476 



Theory of the Average. 



[CHAP. xix. 



be situated at the mean, representing the true result, 

 or that given when five of the causes of disturbance just 

 neutralize the other five. Again, 210 will be at what we 

 will call one unit s distance from the mean, or that given by 

 six causes combining against four ; and so on ; until at the 

 extreme distance of five places from the mean we get but 

 one result, since in only one case out of the 1024 will all the 

 causes combine together in the same direction. The set of 

 1024 efforts is therefore a fair representation of the distri 

 bution of an infinite number of such efforts. A graphical 

 representation of the arrangement is given here. 



11. This representing a complete set of single ob 

 servations or efforts, what will be the number and arrange 

 ment in the corresponding set of combined or reduced 

 observations, say of two together? With regard to the 

 number we must bear in mind that this is not a case of the 

 combinations of things which cannot be repeated; for any 

 given error, say the extreme one at F, can obviously be 

 repeated twice running. Such a repetition would be a piece 

 of very bad luck no doubt, but being possible it must have 

 its place in the set. Now the possible number of ways of 

 combining 1024 things two together, where the same thing 

 may be repeated twice running, is 1024 x 1024 or 1048576. 



