372 Prof. Karl Pearson on lestinp the 



where we have 



-M«'(H} +2S -{*?} 



= s 1 (^-)+i( ( ^ 1 +^ 2 +...+^_ 1 ) 2 . 



y n s ' 7iq 

 Now S(n # ) = N, S(n # ) ==N ; 



and therefore 



#l+#2 + • • • +*«-l + «j = 



and (a? 1 +o? 2 + ... -r^-i) 2 = #*. 



Thus finally we have for % 2 



X 2 



- ■£> 



where the summation is for all values of x s from 1 to q. 



There are, however, only </ — 1 independent variables, and 

 accordingly the chance that a sample of N will occur with a 

 value of the deviation-complex of magnitude % 2 or greater is 



P = 



x <,-* e -ix*d x 



JO 



This is the value of P calculated in the tables of the 

 " goodness of fit/' and accordingly all turns on the deter- 

 mination of 



*-s{k==*} 



2. On the Probable Error of the Determination of 

 Goodness of Fit. 



If 



% 2 = Si g| fa-<) 2 1 



then in a recent memoir * the probable error of % 2 has been 

 found, or rather cr %2 . 



* Young and Pearson : Biametrika, vol. xi. part ii. 



