JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Values of t required by the Examples Given Degrees of Probability Freedom 0.8 0-7 0-1 0.05 0.01 I 3 0-277 0.424 2.353 3.182 5-841 5 0-267 0.408 2.015 2.571 4-032 I0 0.260 0.397 1.812 2.228 3.169 11 1-796 2.201 3.106 15 1-753 2.131 ' ' Values of F required l•y the Examples Given N• is degrees of freedom of larger variance. Ns is degrees of freedom of smaller variance. p = 0.20 • = 0.001 '•'-'-• N• 4 5 2 10 1.8 . 1.8 14.9 15 1-7 1-7 11.3 18 1-67 1.64 10.4 19 1.66 1-63 10.2 Values of X 3 required by the Examples Given Degrees of Probability Freedom 0.2 0.1 0.02 0.01 1 1.642 i 2-706 5.412 6:635 2 3.219 I 4-665 7.824 9.210 4 5.989 7.779 I 1.668 13.277 Value of r required by .Example Given Degrees of Probability Freedom 0-01 0-001 2 0.990 I .- 0-999 VARIANCE There are several ways in which the .variability of a set of results ca'• be expressed, of which the more useful are the range and the vananc The ran e of a series of results is the difference between the highest and th•'• lowest restfit of the series. ' The variance is the sum of the squares of the de•ation of each individ½•'"•)•( result from the arithmetical mean of the series divided by the numbe results in the series when there are 30 or more results, divided by one le•'•:}• than the number of results when the series consists of less than 30 resul•2:•

STATISTICAL METHODS IN THE COSMETIC INDUSTRY variance is universally denoted by ,' (sigma squ•red). The divisor of the squares of the deviations is called the "degrees of ' associated with the variance and can be regarded as the number deviations from the mean that can be selected at random. Since the of the deviations from the mean of n results must be zero, only n -- 1 the results can be selected at random, the series having then n -- 1 degrees freedom. Symbolically, z (x - The standard deviation is the square root of the variance and is thus by .. ':'•d?•i-::::i The uses of the standard deviation and of the variance will become ?/evident from a study of the examples of the application of statistical methods . •,½:: which follow. ' CALCULATION OF Wm*NC ' • (ii"iii I:: Consider a series of results obtained for the weight of the contents of i•:i•:?bottles being filled on an automatic filling line. These were 37.6, 38-2, iii !35.8, 36.2, 36-4, 37-3 37-4, 36-9, 35-8, 36.7, 36-7, 36.3 grams. It is required ii•i':to know ihe mean l•ottle content of the whole of the population (i.e., the •}111Whole of the bottles filled under the conditions of sampling), from which the :.: ß :i: samples weighed were selected at random during the filling, and further, ?the limits of the mean such that the probability is 19 times out of 20 times :•il that the true mean of the population will be within these limits. By definition the variance of the sample n--1 The calculations are best tabulated, giving' • • - • (• - •)' 37-6 I 0.825 0.680625 38-2 1.425 '2-030625 35.8 -- 0-975 0-950625 36-2 • 0-575 0.330625 36.4 . -- 0.375 0-140625 37-3 0.525 0-275625 37.4 0.625 0.390625 36.9 0.125 0-015625 35.8 -- 0-975 0.950625 36-7 -- 0-075 0-005625 36.7 -- 0-075 0.005625 36.3 -- 0-475 0.225625 441.3 0.000 [ 6-0oe2500 Sum (oe) = No. of l•esults (n) = 12. Mean (x) = 36.775. 235