16 STATISTICAL METHODS. 



This measure is known as the standard deviation. It 

 is a concrete number expressed in the units of the classes. 

 This, the best measure of variability, is expressed geomet- 

 rically as the half parameter, or the abscissa of the point on 

 the frequency curve where the change of curvature (from 

 concave to convex toward the centre) occurs. 



The probable error of the standard deviation is 



+ n.B74K ^andard deviation _ 0-6745 * 

 V2 X number of variates \/2n 



Other Indices of Variation. The average deviation, 

 or average departure, is found thus: 



_ sum of [deviations of class from mean X frequency] 

 number of variates 



The average deviation is equal to .7 97 9 X standard deviation, or 



The probable (or mid) departure is the distance from the mode 

 of that ordinate which exactly bisects the half curve QMX or OMX 1 , 

 Fig. 5, it is equal to 0.6745 X standard deviation = 0.6745o. Neither 

 of these last two indices of variation is as good as the standard devia- 

 tion when n is rather small. 



The standard deviation, like the other indices of variation, 

 is a concrete number, being expressed in the same units as 

 the magnitudes of the classes. The standard deviation of 

 one lot of variates is consequently not comparable with the 

 S. D. of variates measured in other units. It has been pro- 

 posed to reduce the index of variation to an abstract number, 

 independent of any particular unit, by dividing the index of 

 variation of any variates by the mean; the quotient multi- 

 plied by 100 is called the coefficient of variability. In 



a formula, C=-^XlOO% (Pearson, '96; Brewster, '97). 



The probable error of the coefficient of vari- 

 ability is given by Pearson as: 



