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

JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS or'-- Z(x -- •)' = .6'0025 6'0025 __ 0-54568 -- n --1 12 -- 1 11 Note that x -- • is checked by addition, when it should total zero. ß It is seen that the arithmetic is laborious even with the aid of a table of squares. However, this work can be reduced considerably by "coding" the results by deducting a convenient value and multiplying to remove the decimal point. The mean of the code'd values is calculated and transformed into the original units by reversing the coding process to give the true mean. The variance is calculated by making use of the algebraically equivalent formula for it, i.e., (,' _ z(x)' - ( zx)'l, The calculations are then' Coded results, i.e. x (x -- 35-0) 10 Called ¾ 37.6 26 676 38-2 32 1,024 35 '8 8 64 36.2 ' 12 144 36.4 ' 14 196 37.3 23 529 37.4 24 576 36-9 19 361 35.8 8 64 36.7 17 289 36.7 17 289 36.3 13 169 441.3 213 4,38.1 coded • ß 4,381 -- 213' 12 11 , 4,381 -- 45,369 12 11 4,381 -- 3780.75 11 600.25 -- -- 54.568 11 ., 236