SECT. 27.] Theory of the Average. 491 



the objects of choice 1 ; as when, for instance, we choose digits 

 at random out of a table of logarithms. 



The reader who likes to do so can without much labour 

 work out the result of taking an average of two or three 

 results by proceeding in exactly the same way which we 

 adopted 011 p. 476. The curve of facility with which we 

 have to start in this case has become of course simply a 

 finite straight line. Treating the question as one of simple 

 combinations, we may divide the line into a number of equal 

 parts, by equidistant points ; and then proceed to take these 

 two and two together in every possible way, as we did in the 

 case discussed some pages back. 



If we did so, what we should find would be this. When 

 an average of two is taken, the curve of facility of the 

 average becomes a triangle with the initial straight line for 

 base; so that the ultimate mean or central point becomes 

 the likeliest result even with this commencement of the 

 averaging process. If we were to take averages of three, 

 four, and so on, what we should find would be that the 

 Binomial law begins to display itself here. The familiar bell 

 shape of the exponential curve would be more and more 

 closely approximated to, until we obtained something quite 

 indistinguishable from it. 



27. The conclusion therefore is that when we are 

 dealing with averages involving a considerable number it is 

 not necessary, in general, to presuppose the binomial law of 

 distribution in our original data. The law of arrangement of 

 what we may call the derived curve, viz. that corresponding 

 to the averages, will not be appreciably affected thereby. 

 Accordingly we seem to be justified in bringing to bear all 



i i e as distinguished from acting ter on Kandomness, may result in 

 upon them indirectly. This latter giving a non-uniform di 

 proceeding, as explained in the chap- 



