THE METHOD OF MEANS. 417 



lying among, or in the middle of, the several quantities. 

 Before however inquiring fully into the grounds of this 

 procedure, it is essential to observe that this one arith- 

 metical process is really applied in at least three different 

 cases, for different purposes, and upon different principles, 

 and we must take great care not to confuse one applica- 

 tion of the process with another. A mean result, then, 

 may have any one of the following significations. 



(1) It may give a merely representative number, 

 expressing the general magnitude of a series of quantities, 

 and serving as a convenient mode of comparing them 

 with other series of quantities. Such a number is properly 

 called The fictitious mean or The average result. 



(2) It may give a result approximately free from 

 disturbing quantities, which are known to affect some 

 results in one direction, and other results equally in the 

 opposite direction. We may say that in this case we get 

 a Precise mean result. 



(3) It may give a result more or less free from unknown 

 and uncertain errors ; this we may call the Probable 

 mean result. 



Of these three uses of the mean the first is entirely dif- 

 ferent in nature from the two last, since it does not yield 

 an approximation to any natural quantity, but furnishes 

 us with an arithmetic result comparing the aggregate of 

 certain quantities with their number. The third use. of 

 the mean rests entirely upon the theory of probability, 

 and will be more fully considered in a later part of this 

 chapter. The second use is closely connected, or even 

 identical with, the Method of Reversal already described 

 (p. 410), but it will be convenient to enter somewhat fully 

 on all the three employments of the same arithmetical 

 process. 



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