JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS the X' test, n the successive steps in the calculation being shown by the headings of Table VI. TABLE VI No. of % Actual No. of (Col. 6) • Log particles Particles Particles Col. 4- (Col. 6) • diameter measured expected Col. 5 Variance Observed Expected expected -- 0.279 2,000 44.0 844 880 -- 36 1,296 2.63 0.591 2,000 65.2 1,324 1,304 20 400 0.88 0'771 2,000 76'1 1,526 1,522 4 16 0.04 0-898 2,000 81.9 1,644 1,638 6 36 0.12 0.996 2,000 86.0 1,720 1,720 0 0 0.00 1.076 2,000 89.1 1,774 1,782 -- 8 64 0.30 -- X 2 ---- 3'97 The denominator in col. 8 is npq where n is number examined, p the probability of occurring, q the probability of not occurring (i.e. 1 --p). For log diameter 0.279, this is 880 (1 -- 0'44) = 493, the other variances being calculated similarly. x 2 has two degrees of freedom less than the number of groups, --• 4, since two have been used in estimating the equation to the log diam. --probit line. The probability of obtaining a x 2 as large as 4 is 0.45, which suggests that the log particle diameters are distributed accord- ing to the Gaussian curve and that the probit of the line drawn is a satisfactory representation of the particle size. The probit line might be badly drawn when x 2 would have a low probability, but an alternative method of calcula- tion allows for this and permits the best fitting line to be drawn from its calculated equation (see Arithmetical Analysis, p. 166). We will want to know the confidence limits that apply to the mean log diameter (or to any other selected log diameter), and tables of weighting coefficients (w) have been prepared, m•8,•* which enable these to be deter- mined. The weighting coefficient is obtained from the expected value of the probit by reference to the standard tables, the part relevant to these calcula- tions showing: 0.0 O. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.627 O.634 0.637 0.634 0.627 0.616 0.601 0.581 0.558 0.532 0.503 0.471 0.439 0.405 0.370 0.336 The successive steps in the calculation are shown by the column headings of Table VII. 164

STATISTICAL METHODS IN THE COSMETIC INDUSTRY TABLE VII Log No. of Expected Weighting diameter particles probit co-elficient nw nw:v nw:v • measured x n Y w 0'279 2,000 4.85 0' 630 1,260 352 98 0.591 2,000 5.39 0.601 1,202 711 420 0'771 2,000 5.71 0.530 1,060 817 630 0'898 2,000 5'91 0.468 936 841 755 0'996 2,000 6.08 0.412 824 820 817 1.076 2,000 6' 23 0.360 720 775 834 TOTALS : 6,002 4,316 3,554 The mean log diameter, •, used can be found by dividing Y, nwx by r. nw, giving ß -• 0.719. Now the variance of the mean (m) is (1/Y. nw q- (m-•)'/Znw (x-•)')/b'. We have already found m -- 0.366 (Table IV) and b ---- 1-72, and now' Enw ---- 6,002 Znw(x-•) • •- Znwx • -- (Znwx)'/Enw ---- 3,554 -- (4,316)•/6,002 = 452 Therefore variance of m -- (1/1.72 •) (1/6,002 q- (0.366 -- 0.719),/452 = (1/2.96)(0.000167 + 0.353'/452) • 0'338 (0'000167 q- 0'000276) = 0'000131 whence s• -- %/variance = 0.036. The standard error of the ED50 of the diameter in microns is given by the formula ß S.E.: l0 rn loge 10 x Sm = 2.32 x 2.30 x 0.036----0.192 Thus the ED50 is 2.32 q- 0.192 microns. Care must be taken that the logarithms are used when calculating the standard error of any diameter. Thus the p = 0.1 limits of the mean are not 2.32 q- 1.645 x 0.192 ---- 2.32 q- 0.316, i.e., 2.00 to 2-64 (the value of t at p = 0.1 is 1.645), but are calculated from the logarithmic values, i.e., 0.366 q- 1.645 x 0.036---0'366 + 0.059, i.e., 0.307 to 0.425, which gives in microns 2.03 to 2.66--which are not symmetrical about the mean and would not usually be as close to the incorrect answer first given as they are here. If we wish to find the confidence limits at any other diameter than the mean diameter, we substitute log of that diameter for m and continue calculations in exactly the same way. In cases where the points are irregular and it is difficult to draw a satis- factory graphical line, the accuracy of the analysis can be very greatly improved by further arithmetical calculations. 165