SUPPLEMENTAL DOCUMENT SD-6

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 three-place 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 three-place 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 root-sum-square (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, (Student’s 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 laboratory’s 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 25-gram 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 Student’s 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 = 1-a).  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:

H1 - Hypothesis that ≥ 48 bags contain cocaine

H0 - 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 (H0). 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 p-values (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 H0 and accept H1, 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, W48, can be extrapolated from the average net weight per unit (obtained from Example 1):

 

W48 = 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 Student’s 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, W48, 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 25-grams 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 155-156) 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 used5:

 

Uncertainty for one-place balance (0.1 g readability),  = 0.35810 gram

Uncertainty for four-place 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 laboratory’s 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 one-place balance (with uncertainty of 0.3581 gram).  A sample size of 10 tablets from each group is randomly sampled for individual weighing using a four-place 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, McGraw-Hill.

 

D.2         ENFSI-DWG Guidelines on Sampling of Illicit Drugs for Qualitative Analysis (2016). http://enfsi.eu/wp-content/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 978-0-948926-30-3.

 

D.4        Guide to the Expression of Uncertainty in Measurement (GUM). ISO, Geneva (1993). (ISBN 92-67-10188-9) (Reprinted 1995: Reissued as ISO Guide 98-3 (2008), also available from http://www.bipm.org as JCGM 100:2008).

 

D.5        ASCLD/LAB. 2013. ASCLD/LAB Policy on Measurement Uncertainty. AL-PD-3060 Ver 1.0.

 

D.6        Mario, J. 2010. A probability-based sampling approach for the analysis of drug seizures composed of multiple containers of either cocaine, heroin, or Cannabis. Forensic Science International 197: 105-113.

 

           

 



[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 SD-3 for discussion on weighing processes (dynamic and static) and measurement uncertainty.

[6] The laboratory’s 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.