226
Anal. Chem. 1907, 59,226-232
Visman Equations in the Design of Sampling Plans for Chemical Analysis of Segregated Bulk Materials Dean Wallace* Oil Sands Research Department, Alberta Research Council, Edmonton, Alberta, Canada T6H 5 x 2
Byron Kratochvil Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G 2G2
From prelhnkrary measurements, guidellnes are given for determining whether a bulk population is well-mixed or segregated and for the deslgn of experiments to estimate sample size and number. For wetl-mixed populations, the total amount of sample collected determines the degree to which the population is represented. For segregated populations, sampling uncertainty is reduced prharliy by increadng the number of sample increments collected. Methods for estimating Vlsman’s Sampling constants for segregated popyiatlons are discussed, abng with a way of calculating a theoretkal average particle mess for nonpartfculatematerials. An average theoretical partlcle mass of 20 g was calculated for unmixed samples of Athabasca oil sand. This large value, confirmed by using the Benedetti-Pkhler equatlon, explains the extensive sampllng required to estimate rellabty the bitumen and water in oil sand.
In chemical analysis the analyst aims to collect data under conditions where random and systematic errors are in statistical control. In attempting to meet this criterion, analysts often overlook the major contributions to error caused by the sampling stages which precede delivery of the laboratory sample. Users of the data, however, generally require an estimate of the overall error, which includes that introduced by sampling. Preparation of sampling protocols requires effort-so much, in fact, that time or cost is sometimes used as justification for not carrying out preliminary sampling. Why, for example, should a data user determine sampling constants while sacrificing the &moreimportant” goal-a precise estimate of the composition of the population a t the time of sampling? Sampling experiments must be considered as investments that pay off whenever the resulting protocol indicates a reduction in sampling is appropriate. But even if a developed protocol does not permit reduction in sampling, the extent of the sampling uncertainty has been established. Youden defined the overall random error of an analysis, so, as a function of the random errors in each stage of the analysis (1) by Equation 1 gives the overall error in terms of the variances contributed by the sampling, subsampling, and analytical operations. The expression can be modified to incorporate any number of stages. Reduction of the overall standard deviation, so, to acceptable levels is best achieved by assessing the magnitude of the random error in each stage of the analysis. This can be done by first performing replicate analyses of reference materials to yield estimates of analytical uncertainty, so. Next, test portions of a homogenized laboratory sample are analyzed to 0003-2700/87/0359-022680 1.50/0
provide an estimate of the subsampling plus analytical uncertainty, s,+,, which is used to isolate the subsampling term by Equation 2 requires that sa be the same for both the reference material and the material being analyzed. If no suitable reference materials exist, s, cannot be separated from sa and the combined quantity s,+, is used in further calculations. The sampling uncertainty, sg, is similarly egtimated by analysis of several increments taken directly from the parent population to obtain a value for so, followed by substitution into eq 1. Collection and analysis of more test portions from the laboratory sample, or adoption of more precise subsampling and analytical methods, may be needed if random errors in subsampling, sample treatment, and measurement are the major contributors to the overall uncertainty. The presence of a large s, term indicates significant variability in the population. To control sampling uncertainty, a sampling protocol is required that describes the variability for each property to be measured. Sampling constants such as those developed by Ingamells ( 2 )for well-mixed materials and by Visman (3) for segregated populations may be used to calculate the number of sample increments that will hold s, to a specified level. For a given population, values for these constants are obtained by initial testing until they are considered typical of samples to be collected in the future. In this paper we evaluate the relationship between Ingamells’ and Visman’s approaches to sampling and discuss methods for estimating sampling constants that describe segregated bulk materials.
BACKGROUND Determination of Increment Size in Well-Mixed Materials. Ingamells and Switzer proposed a sampling constant, K,, defined as the weight of a single increment that must be withdrawn from a well-mixed material to hold the relative sampling uncertainty, R, to 1%at the 68% level of confidence (2). Ingamells’ sampling constant can be estimated by considering the material to consist of a mixture of two kinds of particles of known volume, density and composition ( 4 ) ,but such detailed knowledge of the material is not necessary. Instead, K,can be determined from the relation
K, = wR2
(3)
where R is the relative sampling uncertainty determined from the analysis of a set of increments of weight w. Application of Ingamells’ constant to a bulk material is valid only when the particles in the sample are well-mixed; under these conditions sampling uncertainty associated with collection of a single increment of weight w will be the same as uncertainty associated with collecting x increments of weight w / x . It is possible to determine with high confidence whether a material is well-mixed by analyzing two sets of increments 0 1987 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987
of widely differing weights, wrs: and wsm,for under well-mixed conditions SLgPWLg
= SSm2WSrn
(4)
Here sLgand sSmare estimates of uLgand uSm,the true population standard deviations, and are themselves relatively imprecise unless large numbers of increments are analyzed. A one-tailed F test should be used to determine whether uLg and differ significantly at a specified confidence level. The consequences may be severe if one decides a sample is yell-mixed when it is not; therefore the test should be applied at a 90% or lower level of confidence. If the null hypothesis is rejected, uLg > ( W S ~ / W L ~ ) ~ / ~ Uthe ~ , ,sample is considered to be segregated, and ssmand sLgcan be used to estimate Visman’s sampling constants for a segregated material as shown in the next section. Determination of Increment Size and Number in Poorly Mixed (Segregated) Materials. Visman developed an equation s,2
A =-
+ -B wn n
(5)
that describes sampling variance,, :s as the sum of random (A term) and segregation (B term) components. When values for the constants A and B are known for a material, the sampling variance for n increments, each weighing w units, can be calculated. Equation 5 was derived by calculating the uncertainty associated with increments of varying sizes measured on a 100 X 100 grid of sites and 2500 lead balls. The balls were placed on the sites in various systematic or random patterns, samples of 1, 3 X 3, or 9 X 9 sites were collected, and the number of lead balls in each sample was recorded (3). Examples included a population that was completely segregated (Le., the degree of segregation, z,equals 1, and all the lead balls occupied adjacent sites in a corner), and populations with values of z equal to 0.33, 0.20, and 0.11. These distributions are distinguished from random distributions by the presence of patches or strata. For a fixed composition, the average size of the patches increases and the number decreases with increasing segregation. Visman’s approach can be used to define the degree of segregation of a chemical substance in a bulk material. It addresses “worst case” segregation from a sampling perspective, that is, the case where the patches are themselves randomly distributed. It will result in oversampling if a trend exists in the concentration of the component through the population. Values for the sampling constants A and B can be determined by analyzing sets of large and small increments as described previously. If the sizes of the increments differ sufficiently, then
227
and Duncan (6,7)led to a second method of estimating values for A and B, applicable if an average particle mass can be obtained. In this method increments of a fixed weight are collected in sets and analyzed. For the case where increments are collected in pairs, an intraclass correlation coefficient, r, may be calculated for the measurement of interest by
r=
2C(x - X ) ( x ’ - z)
C(x - $2
(9)
+ C(x’- $2
where z is the mean of all observations and the terms x and x‘are the individual data points within each pair (8). Alternatively r can be calculated from within- and between-set variances, sw2and sb2, from an analysis of variance (ANOVA) of the data. Then
r=
(sb2
- sw2)/(sb2+ ( n - l)sw2)
(10)
where n equals the number of increments in each set. Equation 10 reduces to
r = (sb2- sw2)/(sb2 + s,2)
(11)
when increments are collected in pairs (Le., n = 2). In cases where the number of pairs, p , is small, the value of r calculated from ANOVA data more closely approximates the rigorous expression of eq 9 when eq 11 is modified to
r = [ 0,- l)sb2 - psw2]/ [ (p - l ) s b 2
+ psw2]
(12)
Once the value of r is known, it may be used to calculate A and B by simultaneous solution of eq 7 and
r = B/Am
(13)
The degree of segregation, z, equals r1j2. Limiting values for p, the population intraclass correlation coefficient, are 0 and 1. The estimator, r, of p can sometimes have negative values. Such values are assumed to be a result of the random selection of increments and to indicate a value for p of zero. Under these conditions eq 13 yields B = 0 and the population is considered to be well-mixed. The major drawback to the use of r and eq 13 is the difficulty in estimating an average particle mass. For a moderately uniform particulate mixture a value may readily be obtained by particle sizing techniques such as sieving along with knowledge of the average density of the material. But for many bulk materials, such as oil sand and soil, particles are not clearly defined, and an average particle mass cannot be measured.
APPLICATIONS OF SINGLE-CONSTANT SAMPLING APPROACHES TO SEGREGATED POPULATIONS A valid sampling protocol for a segregated population can be set up using a single sampling constant as long as the increment weight chosen to initially evaluate the variability of the population is not altered in subsequent sampling. This caveat applies to the use of Ingamells’ constant as well as the general equation from classical statistics
t%,2 n = -(lo4) and z = (B/Am)ll2 (8) where m is the reciprocal of the average particle mass of the material and has the same units of mass as A. Duncan has pointed out that the assumption made in eq 6 and 7 that variance due to segregation is independent of increment size may not be true (5). Instead, the apparent degree of segregation, which can be related to the intraclass correlation coefficient for sets of samples, decreases with increasing increment size. Ensuing discussions between Visman
R2R2
In eq 14,n is the number of increments that will hold the relative sampling deviation, R, to a level of confidence represented by t from Student’s t table (9).The terms 2 and s, represent the best available estimates of the composition and variability of the population based on analyses of preliminary samples. A preliminary value for n is determined by substituting a value for t representing n = 03 at the desired level of confidence. A new value for t based on the estimate of n is then substituted in the equation and the process is iterated until a constant n is reached.
228
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987
Time
segregated population may simply result in the collection of larger segments of a small number of strata; under these conditions s, is affected only slightly, if at all. The same trend is depicted in Figure lc,d, which represents concentration as a function of time as a bulk material is transported by conveyor past a fixed detector, or concentration as a function of distance as a detector transects a surface or is lowered in a hole. Effect of II on Sampling Uncertainty. In a sampling operation whose cost is controlled by the total amount of sample rather than the number of increments collected, reduction in sampling uncertainty can be achieved while minimizing the increase in expense by collecting more but smaller increments. Increasing the number of increments, n, without changing the total weight collected, wn, will not alter the variance for well-mixed populations but will reduce it for segregated populations. Figure la,c illustrates the effect of increasing n while decreasing w for a well-mixed sample. For n increments
-
Time
s: = A / w n
--+
Figure 1. Representation of effect on sampling uncertainty of increasing increment size and number collected from well-mixed (a, c) and segregated (b, d) populations. a and b represent sample increments of different size on a two-dimensional surface; c and d represent sampling on a belt or transect.
A value for K, calculated from eq 3 for a segregated population has a different meaning than K , for a well-mixed population. In the latter K, represents the weight of a single increment that will hold R to 1% a t the 68% level of confidence; in the former wR2 represents the weight of a composite made up of increments of the weight used to calculate the sampling constant that will hold R to 1%at the 68% level of confidence. When sets of increments from a segregated sample are collected, the numerical value of wR2 from eq 3 increases with increasing w. Values for wR2 increase by a factor of approximately 5 upon increasing, from 50 g to 500 g, the weight of increments of Athabasca oil sand collected for determination of bitumen, water, and solids content (Table 3, ref 10). In view of these results the relationship between Ingamells’ constant K, and Visman’s constant A, given by
A =1 0 - 4 ~ ~ ~ 2
(15)
in ref 4, is valid only for well-mixed materials where Visman’s B term equals zero. When B equals zero
sS2= A / w n and
When n equals 1,K, = 1 0 4 A / f 2 ,which rearranges to eq 15. Thus, when B does not equal zero, eq 15 does not apply. For segregated populations the effect on s,2 of a change in increment weight cannot be determined unless estimates of A and B are known. Changes in w do not reduce the contribution of the segregation term B to the sampling variance. From eq 5, when w is changed by a factor x , the sampling variance changes from
A B A B -+-to-+-,notto wn n wxn n This point is illustrated in Figure la,b, which represents distributions of a component in a well-mixed and a segregated population. Increasing the size of increments collected from a well-mixed population reduces the uncertainty due to random distribution. Increasing the size of increments from a
and for xn increments, each weighing w / x A A =ss2 = (w/x)(xn) wn Increasing the number of increments from a segregated population without altering the total weight of sample collected reduces the sampling variance when x > 1 because A + -B= - +A- < -B+ - A B (w/x)(xn) x n wn rn w n n This situation is depicted in Figure Ib,d. As the number of increments is increased for a segregated population, more strata are sampled and the composite is more representative of the population than a single increment of the same total weight. But when x < 1 (indicating a decrease in the number of increments for a fixed total weight), then
-A+ - >B- + -A
B
wn xn wn n and some sampling precision is lost. Effect of Increment Weight on Sampling Uncertainty. The fraction of overall sampling error arising from the segregation variance term in Visman’s equation is independent of the absolute values of A and B for a given sampling protocol. For example, when increment weights equal to AIB, defined as the optimum increment weight, wept, are collected from either a particulate or nonparticulate material, 50% of the sampling variance will arise from segregation regardless of the absolute values of A and B . The percentage of overall sampling variance due to segregation as a function of wopttimes a factor, x , is plotted in Figure 2. The relative reduction in sampling variance with changing increment size, if wopt is known, can be determined from the figure. Consider the case where increments weighing wept are collected from a bulk sample. The contribution of A to the total variance is 50% (say 50 units). For w = 0.25woPt,A contributes 50/0.25 = 200 units and for w = 5wOpt,A contributes 5015 = 10 units. Because the segregation variance contributes 50 units whether w = 0.25wOptor 5wopt,the reduction in total sampling variance by increasing increment weights from 0.25wOp,to 5wOptis (200 + 50) - (10+ 50)/250 = 0.76 or 76%. This observation is tested in Table I, which lists the sampling variances for the determination of oil, water, and solids on 950-g and 50-g increments of Athabasca oil sand (IO). These weights correspond to approximately 5wOptand 0.25wOptfor these components in this material. The ratio s,,g502/ss,502 falls in the range 0.22-0.51, indicating that Figure 2 and the associated calculations reliably estimate reduction
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987
229
1.0
0.8
C
'2
e
60-
e 5
50-
'
0.6
5 5
% lh
0)
g
' .-rn
40-
0.4
3020 -
0.2
10-
0 0.001
0.01
1
0.1
10
100
1000
0 0.0
Fraction of Optimum Increment Weight
Figure 2. Percentage of error due to segregation as a function of increment weight. The relation is described by log y = O.OOlO(log x ) ~ - 0.1335(log X ? 0.4972 log x 1.6035. The correlatlon coefficient is 0.9978.
+
+
Table I. Sampling Variances for Single Increments Collected from Athabasca Oil Sand Weighing 950 g (-5w,,) and 50 g ( - 0 . 2 5 ~ ~ ~ ~ )
high-grade oil sand bitumen solids water low-grade oil sand bitumen solids
water
sa~~~02/sa~~2
s,>9M2
sB3R2
0.96 0.24 0.46
3.45 0.55 2.07
0.28 0.44 0.22
0.49 1.00 2.02
1.74 1.96 4.00
0.28 0.51 0.51
in sampling variance, within the limits of the uncertainties in the estimates of sampling variance and wOpv It can be concluded that, in general terms, changing w has a significant effect when w falls in the range of 0.1 to 1OwOpt.. Outside this range the variance arising from random distribution is either so small or so large relative to the segregation variance that changing w has no significant effect on s,. An alternate method for determining the effect of changing increment size on sampling variance is limited to particulate materials. Equation 16, which is derived in the Appendix,
-ULg2 - uSm2
Wsmmz2
1
+ wSmmza
+
%( wLg
I
1
)
(16)
+ wSmmz2
describes the relation between uLg2/uSm2and wSm/ wLg for particulate materials over a range of particle size and degree of segregation. The equation was used to generate the bottom three lines in Figure 3. These plots illustrate the relationships between ak2/uSm2and w ~ ~ for / populations w ~ ~ with degrees of segregation of 0,0.5, or 0.8 and from which s m a l l increments containing one particle are Collected. The third line from the bottom of Figure 3 suggests that if increments containing one particle (wSmm = 1) had historically been collected from a population with a degree of segregation of 0.8, an increase in increment size to two particles (wsm/wk = 0.5) would reduce the sampling variance to 69% of its original value. Figure 3
0.2
0.4
0.6
0.8
1.0
WSm
Figure 3. Effect of wSm, m , and z on the reduction of sampling variance due to increasing increment size.
and eq 16, therefore, support the concept originally presented in Figure 1that an increase in increment weight by a factor x causes a reduction in sampling variance by some factor less than x if a population is segregated. The extent of reduction depends on the number of particles in the small and large increments as well as the degree of segregation. The top three lines in Figure 3 were generated from eq 16 and describe the reduction in sampling variance resulting from increasing increment size when the original increment contains 1, 10, or 100 particles and the population has a degree of segregation of 0.8. A large reduction in sampling variance occurs only if the original increment contains few particles. The relationship between uLg2/usm2and wSm/wLg can be established for any combination of wSm, wLg, m, and z by substitution of the appropriate values into eq 16. Figure 3 also provides the basis for a third test for segregation. Two tests have been discussed previously. In the first an F test was used to determine whether data collected from analyses of large and small increments are equivalent. In the second, an intraclass correlation coefficient was calculated for pairs of increments. If the null hypothesis is accepted in the F test, or if r is found to not be significantly different from zero by the appropriate statistical test, the population is considered to be well-mixed. The third test involves analysis of several sets of increments over a range of weights varying by at least a factor of 10. Approximately 30 data points must be collected to hold the error in s (and therefore R ) to 25% relative at the 95% level of confidence (11). Rearrangement of eq 3 to R2 = K , ( l / w ) suggests that for well-mixed materials a plot of R2 against l / w on a linear scale will yield a straight line of slope K, and an intercept of zero. As a result it is tempting to decide the population is well-mixed if the standard deviation in the slope is small. However, the y intercept of the line is the measure of segregation. If the confidence limits around the y intercept do not encompass the origin in a plot of R2 vs. l / w , the population may be considered to be segregated. When the data are plotted as in Figure 3 and m is known, confidence limits can be assigned to z regardless of its value.
290
ANALYTICAL CKMISTRY. VOL. 59, NO. 2. JANUARY 1987
each kind of particle and the average concentration, while p represents the probability of finding any one kind of particle. If an average theoretical particle of oil sand were to weigh 20 g, there would be little to distinguish between the two kinds of particles in the mixture. Then d, d2 d = 2.0 g/cm3 and p = 0.5. For an oil sand with an average bitumen content Pof8%
- -
Figure 4. Miuoscopic sbuclure showing me nonpancuble malure of Alhabasca 011Sand (131. The dameter 01 a 'typical' sand grain Is In
me range of 200 #m.
GUIDELINES FOR DEVELOPING SAMPLING CONSTANTS Just as knowledge of laboratory skills yields high-quality measurements with least effort and expense, 80 familiarity with designs for obtaining optimum estimates of sampling constants yields high-quality samples. This section contains guidelines for such estimations. Estimation of Visman's Constants. If it has been established that a population is segregated by the method described in the preceding section, then the data necesaary to determine Visman's constants using the two-increment-size approach summarized by eq 6 and 7 is already available. If, however, it was decided a priori that the population is segregated. and preliminary work is needed for the purpose of establishing the devee of segregation. then the statistically more rigorous 'increment-pair" approach should be conridered. This approach, which uses the intraclass correlation coefficient to estimate A and E , requires an estimate of the average particle mass, l/m. of the material. Solids such as coal, ores, and chemical solids may be screened and weighed to determine particle mass. Because larger particles in a population tend to influence sampling uncertainty most, the value for average panicle mass should be based upon the size of screen upon which 5% of the material is retained, as recommended by Cy (12). Some materials, including most biological materials and those containing viscous semisolids such as oil sand, cannot be sieved. For example, a cross section of the microscopic structure of Athabasca oil sand. Figure 4, shows a matrix of m a l l sand grains loosely held in a continuum of bitumen and water (13). Although the masa of an average sand grain can be measured after removal oft he bitumen and water, a particle of the overall material m o t be defined by physical methods. Under these conditions, is a value for m based upon average clean sand grain size valid? The answer was determined by analyses of both large and small increments and pairs of increments taken from a barrel of oil sand. A, E, and r were estimated from eq 6,7, and 9 and substituted into eq 13. The theoretical particle mass, l/m, fell in the range of 20 g. Therefore, the roughly 15OOO-g composite of oil sand required to hold R to 1% for a bitumen determination consists of approximately 750 "theoretical" particles. This conclusion was confirmed by means of the Benedetti-Pichler equation (14)
Here n is the number of particles in a mixture of two kinds of particles that must be collected to hold R to a specified level. The terms d,, dz. and a represent the densities of the individual kinds of particles and their average density. P,,P2, and P a r e the concentrations of the sought-for component in
and the difference in composition between the two kinds of particles, PI- P2,is approximately 4.5%-a reasonable value. On the other hand if the average particle mass of the whole oil sand is assumed to equal the maos of an averagesized grain of silica and the sample is a mixture of particles of bitumen and silica, the values substituted into the Benedetti-Pichler equation would be as follows: d, (density of bitumen), 1.00 g/mL; dz (density of silica), 2.6 g/mL; d (density of oil sand), 2.0 g/mL; P (average concentration of bitumen in oil sand), 8%;PI- P2,100% - 0% = 100%. A 200 pm diameter particle of silica weighs approximately 11 pg; therefore a 15ooO-g composite required to hold R to 1% would contain approximately 1.4 X lo9 particles. Under these conditions
P ( 1 - P) =
1.4 x 109
= 530
Because p(1- p) cannot exceed 0.25, the conditions as described are theoretically impossible and the hypothesis that the average size of oil sand equals the average particle size of a sand grain must be rejected. In the case of Athabasca oil sand, the assumption that an average particle weighs 21 pg rather than 20 g would yield an unreasonably large value for Visman's B term. Thus it is safer in such instances to use the two-increment-size approach for calculating A and B rather than risk choosing a wrong value for m. Nevertheless, calculation of a theoretical particle mas illustrates why such large samples must be collected from some populations if R is to be held to a specified level. Estimation of r . An intraclass correlation coefficient can be calculated from the analysis of pairs of increments either rigorously by eq 9 or more approximately by eq 10 and 11. Errors in r caused by using the latter usually result in negligible errom in A and B and so have little impact on sampling protocols, especially when l/mrw is significantly less than 1. Since
..
-
I
,
.. .",
then, if the average particle mass, l/m, is small and rw is Large, 1 + l/mnu 1 and errors in rare inconsequential. For larger values of l/mrw,use of eq 9 to determine r reduces the error in E. A difticulty that arises in obtaining an accurate estimate of p lies in the appropriate selection of increments. Both the location of each pair and the relative orientation of the members within each pair of increments must be randomly selected. Figure 5a,b illustrates orientations that could lead to significant errors in r. In Figure 5a the pairs tend to be oriented along straw, this leads to low estimates of ow and high estimates of eb. Then, r provides a high estimate of p and oversampling would be prescribed for a specified level of .s Figure 5b illustrates orientation at right angles to prevailing stratification; this leads to low estimates of p and so to undersampling. "Directional" segregation or banding in a population can be identified by collecting increments in sets of four rather
B'
r'i . . 1.
.
*
..a.
.
.*
.
.
*
.
.
........ .
...'..* .... ... .... ... . ... ...*:
*..
dn
. *
.. .'., . .. ....
. ... .. .. ... ..*. .
and
WSm
*.
9
'
sS2
s,P(wLg
WLgWSm
-
- WSm)
S,'W(WL~
- WSm)
1
(19)
These equations are first derivatives of the equation
a. *
*
. .....
'
* . .
*'*
... . ... . ::m .. . :.. . ,: .. . . . :? .: . .*... . * . ,. ..:'. ... .
.a, *
e . . .
*
*
*,
*
'
...a
8
I
a:.:.,
Flgure 5. Orientations of pairs of increments in a segregated population that lead to overestimation (a) and underestimation (b) of the degree of segregation. Arrows in section c indicate dlrections in which pairs of increments may be evaluated to assess effects of orientation.
than in pairs (Figure 5c). Values of r for four orientations can be calculated and compared. They will be approximately equal if segregation occurs in patches but will differ if banding is present. Banding in a sample having a significant third dimension can be identified by collection of an additional pair of increments in the z axis. Orientation of increment pairs is not a problem when sampling from a flowing stream or moving belt, provided the sampling device selects increments incorporating a complete cross section of the stream. Collection of only a fraction of the stream cross section at any one time is not recommended because lateral or vertical segregation may bias the results. Selection of Increment Sizes in the Two-IncrementSize Approach to Calculating Visman's Constants. When Visman's original method is used to calculate A and B, increment size (wk and wSm) and the number of increments in each set controls the precision of A and B. First consider increment size. The ratio w L g / w S m should be kept as large as possible to ensure the relative error in the difference sSm2 sk2 is small. A ratio of 10 is about the minimum acceptable. Judicious selection of increment weights when testing for segregation by the y-intercept method can provide a number of independent estimates of A and B. The term w w s m /( w k wsm)should also be held to as small a value as possible to minimize the error in the product [wLgwSm/(wLg- W S ~ ) ] ( S S-~sL2). ~ This is accomplished by keeping wsm small. The limit for wSm is determined either by the minimum size of test portion required for analysis or by limitations in the sampling equipment and operation. Next consider increments. Analysis of equal numbers of increments in each set does not guarantee the most precise estimate of A and B. Proof of this unexpected fact follows. The change in the number of increments that will hold s,2 to a selected value as a function of sSinor sLgis given by
-
-
A n=ws,2
.. . 1.'. .
.*:*. . *
+ -s,2B
with A and B defined by eq 6 and 7. Comparison of the values for dnldssm and dnldsLg will indicate whether the number of increments in the prescribed protocol is more sensitive to errors in sSm or sLg in the original sampling experiments. Emphasis can then be placed on obtaining a more precise value by analyzing more increments for the set that controls n to the greater degree. For example, in the sampling of a high-grade Athabasca oil sand for determination of bitumen content, substitution of the values w& = 951.95 g, wsm = 50.62 g, S L = ~ 0.98, ssm = 1.86, w = 163 g (i.e., wOpJ,and s, = 0.14 (1% relative) into eq 18 and 19 yields dn/dss, = 50 and dnlds, = 70. An error of 0.1 in the estimate of ss, or s L g would cause errors of 5 and 7 in the estimate of the number of increments n that would hold s, to 1%relative. Either error is relatively small, considering that the calculated value of n for increments weighing wept falls in the range of 80. The values dn/dssm and dn/dsLgneed not be similar. If, for example, sLg was 1.25 in the case described above, wept would have been 70 g and values of 134 and 37 for dnldss, and dnlds, would have been found. More effort would have been required to obtain a precise estimate of gsmthan of uLg. A recommended strategy for calculating A and B entails collection and analysis of 20-30 increments of each weight to determine reasonable estimates of s h , sk and woPv Additional samples can later be analyzed to reduce the error in the term more strongly affecting n. Use of Composites to Estimate A and B. It is possible to develop sampling protocols while simultaneously establishing a precise estimate of the mean composition of a population through the collection and analysis of composite samples. Both large and small increments can be cornposited to some degree to permit collection of more increments than would be possible if the increments were analyzed individually. This is because Visman's equation can be modified to incorporate data from the analysis of composites as follows: In eq 5 the uncertainties associated with collection of one (n = 1)small and one large increment are A B and sLg:!= A B SSm2
=-
+
WSm
+
WLg
Rearranging
A A B = sSm - - and B = sLg2- WSm wb!
(7)
Simultaneous solution for A yields
Analyses conducted on composites of y small increments of weight w1or x large increments of weight w2, selected so that ywl = wSm and xw2 = wLg, give variances of
232
ANALYTICAL CHEMISTRY, VOL. 59, NO. 2, JANUARY 1987
From these relations one obtains
and
This method still requires the analysis of 20-30 composites from each set of increments, but a larger number of individual increments has been collected. This provides better representation of the population for the purposes of estimating composition a t the time of determining A and B.
CONCLUSIONS A single sampling constant such as Ingamells’ constant, K,, for well-mixed materials can be used to describe the variability of a segregated material. The sampling variance associated with the collection of a specified number of increments can be estimated only if the increments are of the same size used to calculate the constant. Visman’s sampling theory, which incorporates a segregation constant, permits estimation of variability at different increment-size levels. The degree of segregation, z , and the confidence limits in the estimate of z for a segregated population can be estimated by plotting ak2/uSm2against wSm/wLgover a range of increment weights. Sampling uncertainty is little affected by changing the increment weight, w, if w is less than one-tenth of or greater than 10 times the optimum increment weight. Visman’s two approaches for calculating the sampling constants A and B can be combined to determine a theoretical particle mass for a nonparticulate material. Use of the theoretical particle concept explains the need to collect larger samples from some populations if they are to be representative. The concept also demonstrates limitations in the use of an intraclass correlation coefficient-average particle mass relation to calculate A and B. Therefore, the two-increment-size approach of Visman, even though not statistically rigorous, is preferred when sampling nonparticulates. Procedures for estimating the errors in A and B and for the analysis of composites to estimate A and B have been developed. ACKNOWLEDGMENT The authors thank Jan Visman of Vortek Engineering, Edmonton, Alberta, for invaluable discussions of this work. APPENDIX Equation 16 is derived from the relation z 2 = B/Am by A isL;
mz2 = B -= A
-WLg
A
-- -VLg2_ _1 A
-
%g
oLg2/oSm2
A/uSm2
_-
1
WLg
uLg2/gSm2
1
[ mz2:+
1 ]
1
wSmmz2 + %[ 1 + wSmmz2 W L ~ 1 + wSmmz2
(15)
The slope and intercept of the line describing aLg2/aSm2 as a function of wSm/wLg are 1/(1 wSmmz2)and 1 - 1/(1+ wSmmz2)(or wsmmz2/(1 + wSmzm2)).
+
LITERATURE CITED Youden, W. J . Assoc. Off. Anal. Chem. 1967, 50, 1007-1013. Ingamells, C. 0.;Switzer, P. Talanta 1973, 20, 547-568. Vlsman, J. Mater. Res. Stand. 1969, 9 , 9-64. Ingamells, C. 0. Talanta 1978, 27, 141-155. Duncan, A. J. Mater. Res. Stand. 1971, 7 1 , 25. Visman, J.; Duncan, A. J.; Lerner, M. Mater. Res. Stand. 1971, 1 7 , 32-37. (7) Visman, J. J. Mater. 1972, 7 , 345-350. (8) Snedecor, G. W.; Cochran, W. G. Statistical Methods, 6th ed.; Iowa State Unlversity Press: Ames, IA, 1967; p 295. (9) Kratochvll, 8.; Taylor, J. K. Anal. Chem. 1981, 53, 924A-938A. (10) Wallace, D.; Kratochvil, B. AOSTRA J . Res. 1986, 2 , 233-239. (11) Natrella, M. G. Experimental StaNstics : National Bureau of Standards: Washington, DC, 1966; Handbook 91, p 2-12, Figure 2-2. (12) Gy, P. M. Sampling of Particulate Materials; Elsevier, Amsterdam, 1979; pp 256. (13) Takamura, K. Can. J . Chem. Eng. 1982, 6 0 , 538-545. (14) Benedettl-Plchler, A. A. In phvslcal Methods in Chemical Analysis; Bert, W. G., Ed.; Academic Press: New York, 1956; Vol. 111, pp 183-217. (1) (2) (3) (4) (5) (6)
RECEIVED for review May 19,1986. Accepted September 11, 1986. The support of the Alberta Oil Sands Technology and Research Authority through the Oil Sands Sample Bank Project, of the Department of Chemistry, University of Alberta, and of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
