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Part III. — Eighteenth Annual Report 



to use percentages, and in order to do so, and avoid at the same time the 

 above-meutioned difficulty, the standard dimension should be chosen as 

 near 100 as possible. 



Certain parts of the body, again, may be grouped together if they are 

 related to one another. Thus, the height or breadth of the tail is usually 

 expressed — when it is taken — in terms of the tail-length, and the breadth 

 and height of the skull are expressed in percentages of the skull-length. 

 For the present work forty characters in all have been taken, but only a 

 few of these have been chosen for the present paper, as it was thought 

 better to direct more attention to the thorough working out of the method 

 in a few cases than to the superficial display of the variations of many 

 characters. The characters chosen are probably the most valuable for the 

 purposes required, but the value of the others has yet to be decided. 



When the percentage-values of a character are arranged in order it is 

 found that a varying number of individuals come under each percentage. 

 Each percentage-value is called a " variate " or " variant," and the number 

 of individuals under each variant is the " frequency " of the variant. The 

 variant having the largest frequency is called the " mode." When the 

 average or mean has been calculated these variants are commonly spoken 

 of as " deviations." More strictly used, the term " deviation " applies to 

 the distances that the variants are from the mean or average. When 

 these distances have been calculated their average may be obtained, and 

 this is called the "average deviation" of the variants. Within recent 

 years, however, biologists, following the advice and example of Pearson, do 

 not employ the average deviation, but the " error of mean square " of the 

 deviations. This is obtained from the formula 



n 



where x is any deviation, / its frequency, and n the total number of 

 individuals, cr, the so-called " error of mean square " of the physicists, is 

 called the "standard deviation" or "index of variability" by biologists. 

 It is employed to represent the range of variations or the "variability" 

 of a character, and is useful in comparing the variability of different organs 

 n the same species or the same organ in different species. 



A short method of obtaining cr, given by Pearson and described by 

 Duncker, consists in taking the " mode " as the approximate average, 

 calling it 0 and the deviations from it + 1, + 2, + 3, etc., on the one 

 side, and — 1 — 2 — 3, etc., on the other. Calculating the "average devia- 

 tion " of these (having regard for the signs) we obtain the distance of the 

 true average of the variants from the mode. The sum of the squares of 

 these deviations is then calculated, according to the formula 2 (n 2 /), and 

 cr is then obtained from the formula — 



where x is any one of the integral deviations, and / its frequency as before. 

 Since this mode of working involves only whole numbers, it is obviously 

 much simpler than the older method of employing the actual deviations 

 from the true average, which meant using numbers of two and three 

 figures where now 0, 1, 2, etc., are used. 



The further processes in the development of the mathematicaljmethod 

 are borrowed from the mathematical theory of probabilities. Without 

 entering into the reasoning of their mathematical aspect, which may be 

 found in the writings of Galton, Weldon, and Heincke (I.e.), it is necessary 

 to show their biological meaning. The individuals examined represent 

 only a very small fraction of the total number of individuals in the group. 

 Hence the average found for the observed individuals is only probably 



