SERIATIOtf AKD PLOTTING OF DATA. 15 



measured, but the accuracy does not increase in the same ratio 

 as the number of individuals measured, but as the square root 

 of the number. The probable error of the mean decreases as 

 the standard deviation decreases. 



The index of the variability, cr, of the variates when 

 they group themselves about one mode is found by adding 

 the products of the squared deviation-from-the-mean of each 

 class multiplied by its frequency, dividing by the total 

 number of variates, and extracting the square root of the 

 quotient, thus : 



sum of [(deviation of class from mean) 8 



X frequency of class] 



number of variates 



n 



This measure is known as the standard deviation. 

 The probable error of the standard deviation is 



standard deviation cr 



0.674o = 0.6745 . 



y 2 X number of variatts \/2n 



Other Indices of Variation are the average deviation, or aver- 

 age departure, which is found thus: 



of [deviations of class from mean x frequency] 



number of variates 

 The probable error is the distance from the mode of that ordinate 

 which exactly bisects the half curve OMX or OMX*, Fig. 23; it is equal to 

 0.6745 X standard deviation = 0.6745<r. Neither of these last two indices 

 of variation is as good as the standard deviation 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 proposed to re- 

 duce the index of variation to a concrete number, independent 

 of any particular unit, by dividing the index of variation of any 

 variates by the mean ; the quotient multiplied by 100 is called 



the coefficient of variability. In a formula, GV ^. 

 (Pearson, '96 ; Brewster, '97. ) 



