SECT. 9.] Averages. 445 



value. It gives a &quot;middle point&quot; of some kind, but says 

 nothing whatever as to how the original magnitudes were 

 grouped about this point. For instance, whether two mag 

 nitudes had been respectively 25 and 27, or 15 and 37, they 

 would yield the same arithmetical average of 26. 



9. To break off at this stage would clearly be to leave 

 the problem in a very imperfect condition. We therefore 

 naturally seek for some simple test which shall indicate 

 how closely the separate results were grouped about their 

 average, so as to recover some part of the information which 

 had been let slip. 



If any one were approaching this problem entirely anew, 

 that is, if he had no knowledge of the mathematical exi 

 gencies which attend the theory of &quot; Least Squares,&quot; I ap 

 prehend that there is but one way in which he would set 

 about the business. He would say, The average which we 

 have already obtained gave us a rough indication, by as 

 signing an intermediate point amongst the original magni 

 tudes. If we want to supplement this by a rough indica 

 tion as to how near together these magnitudes lie, the 

 best way will be to treat their departures from the mean 

 (what are technically called the &quot;errors&quot;) in precisely the 

 same way, viz. by assigning their average. Suppose there 

 are 13 men whose heights vary by equal differences from 

 5 feet to 6 feet, we should say that their average height 

 was 6G inches, and their average departure from this average 

 was 3 T 3 g- inches. 



Looked at from this point of view we should then pro 

 ceed to try how each of the above-named averages would 

 answer the purpose. Two of them, viz. the arithmetical 

 mean and the median, will answer perfectly; and, as we 

 shall immediately see, are frequently used for the purpose. 

 So too we could, if we pleased, employ the geometrical 



