.. STATISTICAL METHODS IN THE cOsMETIC INDUSTRY ANALYSIS OF VARIANCE ß _ . i .The t test is a rapid method of determining the relation between factors by using the appropriate procedure described above. However, three or more factors have to be compared it is laborious to analyse respective relations in pairs, and the work can be considerably reduced '•he}-r-:-• "Analysis of Variance" techmque: ....... • •-'o•veral factors • •i:as' the valuaate propert• This anows the total variance to be factor contributing to me t• ß so as to give the variance due to each factor. discussing the t test, sections (a, b), an example was given in which Mcohol in six samples of an essential oil was determined by three methods, and the effect of the method evMuated. It will be useful use this example again to illustrate the cMculation of the simplest case of of variance. For the purposes of this example, it six determinations were made by each of three methods on one bulk 'a½:•of od.. o, Alcohol Content /o Determination Method 3 No. Method 1 Method • Y 1 26 15 52 2 8 10 24 3 14 6 35 4 24 31 43 5 8 19 34 6 17 22 48 -• 103 236 • 97 •:• :: No. •esults eac• method •: Total •o. of results Grand Total = 436 • •x, y, z. are dealt with by the following procedure' •:•':•(:•':/' (1) Square the total of aU individuals and di•de by the total number (•x,v.,z)' (436)' 10,561 (This is the Correction Term }?:of:individuals = - N 18 ?•'(C.T.) which is subtracted from each sum of squares.) '•:•:.. (2) Square each individual and add = zx,, z = + 8' + ....... + + 48 the number ?.?: (3) Square the total of each method and divide by • individuals in it and add ' 97 • + 103 • + 236 • 75,714 _ 12,619. . ?This leads to the following Analysis of Variance table :':' 241 . 17 :.

'JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Source of Variance , Between Methods Within Methods Total Degrees of Freedom No.' of Methods -- 1 -----2 17 -- 2 = 15 , No. of Individuals --1=I7 Sum of Squares (3) - C.T. = 2,058 3,285 --' 2,058---- 1,227 (2) - c.r. = 3,285 Mean Square col. 3 .'-- col. 2 1,029 81.8 Components of Variance •2 The Between Methods Mean Square may or may not be significantly different from the Within Methods Mean Square, .and. this is estimated by means of the F test in which the Mean Square for Between Methods is dividei :, by that Within Methods to give the Variance ratio, E. This is 1,029/81'8 -- 12-6. Reference to tables of F, with N• the degrees of freedom of the larger Mean Square (two) and N,. the degrees of freedom of the smaller lVIean Square (fifteen), shows that at p = 0'001, F has a value of 11.3. Our value is greater than this, showing the probability that the value found would be obtained':71 as a matter of chance to be less than 0.001 meaning that there is little doubt that there is a significaht difference in the variance between methods when compared with the var/ance within methods. The internal "Within Methods" component of the Variance (= %" 81.8) is the me•ure of the error of the determinations, whilst the = :½ .. "Between lVIethods" component of variance (na n' q- a•' = 1,029) gives the variance i! .. due to the method. For the latter we have: 6=• 2 + 81.8 = 1,029 . .'.a• • -- 158. The total variance of the sample is the sum of that due to method, a'- -- 158 and that due to error, %• = 82. It is apparent that variance due to the method makes a bigger contribution to the total variance/• than does that due to the error. If it is desired' to know the true alcohoF• content of the oil, more improvement would be given by steps taken ti5 : * reduce the variance between the methods than by reduction oI the error inherent in any one method. (b)'In the t test, section (a), the alcohol was determined by each of "methods, on six samples. To make most use of the data, it can be analys•e ( to show the variance due to the Method, to the Sample and to the Errol: The results are as shown in table opposite ß • ß giving the-calculations' " 436' (1) Correction Term -- -- -- 10,561. ß (2) Squar• each individuk['and add ' 26" q- 82 q- ....... (3) Square the total of each Method and divide by No. of results :i: 242