APPENDIX 



683 



the average most commonly employed, and one of the strongest arguments 

 advanced to justify this method is its universal acceptance. It is worth 

 while, however, to call attention to one of the abuses of the arithmetic 

 mean. For instance, if a very few (say four) measurements have been 

 made of a certain character, the arithmetic mean has often been taken as 

 a good index of their meaning ; but if these few measurements differ 

 widely, to report their arithmetic mean is to furnish a very misleading and 

 untrustworthy piece of information. This has often been done by those 

 who have given no thought to statistical methods. 



There is a sort of commercial point involved in the arithmetic mean 

 which should not be overlooked. For instance, if a real estate dealer sells 

 a hundred lots at various prices, of which the arithmetical average is $800, 

 this assures us that if the seller had sold each of the lots for $800, instead 

 of selling at different prices, he would have realized precisely the same 

 from the sale of the whole number of lots as he has realized from selling 

 at varying rates, even though we have no information as to what any indi- 

 vidual has paid for a lot. 



Weighted arithmetic mean. A slight modification of the above method 

 is often used. To illustrate, the thousand measurements of lengths of ears 

 of corn may be arranged, let us say, in half-inch groups as follows : 



where, for instance, the 6-inch group includes all ears whose lengths are 

 between 5.75 and 6.25. In general, if v v v z , . . ., v r represent the class marks, 

 and/^,/2, "'if r represent the corresponding frequencies, then 



weighted arithmetic mean = /i y i +/" + ' ' ' +/^ . 



/I +/2 + ~'+fr 



Stated in words, this mean is obtained by multiplying each mark of a 

 class by the corresponding frequency, and dividing the sum of the products 

 by the total population. 



This kind of average is used a great deal in our work and is approxi- 

 mately equal to the ordinary arithmetical average if the groups are fairly 

 narrow. Its advantage over the ordinary arithmetic mean lies in the fact 

 that it is more easily computed. In reporting the mean daily temperature, 

 the average length of ears of corn, the average height of a certain class of 

 men, one of the above kinds of averages is meant. We use these averages 

 so much in this work that we speak of them as " the mean," for brevity, 

 so that when the term " mean " is used without a limiting adjective, it is to 

 be understood that an arithmetic mean is meant. 



The geometric mean. The geometric mean of n numbers is found by mul- 

 tiplying the numbers together and extracting the nth root of the product. 



