TYPE AND VARIABILITY 431 



We have therefore as a correction on account of the true mean 

 1.6758 0.0292+ = 1.6466. 



This agrees very nearly with the value 1.6465 previously 

 found, but this shorter method is the more accurate, because 

 fewer decimals have been lost. The square root of 1.6466 is 

 1.28+, the standard deviation sought, agreeing perfectly with 

 the former value and derived by a very much shorter method. 



The student will note that the difference in the two methods 

 is essentially this : in the latter we deal only with deviations, 

 while in the former entire values are involved. It is true that 

 deviations are taken from an assumed mean, but the correction 

 is accurately made, and the whole operation can be carried 

 forward not only with smaller numbers but also without the 

 loss of decimals necessarily involved in the more direct but far 

 more laborious and on the whole less exact method first given. 

 The first method is useful for expounding the principles in- 

 volved, but the later is far preferable for actual use, not only on 

 account of its brevity, but on account of its increased accuracy 

 as well. 



Average deviation and standard deviation contrasted. These 

 two expressions for variability rest upon the same arithmetical 

 principle, but the latter has decided mathematical advantages 

 over the former for many purposes and is the one universally 

 used by mathematicians. The only advantage in the average 

 deviation lies in the simpler calculation, as neither squares nor 

 roots are involved. With the shortened method of finding the 

 standard deviation, however, this advantage is slight. 



It makes little difference which is used in practice, provided 

 the same method is always employed. The results obtained differ 

 considerably (0.97+ as compared with 1.28+). The standard 

 deviation is always larger than the average deviation because of 

 the squaring of the several deviations. It thus exaggerates the 

 wider departures from type as compared with the methods 

 employed in finding the average deviation. This being true, 

 results obtained by the two processes can never be compared ; 

 that is, when dealing with values designed to express variability 

 we must always know whether average deviation or standard 

 deviation is mpant. 



