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

STATISTICAL METHODS IN THE COSMETIC INDUSTRY the coding involves a multiplication factor (in this case 10), the ce must be divided by the square of that factor to give the variance original data, thus' 54'568 • --• -- 0.54568 100 ß v•ith the value found directly. The mean of the coded results 213 -- 17-75 12 itch must be divided by the multiplier (10.) and 35'0 added to the result the mean of the original data, i.e., 17.75 -- q-35.0---- 36.775 10 Standard Deviation is the SCluar e root of the variance, in the above being •/0.54568 = 0'7386, or 0.74 useful purposes. This relates to any individual result. The standard of fhe mean of the series of results is given by the formula 0-74 Thus our a,• ¾'12 0-21 •:71:. If the distribution of the results is such that there are a i•aximum number •iii!about the mean, tailing off on either side of it,' then in practice the distribution !}!i}an be assumed to be normal (or Gaussian), for which tables have been con- !?'Structed showing how often a variation (d) from the mean expressed in terms iii(•øf the standard deviation of the mean can be expected to occur as a matter (•?of chance. This relation d/, is termed Student's t, and tables of this value ?are quoted in most standard works on statistics. !ii.ii!11i.:: In this example, it is desired to know the limits of the true mean of the ?i population at a probability level p := 0,05. :!!:.: Reference to a table of Student's t (see above) shows that for the example ::':With 11 degrees of freedom, at p -- 0-05, t is 2.20. i?., d d !11!:: Now, by definition, t - , thus 2.20 -- ?:?• er,• 0'21 :½:'/' d 2'20 X 0'21 = 0'462, which is the deviation either side of the mean '• with}n which the probability is 19 that the true mean of the population lies. -- 2O 237