418 THE PRINCIPLES OF SCIENCE. 



The significations of the terms Mean 

 and 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 desirable 

 to ascertain carefully what significations we ought to 

 attach to them. The English word mean is exactly equi- 

 valent to medium, being derived perhaps, through the 

 French moyen, from the latin medius, which again is un- 

 doubtedly kindred with the Greek pca-os. 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 a mean in a mathematical point 

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

 kinds of means. The old arithmeticians recognised at 

 least ten kinds, which are stated by Boethius, and even 

 an eleventh was added by Jordanus d . 



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 any series of quantities divided by their 

 number, and may be represented by the formula ^ (a + 1). 

 But there is also the geometric mean, which is the square 

 root of the product, *^/cTx~l>, or that quantity the logar- 

 ithm of which is the arithmetic mean of the logarithms 

 of the quantities. There is also the harmonic mean, 

 which is the reciprocal of the arithmetic mean of the 

 reciprocals of the quantities. Thus if a and I be the 



d De Morgan, Supplement to the ' Penny Cyclopaedia/ art. Old Appel- 

 lations of Numbers. 



