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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  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)
.
/; = 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16 samples, withdrawn at random from an unmixed powder, gave a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 . 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```