272 PROCEEDINGS OF THE AMERICAN ACADEMY. 



In like manner, the sum of all terms less than the mean is 



. — d — c — b 



na n 



n 

 2 



One half the difference between these is 



25 + 2c + 2d....n 



n 



(Formula 5.) 



But Formula 5 is identical with Formula 4, and is therefore the formula 

 for the average deviation. 



Hence the average deviation may be found by separating the numbers 

 into two groups, — one of which shall contain all quantities greater than 

 the mean, the other all those less than the mean, — and taking half the 

 difference between the means of each group. 



In practice there will usually be several observed values equal to the 

 mean ; these may be distributed to make the two groups of equal size. 

 If the number of observed values is odd, one of these mean values may 

 be neglected. 



In order to compare the average deviation of one group of numbers 

 with that of another which has a diiferent mean, it is necessary to reduce 

 the two to a common measure. This is most simply done by dividing 

 each average deviation by the corresponding mean.* It is proposed to 

 call this ratio the coefficient of variability and to designate it by the 

 symbol 0. V. 



* Tlie justification for this procedure is found in the following considerations: 

 The relative size of the average deviation of two organs depends very largely upon 

 the relative size of these organs. Where the mean dimension is large, we expect 

 a greater average deviation than where it is small. Thus the average deviation 

 of the stature of adult British males from the mean is about 2 inches. An aver- 

 age deviation of 2 inches in the length of the nose, in any race, would clearly 

 indicate a much greater variability in the nose length than in stature. In compar- 

 ing the variability of two such diverse measures as stature and nose length, it is 

 better to compare the ratios of tlie average deviations to the mean dimension. 

 Thus, since the mean stature of adult British males may be taken at 67 inches, 

 variability in stature may be expressed by the ratio ^^ = .02985. This number 

 indicates that the average deviation from the mean stature is about three one- 

 hundredths of the mean stature; which is clearlj' more important than to say that 

 it is 2 inches. Moreover, this method of expression has the advantage that it is 

 independent of the unit in which the dimension is measured, whether feet, milli- 

 meters, grams, degrees, or ergs. — C. B. Davenport. 



