MEASUREMENTS OF TYPE ASD VARIABILITY 5 



2. The Comparative Accuracy of the Mean Square Deviation 



(called by various authors <r, /*, c, S.D., or Standard 



Deviation) and the Average Deviation 



The most burdensome of the ordinary statistical operations is 

 the calculation of the square root of the average of the squares of 

 the deviations of a series of measures from their central tendency, 

 that is, of the mean square deviation. In the case of the author,! 

 at least, practical judgment has long rebelled against the imposi-J 

 tion of this measure upon workers with mental measurements byl 

 the experts in the theory of measurement of variable facts. To call 

 it the standard deviation has seemed to him objectionable. There 

 is apparently no reason whatever for its use except its supposedly 

 greater accuracy. Perhaps because of lack of knowledge of the 

 purely mathematical side of statistics, I am not aware that this 

 greater amount of accuracy has been calculated from theory in the 

 case of typical forms of distribution other than the so-called 'nor- 

 mal.' At all events it will be useful to the non-mathematical stu- 

 dent to learn the facts in the case of empirical samplings from 

 known series. 



The series were A, B, C and D of Table I. The facts concern- 

 ing the divergences from the true average deviation of the total series 

 of average deviations obtained from random samplings, and similarly 

 for mean square deviations, are given in Table II. 



The average deviation and the mean square deviation were cal- 

 culated from an approximate average never over a half of the unit 

 of the scale from the actual average and as a rule from an approxi- 

 mate average less than a fourth of the unit of the scale from the 

 actual average. The Q. was calculated on the basis of the same 

 suppositions as the median. 



So far as these samplings go, the average deviation is nearly as 

 accurate as the meap square deviation. ^TheiaTFer~"is on the whole 

 5 per cent, more accurate, with about one chance in eighteen that an 

 infinite number of drawings from these series would raise this su- 

 periority to 15 per cent. There surely can not be enough superiority 

 of the latter to recommend its use in even 10 per cent, of the< opera- 

 tions involved in present researches in psychology, sociology or ^edu- 

 cation. Indeed it is a question whether the mathematical statis- 

 ticians ought not to recognize the average deviation as approximately 

 equal in accuracy and vastly superior in practical serviceableness, 

 and hence as the measure to be recommended to students. 



There is something to be said in favor of a still simpler measure 

 of variability, the percentile. Galton's quartil^ (Q.). for instance 

 (one half the distance between the 25th percentile and the 75th 



