360 THE PRINCIPLES OF SCIE1 CE. [CHAP. 



(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 Eeversal already described, 

 but it will be desirable to enter somewhat fully into all the 

 three employments of the same arithmetical process. 



The Mean and the Average. 



Much confusion exists in the popular, or even the 

 scientific employment of the terms mean and average, and 

 they are commonly taken as synonymous. It is necessary 

 to ascertain carefully what significations we ought to 

 attach to them. The English word mean is equivalent to 

 medium, being derived, perhaps through the French moyen, 

 from the Latin medius, which again is undoubtedly kindred 

 with the Greek /ieo-o?. Etymologists believe, too, that this 

 Greek word is connected with the preposition //.era, the 

 German mitte, and the true English mid or middle ; so that 

 after all the mean is a technical term identical in its root 

 with the more popular equivalent middle. 



If we inquire what is the mean in a mathematical point 

 of view, the true answer is that there are several or many 

 kinds of means. The old arithmeticians recognised ten 

 kinds, which are stated by Boethius, and an eleventh was 

 added by Jordanus. 1 



The arithmetic mean is the one by far the most 

 commonly denoted by the term, and that which we may 

 understand it to signify in the absence of any qualification. 

 It is the sum of a series of quantities divided by their 

 number, and may be represented by the formula (a + &). 



1 De Morgan, Supplement to the Penny Cyclopcedia, art. Old 

 Appellations of Numbers. 



