SECT. 22.] Theory of the Average. 487 



a knowledge of its determining elements; viz. the position 

 of the centre, and the numerical value of the modulus. In 

 the inverse problem, on the other hand, we have three ele 

 ments at least, to determine. For not only must we, (1), as 

 before, determine whereabouts the centre may be assumed to 

 lie; and (2), as before, determine the value of the modulus 

 or degree of dispersion about this centre. This does not 

 complete our knowledge. Since neither of these two ele 

 ments is assigned with certainty, we want what is always 

 required in the Theory of Chances, viz. some estimate of their 

 probable truth. That is, after making the best assignment 

 we can as to the value of these elements, we want also to 

 assign numerically the probable error committed in such 

 assignment. Nothing more than this can be attained in Pro 

 bability, but nothing less than this should be set before us. 



22. (1) As regards the first of these questions, the 

 answer is very simple. Whether the number of measure 

 ments or observations be few or many, we must make the 

 assumption that their average is the point we want; that is, 

 that the average of the few will coincide with the ultimate 

 average. This is the best, in fact the only assumption we 

 can make. We should adopt this plan, of course, in the 

 extreme case of there being only one value before us, by just 

 taking that one; and our confidence increases slowly with 

 the number of values before us. The only difference there 

 fore here between knowledge resting upon such data, and 

 knowledge resting upon complete data, lies not in the result 

 obtained but in the confidence with which we entertain it. 



23. (2) As regards the second question, viz. the deter 

 mination of the modulus or degree of dispersion about the 

 mean, much the same may be said. That is, we adopt the 

 same rule for the determination of the E.M.S. (error of mean 

 square) by which the modulus is assigned, as we should 



