SUPPLEMENTAL DOCUMENT SD6
FOR PART IVC
Quality Assurance/Uncertainty
Measurement Uncertainty for Extrapolations of
Net Weight and Unit Count
Table of Contents
Introduction
.
...2
A Example
1: Extrapolation of net weight
3
B Example
2: Extrapolation of net weight in conjunction with a hypergeometric sampling
plan
.
.10
C Example
3: Extrapolation of unit count
...16
D References
..23
Introduction
The following examples demonstrate various approaches for deriving
estimates of uncertainty associated with weight and count extrapolations:
A Example 1:
Extrapolation of net weight
B Example 2:
Extrapolation of net weight in conjunction with a hypergeometric sampling plan
C Example 3:
Extrapolation of unit count
These examples are meant to be illustrative, not exclusive. Laboratories
should develop defensible procedures that fit their operational environment and
jurisdictional requirements. Notes and calculations are provided to clarify
these applications. Weight calculations are based upon assumptions that
populations are normally distributed.[1]
Various terms used in this document are defined in the SWGDRUG
Recommendations Annex A. The following examples should not be directly applied
to methodology used without first considering the specific purpose of the
method and its relevant operational environment.
A Example 1:
Extrapolation of net weight
Scenario:
A laboratory receives an exhibit containing
100 bags of white powder.
Objective:
The analyst needs to determine the total net
weight of the powder in the 100 bags.
This is done by weighing the powder from a sample of the population and
extrapolating to the total population.[2]
Procedure:
A.1
Determine the population size N.
Only bags which have sufficient similar characteristics are placed in
the same population.
In this
example, the contents of all 100 bags are visually consistent in substance
amount (about 0.5 gram) and physical appearance (i.e. color, texture, etc.),[3]
hence N = 100.
A.2
Select the sample size, n, to be weighed.^{1}
In this
example, the analyst chooses a sample size n
= 10. The 10 units are randomly
selected[4]
from the total population.
(Results for
other n values are given later in the
section.)
A.3
Measure the weight of the powder in each of the
randomly selected units.
The weight (X) of the powder in each of the 10 bags
is measured by dynamic weighing on a threeplace balance (with 0.001 gram
readability)[5]
as recorded in table 1.1.
Table 1.1: Individual weights of 10 bags.
Bag 
Wt of powder (X), gram 
Bag 
Wt of powder (X), gram 
1 
0.593 
6 
0.574 
2 
0.509 
7 
0.580 
3 
0.557 
8 
0.540 
4 
0.548 
9 
0.532 
5 
0.569 
10 
0.529 
A.4
Calculate the average weight per unit, , the standard deviation, s, and the relative standard deviation, RSD.
Average weight per unit, 
= 0.5531 gram 
Standard deviation, s 
= 0.02622 gram 
Relative Standard Deviation, RSD[6] 
= x 100% = 4.741% 
A.5
Obtain the standard uncertainty (unexpanded),
, associated with the balance used.^{5}
In this example, the laboratory has determined = 0.00185 gram for
a threeplace balance.
A.6
Obtain the uncertainty associated with the
calculated average weight, . This uncertainty
encompasses the standard deviation as well as the number of measurements
performed.
= 0.008292
A.7
Calculate the combined uncertainty, , associated with the average weight per unit, by
combining the standard uncertainties[7] of the average weight, , and the balance used, ,[8] via the
rootsumsquare (RSS) method.
= 0.008496 gram
A.8
Calculate the extrapolated net weight of the
100 bags, W, and its associated
uncertainty, .
Extrapolated
net weight, W = N * = 100 * 0.5531
g = 55.31 grams
Extrapolated uncertainty, = N * = 100* 0.008496 g = 0.8496 grams
A.9
Obtain the expanded extrapolated uncertainty,
, by using the appropriate coverage factor, k, (Students t value for 9 degrees of freedom).[9] Round up the expanded extrapolated uncertainty,
, to two significant figures.[10]
If a 95% level of confidence is used, (coverage factor k = 2.262),
= * k = 0.8496 g * 2.262 = 1.921 grams » 2.0 grams
If a 99% level of confidence is used (coverage factor k = 3.250),
= * k = 0.8496 g * 3.250 = 2.761 grams » 2.8 grams
A.10 Report the total extrapolated net weight and its
associated uncertainty by truncating the extrapolated net weight to the same
level of significance (i.e. decimal places) as the rounded expanded uncertainty.
When the 95% level of confidence is used:
The amount of powder in 100 bags is 55.3 grams ± 2.0 grams at a 95% level of confidence, determined by
weighing 10 bags and extrapolating to obtain the total net weight.
When the 99% level of confidence is used:
The amount of powder in 100 bags is 55.3 grams ± 2.8
grams at a 99% level of confidence, determined by weighing 10 bags and
extrapolating to obtain the total net weight.
A.11 If the
analyst also performs qualitative analysis on each one of the 10 randomly
selected bags and all are found to contain cocaine (that is, no negatives
found), the following inferences about the population (at the respective 95% or
99% levels of confidence) can be made:
By statistically
sampling 10 bags, it is concluded at a 95% level of confidence, that at least
76% of the population contains cocaine.
By statistically
sampling 10 bags, it is concluded at a 99% level of confidence, that at least
65% of the population contains cocaine.
The above
statistical inferences on the population as well as for other levels of
confidence (depending on laboratorys policy and decision), can be calculated
using the ENFSI DWG Calculator for Qualitative Sampling of Seized Drugs (Reference
D.2). This calculator can also be accessed from the SWGDRUG website at http://www.swgdrug.org/tools.htm).
Appendix 1.1:
Net weights and associated uncertainties
extrapolated for other sample sizes are given in Table 1.2. It is noted that as the sample size n increases, the expanded extrapolated
uncertainty, , decreases. Also,
for a given sample size n, the
expanded uncertainty is larger when a higher level of confidence is used.
Table 1.2: Calculations
for sample sizes of n = 3, 5, 10, 20
and 30.
Sample size, n 
3 
5 
10 
20 
30 
Avg wt of unit, , gram 
0.5530 
0.5552 
0.5531 
0.5514 
0.5510 
Std deviation, s 
0.04214 
0.03086 
0.02622 
0.02860 
0.02759 
% RSD 
7.621 
5.558 
4.741 
5.188 
5.007 
Std uncertainty of avg wt, 
0.024331 
0.013800 
0.008292 
0.006396 
0.005037 
Combined std uncertainty, 
0.024401 
0.013923 
0.008496 
0.006658 
0.005366 
Extrapolated uncertainty, 
2.4401 
1.3923 
0.8496 
0.6658 
0.5366 
Extrapolated wt, W 
55.30 
55.52 
55.31 
55.14 
55.10 
With 95% Level of
Confidence 

Coverage factor, k 
4.303 
2.776 
2.262 
2.093 
2.045 
Exp extrapolated uncertainty, 
10.499 
3.865 
1.922 
1.394 
1.097 
Lower Wt Limit 
44.80 
51.65 
53.39 
53.74 
54.00 
Upper Wt Limit 
65.80 
59.39 
57.23 
56.53 
56.20 
With 99% Level of
Confidence 

Coverage factor, k 
9.925 
4.604 
3.250 
2.861 
2.756 
Exp extrapolated
uncertainty, 
24.218 
6.410 
2.761 
1.905 
1.479 
Lower Wt Limit 
31.08 
49.11 
52.55 
53.23 
53.62 
Upper Wt Limit 
79.52 
61.93 
58.07 
57.04 
56.58 
Raw data of individual sample weights used
are given in Table 1.3.
Table 1.3:
Individual sample weights of 30 bags used in examples.
Bag 
Wt of powder (X),
gram 
Bag 
Wt of powder (X),
gram 
Bag 
Wt of powder (X),
gram 
1 
0.593 
11 
0.583 
21 
0.593 
2 
0.509 
12 
0.510 
22 
0.530 
3 
0.557 
13 
0.540 
23 
0.548 
4 
0.548 
14 
0.582 
24 
0.581 
5 
0.569 
15 
0.552 
25 
0.539 
6 
0.574 
16 
0.530 
26 
0.579 
7 
0.580 
17 
0.509 
27 
0.530 
8 
0.540 
18 
0.580 
28 
0.532 
9 
0.532 
19 
0.520 
29 
0.511 
10 
0.529 
20 
0.590 
30 
0.560 
Step A.7 shows that the combined uncertainty,, has contributions from: the standard uncertainties of
the average weight, , and that associated with the balance used, . If a balance of
a different uncertainty is used, the combined uncertainty will change. Similarly, the distribution of the individual
weights of the population will affect the combined uncertainty. To illustrate the impact of the weight
distribution of the population on the extrapolation of the total net weight,
another 30 bags from a different population (one that has been tested to be
normally distributed) are individually weighed on the same balance. The individual weights of these 30 bags are
given in Table 1.4 below and the associated calculations given in Table
1.5. It is noted that the RSD values
listed in Table 1.5 are all much smaller than those for Table 1.2 (above). This consequentially gives rise to smaller
expanded extrapolated uncertainty,, for all sample sizes in Table 1.5 as compared to Table
1.2.
Table 1.4:
Individual sample weights of 30 bags from a normally distribution population.
Bag 
Wt of powder (X),
gram 
Bag 
Wt of powder (X),
gram 
Bag 
Wt of powder (X),
gram 
1 
0.553 
11 
0.557 
21 
0.552 
2 
0.549 
12 
0.557 
22 
0.554 
3 
0.557 
13 
0.552 
23 
0.555 
4 
0.554 
14 
0.555 
24 
0.557 
5 
0.550 
15 
0.555 
25 
0.551 
6 
0.553 
16 
0.556 
26 
0.557 
7 
0.556 
17 
0.557 
27 
0.557 
8 
0.557 
18 
0.547 
28 
0.556 
9 
0.555 
19 
0.554 
29 
0.551 
10 
0.556 
20 
0.556 
30 
0.552 
Table 1.5: Calculations
for sample sizes of n = 3, 5, 10, 20
and 30.
Sample
size, n 
3 
5 
10 
20 
30 

Avg wt of unit, ,
gram 
0.5530 
0.5526 
0.5540 
0.5543 
0.5543 

Std deviation, s 
0.004000 
0.003209 
0.002789 
0.002886 
0.002728 

% RSD 
0.7233 
0.5808 
0.5034 
0.5206 
0.4922 

Std uncertainty of avg wt, 
0.0023094 
0.0014353 
0.0008819 
0.0006452 
0.0004981 

Combined std uncertainty, 
0.002959 
0.002341 
0.002049 
0.001959 
0.001916 

Extrapolated uncertainty, 
0.2959 
0.2341 
0.2049 
0.1959 
0.1916 

Extrapolated wt, W 
55.30 
55.26 
55.40 
55.43 
55.43 

With 95% Level of
Confidence 

Coverage factor, k 
4.303 
2.776 
2.262 
2.093 
2.045 

Exp extrapolated
uncertainty, 
1.273 
0.650 
0.463 
0.410 
0.392 

Lower Wt Limit 
54.03 
54.61 
54.94 
55.02 
55.04 

Upper Wt Limit 
56.57 
55.91 
55.86 
55.84 
55.82 

With 99% Level of
Confidence 

Coverage factor, k 
9.925 
4.604 
3.250 
2.860 
2.756 

Exp extrapolated uncertainty, 
2.937 
1.078 
0.666 
0.560 
0.528 

Lower Wt Limit 
52.36 
54.18 
54.73 
54.87 
54.90 

Upper Wt Limit 
58.24 
56.34 
56.07 
55.99 
55.95 

B Example
2: Extrapolation of net weight in conjunction
with a hypergeometric sampling plan
Scenario:
The scenario is the same as Example 1, where
the laboratory receives an exhibit
containing 100 bags of white powder.
Sentencing penalty in this jurisdiction increases if the amount of
substance containing cocaine exceeds 25 grams.
Objective:
The analyst
will use statistically based sampling without replacement to determine, to a
99% level of confidence, if the jurisdictional weight threshold is exceeded. This
example does not take purity of the powder into account because it is not
jurisdictionally relevant.
Procedure:
B.1
The analyst needs to determine how many bags must
be sampled to determine if the 25gram threshold weight is exceeded.
To obtain an estimation of the number of bags that must be
sampled to meet the threshold weight, the specified statutory threshold weight
(25 grams) is divided by the average net weight () per unit (obtained from Example 1).
Estimated
number of bags = = 45.1 (46 bags)
The extrapolated net weight of 46 bags results in 25.4
grams ± 1.3 grams (See blue dotted line in figure below. The calculation
to estimate the uncertainty of the measurement is not shown here. Refer to Steps B.3 to B.5 below for
calculation process). The lower bound of
24.1 grams falls below the statutory threshold.
To calculate the number of bags needed for the lower
bound of the extrapolated net weight to exceed the statutory threshold weight,
the specified statutory threshold weight (25 grams) is divided by the difference
between the average net weight () per unit and the confidence interval with
coverage factor k = 3.250 using Students
t value for 9 degrees of freedom based on a sample of 10 bags (see Example
1).
Estimated
number of bags =
= 47.5 (48 bags)
Therefore, a minimum of 48 bags must be sampled to
provide strong evidence that the threshold weight is exceeded. The measurement of uncertainty associated
with weighing 48 bags is 1.4 grams at a 99% level of confidence (see detail
calculation in Step B.4), hence giving a lower bound of 25.1 grams, which is
above the statutory threshold. This is
depicted by the green dotted line in the figure above.
B.2
Determine
the sample size n that needs to be
qualitatively tested to demonstrate that at least 48 of the 100 bags contain
cocaine at a 99% level of confidence.
Method 1: 99% level of confidence corresponds
to an a of 0.01 (level of confidence = 0.99 = 1a). Proceed to use
a hypergeometric sampling calculator to determine the sample size needed. (See Reference D.2)
Using the hypergeometric sampling calculator
and the appropriate parameters
(N =
100, a = 0.01, proportion of positives = 0.48, with
no negatives expected), the sample size is determined to be 6.
or
Method 2: Manually determine the number of
bags, n, to be tested using the established
significance level of 0.01 (corresponding to a 99% level of confidence). In
this instance, there are two possible outcomes or hypotheses:
H_{1 } Hypothesis that ≥ 48 bags contain cocaine
H_{0}  Hypothesis that < 48 bags contain cocaine (null hypothesis)
Since the first chance of failure will occur when 47 out of the 100
bags contain cocaine, we want to obtain enough evidence to reject the
assumption that there are only 47 bags of cocaine (H_{0}). This is done by calculating the probability of
obtaining a positive result at successive sample sizes, n, until it falls below the established significance level (a = 0.01).
= P (all n bags in the sample contain
cocaine)
The following calculations show the pvalues (and resulting levels of
confidence, LoC) obtained for each successive sample tested (with no negatives
found) until a value below 0.01 is obtained (which is sample 6):
0.4700 (53.00% LoC)
0.2183 (78.16% LoC)
0.1003 (89.97% LoC)
0.0454 (95.45% LoC)
0.0203 (97.96% LoC)
0.0090 (99.10% LoC)
The
probability of randomly selecting 6 units and having them all test positive is
too low to occur by chance (below established significance level) if we assume there
are only 47 positives in the population. Therefore, we can reject H_{0}
and accept H_{1}, that there are ≥ 48 bags that contain cocaine
in the population.
Therefore, the number of bags, n, needed for testing is 6.
B.3
A total of 6 bags are randomly selected for chemical
analysis[11]
and confirmed to contain cocaine. Since
all 6 bags are found to contain cocaine, it can be stated, to a 99% level of
confidence, that at least 48 of the 100 bags contain cocaine.
The
total net weight of 48 bags, W_{48}, can be extrapolated from the
average net weight per unit (obtained from Example 1):
W_{48 }=
48 * = 48 * 0.5531 g = 26.5488 grams
B.4
The combined standard
uncertainty, , associated with the average weight per unit
as calculated from Example 1 is:
=
0.008496 gram
The
extrapolated uncertainty for 48 bags, ,
is calculated as
=
0.008496 g * 48 = 0.4078
gram
The
total expanded uncertainty (),
at 99% level of confidence, and rounded up to two significant figures (coverage
factor k = 3.250
using Students t value for 9 degrees
of freedom since the contents of 10 bags were individually weighed in Step A.2)
is
=
0.4078 g * 3.250
= 1.3254 g » 1.4 gram
B.5
The analyst compares the
calculated extrapolated weight of the 48 bags, W_{48}, minus the expanded uncertainty, , (truncated to the same level of significance) against
the statutory threshold of 25 grams.
The
weight of 48 bags is 26.5 grams ± 1.4 grams calculated at a 99% level of confidence. The lower end of the weight
range is = 26.5 1.4 grams =
25.1 grams (which is above 25grams statutory threshold).
B.6
The results
of the analysis can be reported in either of the following ways:
1) A total of 100 indistinguishable bags were received. By using statistical sampling of 6 bags, it
is concluded at a 99% level of confidence that at least 48% of the population
contains cocaine. The extrapolated net weight of 48 bags is 26.5 grams ± 1.4 grams at a
99% level of confidence.
2) A total of 100 indistinguishable bags were
received. Using statistical (hypergeometric) sampling and by testing 6 bags, it
is concluded that cocaine is present in at least 25.1 grams of powder at a
level of confidence of at least 98%.
Explanation on deriving the overall level of
confidence (i.e. at least 98%):
The second report option gives an overall
level of confidence of at least 98% for the weight and identity of the
powder. Each of these parameters is
individually tested at a 99% level of confidence. Where these two statements are not considered
to be independent of each other, the Bonferroni correction (Reference D.1, p
155156) can be used in the calculation of the overall confidence level. This is obtained by determining the value of
(1 0.01 0.01)*100%. If the two
statements are considered independent, the multiplication rule of probability
can be used instead, giving an overall level of confidence of 99%*99% =
98.01%.
Appendix 2.1:
To contrast the practicality of using
hypergeometric sampling to identify a proportion of a population, the following
example is given:
If a
sampling size of 6 is used to determine the content of all 100 bags, the
probability of failure (finding less than 100 bags containing cocaine) =
As
illustrated in this case, if only 6 bags are sampled, the analyst is only 6%
confident that all 100 bags contain a substance containing cocaine.
If a 95% level of confidence is needed for the reporting
of content of all 100 bags, the sampling size needs to be increased as shown
below:
giving a sample size of 95.
Therefore,
it is often practical to report that a certain proportion of the population is
positive instead of reporting on the entire population. This can be achieved by using statistical
sampling. Using the same example of a total population of 100 bags, if the
laboratory only needs to report on the content of 90 bags, the sampling size
would reduce to 23:
As seen
from this example, if the laboratory needs to report on the content of all 100
bags at a confidence level of 95%, a total of 95 bags need to be tested. In contrast, if the laboratory only needs to
report on the content of 90 bags at the same confidence level, the number of
bags to be tested is reduced to 23 (a reduction of 75%).
C Example
3: Extrapolation of unit count
Scenario:
The laboratory receives a large container
with numerous tablets.
Objective:
The analyst needs to determine the total
number of tablets present in the container and its associated uncertainty by
direct weighing of a sample of individual tablets and extrapolating to obtain
the total count.
Procedure:
C.1
Determine whether all the tablets in the
container can be treated as one population.
Since all the tablets in the container are visually
similar, they will be treated as one population.
C.2
Measure the net weight of all the tablets.
The total weight, TW,
of the total population of tablets is determined to be 701.5 grams based on
dynamic weighing on a balance with 0.1 gram readability.
C.3
Choose the number of individual tablets to
weigh.
In this example, the analyst randomly samples and weighs
10 tablets (n = 10).
(Results for other n
values are given later in the section.)
The weight of each tablet X is determined by dynamic weighing on a balance with 0.0001 gram
readability as in Table 3.1.
Table 3.1: Individual weights of 10 tablets.
Tablet 
Wt of tablet (X), gram 
Tablet 
Wt of tablet (X), gram 
1 
0.3084 
6 
0.3437 
2 
0.3225 
7 
0.2918 
3 
0.3349 
8 
0.3116 
4 
0.2981 
9 
0.3077 
5 
0.3293 
10 
0.3426 
C.4
Calculate the average weight per tablet, , the standard
deviation of the tablet weight, s,
and the relative standard deviation, RSD.
Average weight per tablet, 
= 0.31906 gram 
Standard deviation, s 
= 0.018287 gram 
Relative standard deviation, RSD 
= 5.7314 % 
C.5
The number of tablets in the container is
estimated by dividing the total weight of all the tablets, TW, by the average weight per tablet, .
Estimated
number of tablets in container = 2198.6
C.6
Obtain the uncertainty associated with the
two balances used^{5}:
Uncertainty for oneplace balance (0.1 g readability), = 0.35810 gram
Uncertainty
for fourplace balance (0.0001 g readability), = 0.0004840 gram
C.7
Calculate the relative uncertainties of both
weighing processes. The use of relative
standard uncertainties is necessary because the estimated number of tablets is
obtained by a division operation (see C.5).
Relative
uncertainty of the total weight of tablets, :
0.00051048
Relative uncertainty of average weight per tablet, :
= 0.018188
C.8
Combine the two relative standard
uncertainties (and ) to obtain the combined relative standard uncertainty,, associated with the extrapolated tablet count.
=
0.018195
C.9
Determine the absolute combined uncertainty, , for the tablet count by multiplying the combined
relative standard uncertainty, , by the estimated
number of tablets.
number of tablets= 0.018195 * 2198.6 = 40.004
C.10
Expand the combined uncertainty, , using the appropriate coverage factor k.
At a 95% level of confidence for n = 10, the coverage factor k
= 2.262.
Expanded
uncertainty, = * k = 40.004 * 2.262 = 90.489 tablets.
If a 99% level of confidence is used, the coverage factor
k = 3.250.
Expanded
uncertainty, = * k = 40.004 * 3.250 = 130.013 tablets.
C.11
Report the total extrapolated tablet number,
and its associated uncertainty, truncating or rounding to the nearest whole
number per laboratory policy. In this
example, the number of tablets is truncated while the associated uncertainty is
rounded up for a conservative approach.
Number of tablets: 2198 ± 91
The number of tablets is an
extrapolated estimated value based on the individual weights of 10 tablets and
the uncertainty value represents an expanded uncertainty at a 95% level of
confidence.
Number of tablets: 2198 ± 131
The number of tablets is an
extrapolated estimated value based on the individual weights of 10 tablets and
the uncertainty value represents an expanded uncertainty at a 99% level of
confidence.
Appendix 3.1:
Examples of other sample sizes n = 3, 5, 30 and 50 taken from the same
population are given in Table 3.2, together with data from n = 10 for comparison. Raw
data of tablet weights used for Table 3.2 are given in Table 3.3. It is noted
that the extrapolated combined uncertainty, , is smaller as the sample size gets bigger. Also, for a given sample size n, the expanded uncertainty, , is larger when a higher level of confidence is used.
It should be the laboratorys decision and
policy to determine the sample size n
needed for the extrapolation of number of units. Using a smaller n is more time efficient but results in a much larger expanded
uncertainty, . Using a larger n takes more time to complete the
analysis but has the benefit of a smaller expanded uncertainty.
Table 3.2: Calculations
for sample sizes of n =3, 5, 10, 30
and 50.
Sample size, n 
3 
5 
10 
30 
50 
Avg wt per tablet, , gram 
0.32193 
0.31864 
0.31906 
0.32337 
0.32510 
Std deviation, s 
0.013259 
0.015163 
0.018287 
0.017731 
0.019186 
% RSD 
4.1186 
4.7587 
5.7314 
5.4833 
5.9016 
Extrapolated tablet count, 
2179.0 
2201.5 
2198.6 
2169.3 
2157.8 
Std uncertainty of avg wt, 
0.0076551 
0.0067811 
0.0057828 
0.0032373 
0.0027133 
Rel. uncertainty of net wt, 
0.00051048 
0.00051048 
0.00051048 
0.00051048 
0.00051048 
Rel. uncertainty of avg wt, 
0.023826 
0.021336 
0.018188 
0.010122 
0.008478 
Combined relative uncertainty, 
0.023832 
0.021342 
0.018195 
0.010135 
0.008493 
Extrapolated combined uncertainty, 
51.930 
46.985 
40.004 
21.987 
18.327 
With 95% Level of
Confidence 

Coverage factor, k 
4.302 
2.776 
2.262 
2.045 
2.010 
Expanded uncertainty, 
223.403 
130.430 
90.489 
44.963 
36.837 
With 99% Level of
Confidence 

Coverage factor, k 
9.924 
4.604 
3.250 
2.756 
2.680 
Expanded uncertainty, 
515.353 
216.319 
130.013 
60.596 
49.116 
Table 3.3:
Individual weight of tablets for Table 3.2.
Tablet 
Wt of tablet (X), gram 
Tablet 
Wt of tablet (X), gram 
Tablet 
Wt of tablet (X), gram 
1 
0.3084 
21 
0.3152 
41 
0.3580 
2 
0.3225 
22 
0.2763 
42 
0.3090 
3 
0.3349 
23 
0.3058 
43 
0.3251 
4 
0.2981 
24 
0.3014 
44 
0.3459 
5 
0.3293 
25 
0.3376 
45 
0.3054 
6 
0.3437 
26 
0.3313 
46 
0.3195 
7 
0.2918 
27 
0.3388 
47 
0.2802 
8 
0.3116 
28 
0.3192 
48 
0.3463 
9 
0.3077 
29 
0.3323 
49 
0.2802 
10 
0.3426 
30 
0.3348 
50 
0.3356 
11 
0.3476 
31 
0.3462 


12 
0.3450 
32 
0.3317 


13 
0.3196 
33 
0.3322 


14 
0.3171 
34 
0.3272 


15 
0.3321 
35 
0.3305 


16 
0.3441 
36 
0.3383 


17 
0.3435 
37 
0.3456 


18 
0.3240 
38 
0.3456 


19 
0.3293 
39 
0.3106 


20 
0.3155 
40 
0.3408 

Appendix 3.2:
To illustrate the impact of the weight
distribution on the extrapolation of the unit count, three distinct populations
of weights of tablets were evaluated.
All groups contain 50 tablets.
Tablets from each group look visually
similar. The total weight of each group
of 50 tablets is weighed using a oneplace balance (with uncertainty of 0.3581
gram). A sample size of 10 tablets from
each group is randomly sampled for individual weighing using a fourplace
balance (with uncertainty of 0.000484 gram).
The calculations for the extrapolation of tablet count for the three
groups are shown in Table 3.4 below.
The RSD of the sample, and hence the expanded
uncertainty of the extrapolation, depends on the distribution curve. A population with a smaller spread will yield
a smaller standard deviation and hence smaller expanded uncertainty.
Table 3.4:
Calculations for 3 groups of tablets each with sample sizes of 10.

Group 1 
Group 2 
Group 3 
Total Weight of 50 tablets, TW, gram 
16.3 
28.7 
27.9 
Avg weight per tablet, 
0.31906 
0.58253 
0.55591 
Std deviation, s 
0.018287 
0.011608 
0.0052800 
% RSD 
5.73142 
1.9926 
0.94980 
Extrapolated tablet count, 
51.088 
49.268 
50.188 
Std uncertainty of avg wt, 
0.0181877 
0.0036706 
0.0016697 
Rel. uncertainty of total wt, 
0.021969 
0.012477 
0.012835 
Rel. uncertainty of avg wt, 
0.0181877 
0.0063557 
0.0031272 
Combined rel uncertainty, 
0.028521 
0.014003 
0.013211 
Extrapolated combined uncertainty, 
1.45707 
0.68989 
0.66301 
With Level of Confidence
= 95% (k = 2.262) 

Expanded uncertainty, 
3.296 
1.561 
1.500 
With Level of Confidence
= 99% (k = 3.250) 

Expanded uncertainty, 
4.735 
2.242 
2.155 
Figure 1: Histograms showing the spread of weights for the 50 tablets in
the three groups. The spread of the data
in group 1 is larger and further from normality as compared to Group 3.
D References
D.1
Kutner, M. H., Nachtsheim,
C. J., and Neter, J. 2004. Applied Linear Regression Models, 4th edition,
McGrawHill.
D.2
ENFSIDWG Guidelines on Sampling of Illicit Drugs
for Qualitative Analysis (2016). http://enfsi.eu/wpcontent/uploads/2017/05/guidelines_on_sampling_of_illicit_drugs_for_qualitative_analysis_enfsi_dwg_2nd_edition.pdf
D.3
EURACHEM/CITAC Guide. Quantifying Uncertainty
in Analytical Measurement, Third Edition (2012). ISBN 9780948926303.
D.4
Guide to the Expression of Uncertainty in
Measurement (GUM). ISO, Geneva (1993). (ISBN 9267101889) (Reprinted 1995:
Reissued as ISO Guide 983 (2008), also available from http://www.bipm.org as
JCGM 100:2008).
D.5
ASCLD/LAB. 2013. ASCLD/LAB Policy on Measurement
Uncertainty. ALPD3060 Ver 1.0.
D.6
Mario, J. 2010. A probabilitybased sampling
approach for the analysis of drug seizures composed of multiple containers of
either cocaine, heroin, or Cannabis. Forensic
Science International 197: 105113.
[1] Where populations from which samples are selected diverge substantially from a normal distribution, weight extrapolations using small sample sizes (e.g. n = 3) may yield unreliable extrapolations and associated uncertainties.
[2] An alternative approach could be to calculate the total net weight of the powder by subtracting the extrapolated weight of the empty bags from the total gross weight. This will entail different calculations.
[3]
If the bag contents are visually
dissimilar, they need to be separated into different groups before continuing
with the analysis.
[4]
A random sample is defined as the
sample so selected that any portion of the population has an equal (or known)
chance of being chosen. Haphazard or arbitrary choice of units is generally
insufficient to guarantee randomness in SWGDRUG Glossary of Terms and
Definitions, Annex A.
[5]
See SWGDRUG Supplemental Document SD3
for discussion on weighing processes (dynamic and static) and measurement
uncertainty.
[6]
The laboratorys requirement should
ensure that the variability of the measurements is small enough that all
samples can be considered as belonging to the same population. ENFSI Guidelines on
Representative Drug Sampling (Reference D.2), page 48 states In common
practice, an acceptance criterion is that the sampling results are taken into
consideration if the ratio between the standard deviation s and the average
weight of a drug unit in the sample is less than 0.1
(RSD<10%). Otherwise, an increase of the sample size is required in order to
reach the target percentage. In
casework, RSDs of sample weights higher than 10% may be encountered (see
reference D.6). For such cases, when
necessary and feasible, laboratory personnel may evaluate the RSD acceptance
criteria based on weight and type (e.g. pharmaceutical versus illicit) of
sample.
[7]
When a sample size of greater than 10%
of the population is used, a finite correction factor (Q) of should be applied to the combined
uncertainty (Reference D.2). However,
since this correction factor is always less than 1 and decreases as n
increases, it reduces the total uncertainty.
The finite correction factor was not applied to these examples as
omission results in a more conservative estimate of uncertainty.