Powder sampling
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Powder sampling


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likely to have the true value of 
x^ within its limits. The 99%) confidence interval is larger (M= 2.576) but 
is more likely to contain the true value of x^ within its limits. 
For pharmaceutical applications, a value / = 2 is used to denote 
working quality and a value / =^ 3 (99.9% confidence level) is used for 
total quality [35] The statistical reliability of analytical data can be 
improved by increasing the homogeneity of the sample (reducing a), 
increasing the size of the sample, or increasing taken (increasing n). 
In most instances the population standard deviation is not known and 
must be estimated from the sample standard deviation (s). Substitution of 
s for a in equation (1.1) with M = 1.96 does not result in a 95% 
confidence interval unless the sample number is infinitely large (in 
practice n>30). When s is used, multipliers, whose values depend on 
sample number, are chosen from the /^-distribution and the denominator in 
equation (1.1) is replaced by ^{n -1) . 
Assuming a normal distribution of variance, the number of samples 
required, to assume at the 95%) confidence level that the median is known 
to +.4, is given by 
^ts^' 
n- (1.2) 
where ^ = | / i -x^ | is the maximum allowable difference between the 
estimate to be made from the sample and the actual value. 
Example 1 
In many sampling procedures, sub-samples are taken at different levels 
and locations to form a composite sample. If historical evidence suggests 
that the standard deviation between samples is 0.5, and it is necessary to 
know the average quality of the lot to within 0.3, the number of sub-
samples required, at the 95% confidence level, is given by equation (1.2) 
[36]. 
/; = 0.3 
40 Powder sampling and particle size determination 
Example 2 
16 samples, withdrawn at random from an unmixed powder, gave a 
median x^^ = 3.13 |iim by multi-angle laser light scattering (MALLS) with 
a standard deviation of ^ == 0.80 |Lim. Then, in microns, the median lies 
between the limits: 
// = 3.13±2.14- -^^ 
V16-1 
/i = 3.13±0.4 
The multiplier ^ = 2.14 is obtained from a / table for n = 16-1 degrees of 
freedom at the 95% confidence level. Thus we are 95% confident that the 
median lies in the confidence interval (CI): 
CI = 2.69 < 3.13 < ± 3.57 
Based on this data, the number of samples required to give an estimate to 
within ±0.10, (£' = 0.10) is: 
M = 
^2.14x0.8^^ 
V 0.10 
«=293 
After mixing, 16 samples gave a median jc^ of 3.107 |a,m with a standard 
deviation s, of 0.052 fim. 
Then: 
/i = 3.107±2.14 ^0.052^ 
/ /= 3.107 ±0.029 
Thus we are 95% confident that the true median lies in the confidence 
interval: 
CI-3.078 < 3.107 < 3.136 
i.e. the mixing step increases the measurement precision by a factor of 
fourteen. 
Powder sampling 41 
A single sample, run 16 times on a MALLS instrument gave a median of 
3.11 jLim with a standard deviation of 5^ ^ 0.030 |Lim. The total variation 
Isf ] is the sum of the variation due to the measuring procedure Is^ ) and 
the variation due to the sampling procedure (^^). 
It is possible to isolate the sampling error from the measurement error. 
The standard deviation due to sampling (sj is 0.042 |Lim and the standard 
deviation of the measurement technique (s^) is 0.030 |Lim giving a total 
standard deviation (s^) of 0.052 |Lim. As can be seen from this example, 
there is little to be gained in using a measurement technique substantially 
more accurate than the sampling that preceded it. 
+ s: (1.3) 
0.042 
Further, the number of samples required, after mixing, in order to assume 
at the 95% confidence level that the median is known to ±0.10 |Lim is: 
n-\ 
2.14x0.052 V 
0.10 
n = 2.2 i.e. 3 samples. Compounding increments from the unmixed 
powder whilst it is in motion, for example, by riffling in a spinning riffler 
can attain the same accuracy. 
Table 1.2 Means and standard deviations of active ingredients in 
simulated sampling trials 
Sample weight 
(g) 
1 
3 
5 
9 
Mean 
(mg) 
98.20 
99.45 
99.50 
100.20 
Standard deviation 
(mg) 
5.78 
4.78 
3.74 
3.10 
Expected standard 
deviation (mg) 
8.31 
4.83 
3.74 
2.79 
In the pharmaceutical industry there has been a tendency by federal 
agencies to request that blending validation be carried out using samples 
the size of a dosage unit. A theoretical experiment using random numbers 
was carried out to assess the effect of changing sample size [37]. A set of 
42 Powder sampling and particle size determination 
assays of a dosage weighing 1 g containing 100 mg of drug substance was 
extracted from a bulk of 100 g. Four 1 g samples were averaged for each 
of 10 measurements and the standard deviation for each group of four 
determined. The experiment was repeated with 3 g, 5 g and 9 g samples. 
The means and averages for each set of 10 groups are given in Table 1.2. 
The expected standard deviation is based on the premise that it is 
inversely proportional to the square root of sample size. Assuming the 5 g 
sample gives accurate data it can be seen that the 9 g sample is slightly 
worse than expected and the 1 g sample is much better than expected. 
1.9 Theoretical statistical errors on a number basis 
The ultimate that can be obtained by representative sampling may be 
called the 'ideal' sample. A powder may be considered as made up of two 
components A and B. The probability that the number fraction {p) of the 
bulk in terms of ^ shall be represented by the corresponding composition 
ip) of an ideal sample can be computed from the number of particles of .4 
and B in the sample {n) and the bulk (N). 
PO-P) r n 1- (1.4) 
where a; is the theoretical standard deviation, [the variance Var(o;) is 
defined as the square of the standard deviation]. 
For a normal distribution of variance, the spread of data about the mean 
is described by the probability equation 
d^ 
dp 
1 
CTI^IITT 
exp jp-p) 2a' 
2\ 
(1.5) 
Using the transformation a, j = {p- p) 
dip 1 
dy yjlK 
( y-^ 
(1.6) 
(1.7) 
Powder sampling 43 
0.5 
0.4 H 
0.3 H 
0.2 H 
0.1 H 
0.0 
Fig. 1.38 Normal probability function (relative). 
Fig. 1.39 Normal probability function (cumulative). 
44 Powder sampling and particle size determination 
Table 1.3 Cumulative normal distribution 
y 
-4.0 
-3.0 
-2.0 
-1.5 
-1.0 
-0.5 
0.0 
0.5 
1.0 
1.5 
2.0 
3.0 
4.0 
<t> ( % ) 
0.003 
0.13 
2.28 
6.68 
15.87 
30.85 
50.00 
69.15 
84.13 
93.32 
97.72 
99.87 
99.997 
y 
-3.50 
-2.50 
-1.75 
-1.25 
-0.75 
-0.25 
0.00 
0.25 
0.75 
1.25 
1.75 
2.50 
3.50 
d(t>IAy 
0.13 
2.15 
8.60 
18.38 
29.96 
38.30 
39.89 
38.30 
29.96 
18.38 
8.80 
2.15 
0.13 
Table 1.4 Variation in the number of black balls in samples taken from a 
bulk containing 4000 black balls and 8000 similar white balls. 
N 
Upper limit 
3857 
3873 
3889 
3905 
3921 
3937 
3953 
3969 
3985 
4001 
4017 
4033 
4049 
4065 
4081 
umber of black balls 
Lower limit 
3872 
3888 
3904 
3920 
3936 
3952 
3968 
3984 
4000 
4016 
4032 
4048 
4064 
4080 
4096 
Median value (x) 
3864 
3880 
3896 
3912 
3928 
3944 
3960 
3976 
3992 
4008 
4024 
4040 
4056 
4072 
4088 
Frequency of 
occurrence (A«) 
0 
1 
6 
10 
43 
103 
141 
195 
185 
160 
90 
37 
17 
11 
1 
Powder sampling 45 
This differential equation is presented graphically in Figure 1.38 and the 
integrated version in Figure 1.39. From Table 1.3, 68.26% of all 
occurrences lie within ±1 a; (between y = -\ and j = +1) from the mean, 
95.44% within ±2q and 99.94% within ±30;. 
Example 3 
Consider a bulk made up of 8000 white balls and 4000 black balls from 
which 750 are extracted. Substituting in equation (1.4): 
750 
12,000, 
a; =0.0167 
Hence: « ± wo; = (2/3 ± 0.0167) of 750 for the white balls 
n ±na^ =n± 12.5 
where n = 500 or 250 for the white and black balls respectively. Thus 68 
times out of 100 there will be between 487.5 and 512.5 white balls in the 
sample; 95 times out of 100 there will be between 475 and 525 and 26 
times out of 10, 000 there will be either more than 537.5 or less