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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 /; = 0.3 40 Powder sampling and particle size determination Example 2 16 samples, withdrawn at random from an unmixed powder, gave a median x^^ = 3.13 |iim by multi-angle laser light scattering (MALLS) with a standard deviation of ^ == 0.80 |Lim.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