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

JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS ARITHMETICAL ANALYSIS In the above analysis we have employed an equation to a line fitted to points by eye and have introduced errors thereby. It is possible to improve upon this by an arithmetical procedure which should be used when the points are too irregular to draw a satisfactory line to by eye. This makes use of a weighted expected probit called a "working probit," taken from the expected probit read from a provisional line. Finney, TM Fisher and Yates •8 and Bliss TM give tables for the easy calculation of the working probit. We will use Finney's Table IV for our source of working probit, again using Sample No. 3. We calculate the successive columns in Table VIII. TABLE VIII No. of Expec- Working Log particles probit probit dia- exam- meter ined nw nwx nwx • nwy nwy • nwxy • n Y y 0.279 2,000 4.85 1,260 352 98 4.80 6,048 29,030 1,687 0'591 2,000 5'39 1,202 711 420 5'42 6,515 35,311 3,850 0'771 2,000 5'71 1,060 817 630 5'72 6,063 34,680 4,676 0'898 2,000 5'91 936 841 755 5'92 5,541 32,878 4,977 0'996 2,000 6'08 824 820 817 6'08 4,986 30,315 4,967 1'076 2,000 6'23 720 775 834 6'21 4,472 29,882 4,812 -- TOTALS 6,002 4,316 3,554 33,625 192,096 24,969 Note ß Up to and including the column nwx • the calculations are identical with the graphical procedure summarised in Table VII. We now calculate the mean value of x, •, = Y•nwx/2:nw and of y, •7, ---- Y•nwy/2:nw the sums of squares and products about the mean, i.e, Znw(x- •)•= Znwx • -- (Znwx)•/Znz• Znw(y-g)•= Znwy•--fZnwy)'[•' nw and Znw (x-•) (y -- g) = Znwxy -- (r•nwx) (r•nwy)/Znw. We have ß ß = 4,316/6,002 ---- 0.719 g----- 33,625/6,002 = 5.602 Y•nw (x -- :•)•: 3,554 -- 4,3162/6,002 ---- 3,554 -- 3, I02 = 452 Y. nw (y -- g)•---- 192,096 --33,6252/6,002- 192,096 -- 188,377 = 3,719 Zn, (x - - g) = 24,969 - (4,313) (33,625)/6,002 = 24,969 - 24,163 = 806 The constant b (called the regression coefficient) of the equation Y = a -t- bX, is Znw(x -- •)CY -- g)/Znw(x -- •)• = 806/452 = 1.783 and the best line is Y = 5.602 q- 1.783 (x -- 0.719) = 4.320 q- 1.783x. By using the equation to calculate the values of Y at the values of x actually used, we can find • in exactly the same manner as shown in 166