ ro 

.0 

Q) ro N c. 'iii "C Q) 0 

~ 01) 

ro 

c. 

"C 

0 

01) 

0.177 0.124 0.088 0.063 0.044 

DIAMETER, IN MILLIMETERS 

FIGURE 7.-0umulative curve of gold grain-size distribution in a sample of beach sarid. Shown are the various 

measures of particle size for a deposit in which gold grain· size is variable, including the maximum significant. 

size (0.27 mm, equal to the coarsest 5th percentile), the effective particle size as estimated from the distribution 

of analytical data from unsieved 1,000-g portions (0.24 mm), from 100-g portions (0.21 mm), from 10-g portions 

(0.21 mm) and as calculated from the analyses of sieved fr:actions (0.17 mm). 

trast, the effective grain diameter, calculruted by Gy's 

equation in the foregoing section, is 0.17 mm (fig. 7). 

The application of this size to figure 3 indicates that a 

sample of about 360 g would provide adequate 

representation. 

In addition to being calculated from the analysis of 

size fractions, the effective gold-particle diameter was 

estimated from the distribution of analytical data for 

a number of unsieved splits. This method, described in 

the preceding section, should be useful in estimating 

particle size where sieve analyses are not available. 

Three series of unsieved splits, weighing about 10 g, 

100 g, and 1,000 g per split, were analyzed. 

The analy•tic results from ninety-six 10-g splits are 

shown in figure 8. The relative standard deviation of the 

analytical v·alues is 1.69 (table 3), and 49 of the splits 

contained no gold particles coarse enough to be detected. 

This distribution of data suggests (table 1) that the 

number. of particles per split is less than one. More precisely, the value of the relative standard deviation indicates (see fig. 1, value extrapol·ated to values of X less 

than·l.O) an average of effectively 0.36 gol~ particles 

per 10-split or 36 particles per kilogrmn of sample. Th~ 

effective size of these particles, as found by projecting 

this nmnber in figure 2 against ~he average grade of 0.52 

ppm for the 96 splits, is approximately 0.21 mm. Figure 

3 indicates that, for gold of this size from a deposit. 

having a grade of 0.38 ppm (the aver.age of all analyzed 

splits of the sample), a 750-g sample would suffice. 
CI2 SHORTER CON'l'RIBUTIONS TO ECONOMIC GEOLOGY 

z 

0 

:::::; 

.....J 

~ 

~ 

LIJ 

a.. 

(/) 

1-

a::: 

0: 

:; 

,...: 

:::::; 

a. 

en 

0 

LIJ 

N 

~ <( 

z 

<( 

LL 

0 

LIJ 

0 

<( 

a::: 

(.!) 

0 

-l 

0 

(.!) 

10.0 

9 

8 

7 

6 

5 

4 

3 

2 

1.~ 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4' 

0.3 

0.2 

0.1 

0.09 

0.08 

0.07 

0.06 

0.05 

0.04 

= 

= -

-

= 

= 

= 

"" 

--

-

-

-

~"'49 samples 

< 0.04 ppm 

= 

= 

c: 

---

-~-

= 

100 1000 

WEIGHT OF ANALYZED SPLIT, IN GRAMS 

l!"'IGURE B.-Distribution of analytical values for three sample 

sizes. The values shown are for ninety-six 10-g samples, forty 

100-g samples, and nine 1,000-g .samples that were split from 

an 80-pound sample of beach sand. 

The analytical results from forty 100-g splits and nine 

1,000-g splits are also shown in figure 8. The relative 

standard deviations are 0.69 for the 100-g samples and 

0.23 for the 1,000-g samples (table 3). These values indicate (fig. 1) that the 100-g splits contained effectively 

2.1 gold particles per split and that the 1,000-g splits 

contained effectively 19 particles per split. By applying 

these values to figure 2, tli.e effective gold particle size is 

estimated to be 0.21 mm for the 100-g splits and 0.24 

mm for the 1,000-g splits. By using t~ese sizes in figure 

3, it can be seen that adequate sample sizes of 750 g and 

1,100 g are indicated by the analytical results of the 

100-g splits and 1,000-g splits (table 3). 

TABLE 3.-Measu1"es of gold pa1"ticle size for splits of a large 

sample, adequate sample sizes predicted from these data, and 

statistical parameters by which these were calculated 

1(}-g 1QO-g 1,00(}-g splits splits splits 

Number of splits.-------------------- 96 40 9 

Mean gold grade, in parts per million 1_ 0. 52 0. 32 o. 39 

Standard deviation, in parts per 

5,150-g sieved split 

1 

. 0.35 

million. ___________ -------- ___ • ____ • 0. 88 

Relative standard deviation.......... 1. 69 

.0.22 

0.69 o. 09 ---------------- 0.23 ---------------- Effective number of gold particles per 

Efe~:ive · goid.--particie · d'iiillieier;-iii· o. 

36 

millimeters (assuming flake shape). Maximum · significant gold particle 

diameter, in millimeters •• -----------•••••••• ----•••••••••• -·-··-- Adequate sample size in grams (sample stze expected to contain 

2.1 19 

0.21 0.21 0.24 

effectively 20 particles).------------ 750 750 1,100 

0.17 

0.27 

a 360, a 1, 500 

I The mean gold grade of the total14 kg of analyzed sample is 0.38 ppm. This was the figure used, in combination with the measure of gold particle diameter, to determine adequate sample size from figure 3. 2 Based on effective gold particle diameter. a Based on maximum significant gold particle diameter. 

A comparison of the adequate sample sizes obtained 

by using the various estimates of effective particle size 

shows a significant difference between the value, 360 g, · 

obtained frmn analysis of sieved fractions, and the 

values, 750 to 1,100 g, obtained from analysis of unsieved 

splits. In contrast, the adequate sample size given by 

using the maximum significant gold particle size obtained from the analysis of sieved fractions is 1,500 g, a 

figure greater than those given by analyses of unsieved 

splits. 

The reason for the discrepancy between the effective 

gold particle size estimated frmn the analysis of sieved 

fractions and the effective sizes estimated from analyses 

of unsieved splits is uncertain. Among the possible reasons are imperfection in the splitting of samples and 

analytical error. A smrull par.t of the discrepancy could 

be due to the fact that the splits were taken from a 

sample of finite size rather than of infinite size (Deming, 

1950, p. 113). Whatever the reason, the safety margin 

inherent in using the maximum significant gold particle 

size in calculating adequate sample size is evident. 

MATHEMATICAL ANALYSIS 

The minimum sample size considered adequate for 

gold analysis may be defined in terms suitable for statistical analysis as that sample size which allows one to 

have a certain confidence that the true gold content of 

the deposit is within a certain acceptable range of values. 

The confidence may be defined in percentage units as 

o= 100- a), where a, the level of significance as. de-
SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS C13 

fined in statistical usage (Dixon and ~iassey, 1957, p. 

89-91), is the proba.bility that the true gold content of 

the deposit actunJly lies outside the range of values. The 

range of values is the confidence interval, a.nd the limits 

of the range of values are the confidence limits (Dixon 

and ~iassey, 1957, p. 79-80). 

The minimum adequate sn.mple size may be defined 

mn,thema.ticaJly as that sample size for which 

(1) 

where Pr is a notation meaning "the probability 

that ... ",Pis the true mass proportion of gold in the 

deposit, La is the lower o percent confidence limit, and 

U o is the upper o percent confidence limit. Lc and U c 

are in the samP- units asP. A commonly used value foro 

is 95, in which case Lc and Uc are the lower and upper 

95 percent confidence limi~ts. In general, La will be 

smaller than Po, the observed mass proportion of gold 

in the sa.mple as determined by chemical analysis, and 

U o will be larger than Po· 

It is n.dvantageous :for the present purpose to modify 

equation ( 1) to 

PT[Po+Ec-Po<P<Pa+Ec+Po]=c (2) 

where E- c and E+ c n1ay be termed the negative and 

positive relative errors at c percent confidence. The 

relative errors are 1neasures of the probable extent of 

error in the estin1ation of P by P 0 • They are called 

relative because they fl;l'e dimensionless nun1bers that 

1neasure the error relative to P 0 • 

The minimun1 percent confidence and the maximum 

width of confidence intervn.l~that are considered appropriate for gold analyses must be decided arbitrarily. 

These decisions will determine the mini1num adequate 

sample size. 'V\T e will require here that we be 95 percent 

confident thrut Pis in the interval Po+ approximately 

0.5 P0 • In other words, \ve will require tha't the relative 

errors n;t 95-percent confidence be equal to approximately -0.5 and +0.5. It should be noted that relative 

errors of ±0.5 at 95-percent confidence are approximately equivalent in precision to relative errors of 

±0.25 at 68 percent confidence. For studies of reconnaissance nature, a lower precision might suffice; on 

the other hand, certain very detailed studies might 

require a higher precision. 

The statisticn,l ann,lysis of the problem of adequate 

sa,mple size may be grea.tly simplified by assuming that 

(1) the gold particles are of uniform mass, (2) that 

the other pa,rticles are also of uniform ma,ss, although 

not necessarily of tl1e same mass as the gold particles, 

and (3) that the mass proportion of gold in the sample 

is less than 0.01. If these ·assmnptions are true, the mass 

percentage of gold is proportional to the percentage of 

gold by number of grains. Specifically, the mass percentn.ge of gold is equal to the percentage by number 

of grains multiplied by the gold particle mass and divided by the particle mass of the other particles (Prigogine, 1961, p. 12, eq. 4a). Then, for the same sample 

size for which equation (2) was true, it also is true that 

(3) 

where p is the porportion of gold in the deposit in 

terms of numbers of particles, P'o is the observ~d proportion of gold in the sample in terms of numbers of 

particles, and the relative errors are the same as those. 

in equation (2). · 

Equations giving the relation between sample size 

and the relative errors, E-; and Et, in equations (2) 

and (3) may be derived from the binomial distribution. 

The population may be considered to consist of gold 

particles and particles that are not gold, in which gold 

particles form a proportion, p, of the population. 

Strictly speaking, the statistical population is not the 

deposit itself but consists of all the possible samples of 

a given size that could be taken from the deposit. 

The proportion of gold particles in a sample, p0 , is 

equal to X/N, where X is the number of gold particles 

in the sample and N is the number of all the particles 

in the sample. In random samples from this type of 

population, the best estimate of p is Po and the sampling distribution of Po is binomial (Dixon and Massey, 

1957, p. 288). 

The problem of collecting a truly random sample 

from a finely granular deposit is difficult. Grains cannot 

be selected one by one randomly fron1 the entire deposit, 

nor is it possible ·to pass the entire deposit through· a 

sample splitter to obtain a random sample. Rather, all 

the grains in a small part of the deposit must be collected. Samples collected in this way will give a binomial sampling distribution of JJ 0 only if the gold 

grains n,re randomly distributed through the deposit. 

If systematic trends in gold content are present and 

not taken into account by the sampling plan, the sampling distribution of JJ0 may show greater variability 

than does the binomial distribution. The effects of nonrandmn distribution of gold particles can be minimized 

by a proper sampling plan, but such procedures are 

beyond the scope of this paper. It is simply assumed 

here that the sample is a random one from a binomial 

population. 
C14 SHORTER CONTRIBUTIONS TO ECONOMIC GEOLOGY 

If the sampling distribution of Po is binomial and if 

Np>5 and N(1-p)>5, the c percent confidence limits 

for estimating p are 

N [P + Z~-1;2a ±Z fpo(1-po) + Z~-v2a] N+Z~-112a o 2N 1

- 112a"/ N 4N2 

(4) 

where Zt-~a is read from a table of the cumulative 

normal distribution (Dixon and Massey, 1957, p. 229). 

The very large values of N and the very low values 

of Po expected in samples from detrital gold-bearing 

deposits allow simplifications to be made in formula (4). 

The simplifications are valid for commonly used values 

of a. Specifically, they are valid when -~a is at least 

as large as 2.57, or when a is as small as 0.01. 

If N> 103, corresponding to a sample weight of 

about 0. 7 g of medium-grained quartz sand, the factor 

NjN +Z t-~a in formula (4) is approximately equal to 

1 and the factor _~a/4N2 is approximately equal to 

zero. Then, formula ( 4) may be simplified to 

P + Z~-112a ±Z ~ Po(1-po). o 2N 1-1;2a N (5) 

If Po<1o-a, as is true in all but extremely rich gold 

deposits, the factor 1-p 0 in formulas (4) and (5) is 

approximately equal to 1, and the formula for the c 

percent confidence limits for estimating p may be 

further simplified to · 

+ z~-l/2a z !Po 

Po 2N ± 1-1/2a-v JJ" (6) 

Formula (6) may be equated to the terms Po+ 

E-;po and Po+Etpo in equation (3). Solving for 

E-; and Et, it is found that 

E-= ![Z~-I/2a_z !Pol c Po 2N 1-1/2a-v N J (7) 

E+= .!_[Zi-1;2a+Z /Po]. c Po 2N t-1t2a'V N (8) 

Substituting XjN for Po in equations (7) and (8) and 

simplifying, it is found that 

E-=x-112 cz~-1;2a x-112_z J c 2 1-1/2a (9) 

E+ = x-1;2 c~-t/2a x-1t2+Z . J c 2 . 1-1/2a • (10) 

Equations (9) and (10) indicate that, in the range 

of N, Po, and a considered here, the expected relative 

errors at a given percent confidence are functions of 

only one variable, the number of gold particles in the 

sample. In the problem of determining adequate sample 

size, the values of c, E-;_, and E~ are arbitrarily set, 

and equations· (9) and (10) can be used to determine 

t~e value t~~ have:kfrhe minimum adequate 

~ample size, then, is the weight of sample necessary 

f ~or the expected number of gold particles to equal X. 

! ~efore a sample from a gold deposit ·of only roughly 

; known gold particle size and grade is analyzed, the 

, 1expected number of gold particles in the sample can 

! ;be predicted only roughly. After the sample is analyzed 1

1 and .the particle siz.e of its gold determined, the value 

,:,Qf X can be determmed from figure 2. Then, the values 

. of E; and E~ actually resulting from the analysis 

of that particular sample can be determined by equations (9) and (10). 

For the purpose of this study, we have defined the 

minimum sample size as that sample size for which 

E95- and E95+ are equal to approximately -0.5 and 

+0.5. For 95 percent confidence, a=0.05, -~a= 1.960, 

and from equations (9) and (10), 

E95-=X- 112 (1.921X- 112 -1.960) (11) 

and · 

E95+=X- 112 (1.921X- 112+ 1.960). (12) 

The values of E 95- and E 95+ are not precisely equal, but 

for X=20, E95+ is slightly greater than 0.5, being equal 

to 0.54, and E95- is smaller in absolute value than 0.5, 

being equal to 0.34. Therefore, we will accept a sample 

containing 20 gold particles as the minimum sample 

size adequate for gold analysis. If a lower or higher 

degree of precision is required, the minimum sample 

size will contain fewer or more than 20 gold particles 

Figure 1, showing values of the relative errors at 68 

and 95 percent confidence and for a large range of X, 

can be used by those who wish to define adequate sample. 

size for other values of percent confidence and relative 

error (fig. 4). The figure can also be used to evaluate 

the reproducibility of a gold analysis after it is obtained 

assuming that analytical errors are absent and that the 

gold particles are of known uniform mass. It should be 

noted that, although the relative errors are drawn as 

continuous functions of X in figure 1, X may have 

integral values only. 

In equations (9) and (10), when X is greater than 

1,000 and c is 68 percent (or Z1_ 112a=1), the absolute 

values of E'JB and E6s are approximately equal to x-112

• 

In comparison, the relative standard deviation or 

coefficient of variation of X in a binomial population 

,~ 
SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS C15 

having a low value of p is equal to (Np)-112

, where Np 

is equal to the expected value of X (Deming, 1950, 

p. 114-119). It may be shown that the relative standard deviation of X is equal to the relative standard 

deviations of po and Po in the model considered here, 

and that the equations of de Magnee (1956) and 

Prigogine (1961, p. 18, eq 22) for the relative standard 

deviation of Po are equivalent to Deming's equation 

for the relative standard deviation of X. The relative 

standard deviation, s" is plotted in figure 1. 

In the range of N and p considered here, the Poisson 

distribution is a good approximation of the binomial 

distribution and can be used to define relations between 

X, Ec-, and Ec+ essentially the same as those shown 

in figure 1 (Ricker, 1937). The Poisson distribution may 

also be used to define the relation between the relative 

error and the expected number of gold particles, which 

may be called X and is equal to Np. Given a value of X, 

the proportion of all possible samples containing a 

given number of gold particles, X, can be determined 

from the equation of the Poisson distribution. The 

deviation of X from Np may· be defined, as in the 

discussion bn,sed on the binomial distribution, in terms 

of a relative error, E, such that 

X-X E---· X (13) 

Figure 5, derived from a table of the Poisson distribution (Dixon and Massey, 1957, p. 435), shows the 

probability that the relative error, E, will be less than 

or equal to a given value for a given value of Np, or,\. 

This probability is, by definition, the proportion of all 

possible samples in which E is less than or equal to the 

given value. It should be noted that, although the relative errors are drawn as continuous curves in figure 5, 

they have real values only at integral values of X. 

CONCLUSIONS 

To ensure adequate representation, a field sample 

should be sufficiently large to contain at least 20 o-old 

'1 0 part1c es. The number of gold particles per unit weight 

of san1ple depends on the grade and the o-rain size of the 

gold. Figure 3 shows the sample sizes than can be expected to contain 20 particles for different combinations 

of _these two parameters. Smaller particle-per-sample 

ratios may suffice for certain reconnaissance studies but . ' lack the margin for error inherent in the choice of 20 

particles per sample. Figure 4 may be used to establish 

sa1~ple-size requirements for other particle-per-sample 

ratw_s ~nd degrees of precision required of the analyses. 

Preliminary analyses of the material may be necessary 

to establish the range of gold particle size and grade that 

can be expected. 

. The charts (figs. 2, 3, 4) specifically cover the particle 

s1ze and grades that may be expected for detrital gold. 

They readily may be expanded to cover other fields of 

grain size or grade. The charts also may be used toestablish sample-size requirements for other heavy metals 

such as platinum, silver, or mercury, and for other heavy 

minerals (fig. 9). In these cases, it is only necessary to 

change the grain size-mass relationships accordino- to the 

density and grain shape of the material being s~udied. Grain size-mass relationships for minerals other than 

those shown in figure 9 could be developed. To establish 

adequate sample sizes for reliable analysis of materials 

other than gold, however, it is necessary that the srune 

assumptions be 1nade that were made in derivino- there- o 

lation of analytical precision to number gold particles 

in a sample. \Vithin these limitations, the methods presented here apply to any kind of ~nalysis for particulate 

trace elements. 
r 

C16 

>-

t-

~ 0:: 

<!' 

(.) 

u:: 

u 

w 

c.. 

(/) 

20.0 

19.0 

18.0 

17.0 

16.0 

15.0 

14.0 

13.0 

12.0 

11.0 

10.0 

9.0 

8.0 

7.0 

6.0 

5.0 

4.0 

3.0 

2.0' 

1.0 

' 

~Platinum (14-19) j Gold (15-19.3) 

3 - Native mercury (13.6) 3 

- Native silver ( 11.0) 

- Tantalite (7.3). 

- Cassiterite (7.0) 

Columbite (5.3) = Mo.nazite (5.0) 

Zircon (4.6) 

- Rutile (4.2) 

SHORTER CONTRIBUTIONS TO ECONOMIC GEOLOGY 

10,000 1000 100 10 0.1 

PARTICLE MASS. IN MICROGRAMS 

FIGURE 9.-Relationship between pa:rticle mass and g.rain size (diameters of spheres) of different pal'ticulate heavy metals 

and minera•ls as a function of specific gravity. The specific gravity of heavy metals that are likely to occur ·as discrete 

prurticles is shown at the left of the figure. The different relationships can be applied to figures 3 and 6 to determine 

adequate sample size and preconcentration requirements for minerals of any specific gravity, provided the assumptions 

made in relating precision of gold analys·es to number of gold particleS! per sample are also true for deposits of these 

other minerals. If the analysis is for a metal component o·f a particulate mine1~a1 (for example, tin in cassiterite), the 

corresponding grades of the mineral must be calculated from the analytical data before application to figures 3 and 6. 
SAMPLE SIZE AND MEANINGFUL GOLD ANALYSIS C17 

REFERENCES CITED 

BecJ;:er, R. 1\I., 19G4a, Sampling to n items per sample, Pt. 1 of 

Some generalized probability distributions with special reference to the mineral industries: U.S. Bur. Mines Rept. Inv. 

6329, 53 p. 

--- 1964b, Sampling to A amount of items per sample, Pt. 2 

of Some generalized probability distributions with special 

reference to the mineral industries: U.S. Bur. Mines Rept. 

Inv. 0552, 101 p. 

--- 1965a, Computer programs of distribution moments, Pt. 3 

ot Some generalized probability distributions with special 

reference to the mineral industries: U.S. Bur. Mines Rept. 

Inv. 6598, 79 p. 

--- 1965b, Experimental confirmation, Pt. 4 of Some generalized probability distributions with special reference to the 

mineral industries: U.S. Bur. Mines Rept. Inv. 6627, 57 p. 

--- 1966, Theoretical confirmation and applications, Pt. 5 of 

Some generalized probability distributions with special reference to the mineral industries: U.S. Bur. Mines Rept. Inv. 

6768, 60 p. 

Clifton, H. E., 1967, Sample size preconcentration requirements 

for meaningful analysis of gold: U.S. Geol. Survey open-file 

report, July 13, 1967, 1 chart. 

Clifton, H. E., Hubert, Arthur, and Phillips, R. L., 1967, Marine 

sediment sample preparation for analysis for low concentrations of detrital gold: U.S. Geol. Survey Circ. 545, 11 p. 

Deming, ,V. E., 1950, Some theory of sampling: New York, John 

\Viley and Sons, Inc., 602 p. 

Dixon, \V. J., and Massey, F. J., 1957, Introduction to statisticai 

analysis [2d ed.] : New York, McGraw-Hill Book Co., Inc., 

488 p. 

Gy, Pierre, 1954, Erreur commise dans le prelevement d'un 

echantillon sur un lot de minerai [Errors committed in 

taldng a sample from a lot of ore] : Revue de l'Industrie 

1\:finernle, v. 35, p. 311-345. 

---1956, Poids it donner il un enchantillon. Abaques 

d'echantil1onnage [Weight to give to a sample. Nomograph 

of sampling] : Revue de l'Industrie Minerale, v. 38, no. 

636, p, 53-99. 

---1967, Theorie Generale [General theory], v; 1 of 

L'echantillonnage des minerals en vrac [The sampling of 

minerals in bulk] : Bur. Recherches Geol. Minieres Mem., no. 

56, 186 p, 

Krumbein, ,V, C., and Graybill, F. A., 1965, An introduction to 

statistical models in geology: New York, McGraw-Hill Book 

Co., Inc., 475 p. 

Lakin, H. W., and Nakagawa, H. M., 1965, A spectrophotometric 

method for the determination of traces of gold in geologic 

matelials, in Geological Survey research, 1965: U.S. Geol. 

Survey Prof. Paper 525-C, p. C168-C171. 

de Magnee, M., 1956, in Gy, Pierre, Poids a donner it un 

echantillon. Abaques d'echantillonage; Discussion sons la 

presidence de M. Testut [Weight to give to a sample. Nomograph of sampling. Discussion under the chairmanship of 

Mr. Testut] : Revue de l'Industrie 1\iinerale, v. 38, no. 636, 

p. 94-96. 

Miesch, A. T., 1967a, Chart showing number of particles of metal 

expected per 5-pound sample for various levels of grade and 

average pa1~ticle size (regardless of the spatial distribution 

of particles) : U.S. Geol. Survey open-file report, Jan. 9, 

1967, 1 p., 1 chart. 

---1967b, Theory of error in geochemical data: U.S. Geol. 

Survey Prof. Paper 574-A, p. A1-A17. 

Ottley, D. J., 1966, Gy's sampling slide rule: World Mining, 

v.19,no.9,p.40-44. 

Prigogine, A., 1961, Echantillonnage et analyse des minerais 

heterogenes a faible teneur [Sampling and analysis of 

heterogeneous minerals of a weak grade] : Acad. Royale 

Sci. Outre-Mer, C1. Sci. Tech. l\Iem., n.s. v. 15, no. 1, 180 p. 

Hicl.::er, \V. G., 1937, The concept of confidence or :fiducial limits 

applied to the Poisson frequency distribution: Jour. Am. 

Statistical Assoc., v. 32, p. 349-356. 

VanSickle, G. H., and Lakin, H. W., 1968, An atomic absorption 

method for the determination of gold in large samples of 

geologic material: U.S. Geol. Survey Circ. 5~1, 4 p.