SOCIETY OF THE UNIVERSITY OF ABERDEEN. 91 



been measured ? We might take the average value of all the devia- 

 tions from the mean, but this involves the assumption that the 

 probability of a deviation is inversely proportional to its distance 

 from the mean. A little consideration will show that the probability 

 of a deviation occurring decreases more rapidly than its distance from 

 the mean increases. Take as an illustration the case of bullets fired 

 at a target. The bull's-eye here represents the mean position. The 

 marksman who hits the target 1 foot from the bull's-eye is four times 

 as good a shot as the marksman who hits the target 2 feet from the 

 bull's-eye, because the latter requires a target four times the area 

 of that required by the former in order to be sure of hitting. In 

 order, therefore, to estimate the value of a man's hits on a target 

 each hit should be multiplied by the square of its distance from the 

 bull's-eye, and the sum of these products divided by the total number of 

 hits. This expression gives the square of what is called the standard 

 deviation of his shooting, and his skill is, of course, inversely propor- 

 tional to this standard deviation. 



This method of estimating the value of the total deviation from 

 the mean is used in treating anthropometric statistics. Designating 

 the standard deviation by a, the distances from the mean on both 

 sides by JC^K^ . . . JC P , and the frequency at each distance by z^z^ 

 . . . z p , we get 



. . . + ZpX,? ' 



_ 



N 



N being the total number of persons measured, using the symbol 

 for summation, the above expression may be written more shortly : 



N 



To illustrate the manner of carrying out the calculations involved 

 in the method so far explained, let us take as an example the measure- 

 ments of the head lengths and head breadths of 364 peasantry of East 

 Aberdeenshire. 1 



1 See " Physical Characteristics of Adults and School Children in East Aberdeen- 

 shire," Gray and Tocher, Jour. Anth. Inst., 1900, p. 104, 



