STATISTICAL METHODS IN THE COSMETIC INDUSTRY TABLE IV Sample No. 1 2 3 4 5 6 7 ED50, Log Microns 0.106 0.204 0.366 0.260 0.095 0.435 0.463 ED50, Microns 1.26 1.60 2.32 1.82 1.24 2.72 2.90 s, Log basis 0.654 0.538 0.582 0.485 0.675 0.538 0.318 s, comparable to Table II 4.5 3.45 3-• 3.05 4.7 3.45 2-08 The ED50 value and s are more or less the same as those given in Table II, but note that in micron. (iii) The above graphical results are used to provide "expected probits" at the log diameter used experimentally, and these can be either read directly from it or calculated by substitution in the equation for the straight line Y = a + bX (Y being the probit, X the log diameter). This equation is easily found by taking two points on the line, x•ya x,y,, and then b ---- (y•--y,)/(x•- x,) and a--y•- bx•. The equation for the relationship between probit and log. diameter, Y ----- a q- bX, is found to be for No. 3: Y ---- 4.37 q- 1.72X. We have entered the appropriate results for the sample 3 in Column 5 of Table V (below), and will carry out the arithmetical calculations for Sample No. 3. Our data for this sample is now: TABLE V No. of % particles particles % Particles Empirical Expected below stated Log diameter examined below stated probit probit. size calculated .¾ * size Y from expected probit 0.279 2,000 42.2 4.80 4.85 44.0 0.591 2,000 66.2 5.42 5.39 65.2 0'771 2,000 76'3 5-72 5'71 76.1 0.898 2,000 82.2 5.92 5.91 81.9 0.996 2,000 86-0 6.08 6.08 86.0 1.076 2,000 88-7 6.21 6.23 89.1 * Approximately. The numbers are not stated, bug the text states that around 2,000 were measured. (iv) We also want to know whether the assumption is reasonable that the frequency distribution of numbers of particles at a given log diameter can be represented by the normal distribution. To do this we compare the observed frequencies with those to be expected on that assumption, using 163

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