Mr. F. Y. Edgewortlrs Problems in Probabilities. 383 



l & 



that of the latter. Bat the general rule is not so well ful- 

 filled when we similarly consider the Death-rates * in Regis- 

 tration Counties. The death-rates in different counties for 

 the same year are not, as the theory requires, independent, 

 but tend to increase or decrease all together. Accordingly 

 the Modulus for the sums does not shrink in the regulation 

 fashion. It will be less indeed than n times, but greater than 

 sjn times, the Modulus for the ordinary rows. 



A converse f exception may be illustrated by the following 

 supposition. Suppose that, when there is a high death-rate 

 from a particular disease in one year, there is apt to be a low 

 death-rate in the following year or years. Form a table of 

 the rates of deaths from such a disease for a set of counties 

 (like the agricultural) not differing much in respect of healthi- 

 ness. Consider the fluctuation (not now of the horizontal 

 but) of the vertical fringe of sums in such a table. In virtue 

 of the compensatory action between the years the Modulus 

 for the vertical fringe of sums (or means) will shrink more 

 than the general rule requires. 



Both these kinds of exception occur in Banking, as will be 

 shown elsewhere %. Here it need only be explained that in 

 order to verify the rule, or to establish an exception, there is 

 required a great number of observations. Suppose that C x 2 

 is a certain Modulus-squared, and that C 2 2 is another, double 

 Ci 2 . In order to prove that relation, it may fairly be required 

 that C 2 2 should be determined so accurately that its error 

 should not exceed J, or at most f , Cj 2 . In order that the 

 error of C 2 2 should not exceed f C^ 2 , the Modulus for C 2 2 

 should not exceed -A CA But the Modulus for C 2 2 as deter- 



C 2 2 

 mined by n observations is — ;=§. Hence Vn must be 



greater ^ ~ 2 > 10. And n must be greater than 100. 



1 

 The following figures illustrate this theory. Every term 



in the first series stands for one and the same quantity, a 



certain Modulus-squared. And similarly every term in the 



second series stands for another Modulus-squared. 



(1) 162, 18, 200, 162, 2, 8, 288, 32, 8, 32, 50, 162, 32, 

 162, 32, 128, 1250, 1250, 2 



(2) 32, 200, 8, 128, 338, 8, 32, 578, 648, 32, 32, 200, 8, 

 128, 338, 8, 32 



* See my paper " On Methods of ascertaining Bates," Journal of the 

 Statistical Society, Dec. 1885. 



t See the discussion of Virgilian statistics in the paper just referred 

 to. 



X Journal of the Statistical Society, 1886. 



§ By Laplace's formula for the error of the mean-square-of-error. 



