[ 157 ] 



X. On the Criterion that a given System of Deviations 

 from the Probable in the Case of a Correlated System of 



Variables is such that it can be reasonably supposed to have 



arisen from Random Sampling. By Karl Pearson, F.R.S., 



University College, London*. 

 r J 1 HE object of this paper is to investigate a criterion of the 

 probability on any theory of an observed system of errors, 

 and to apply it to the determination of goodness of jit in the 

 case of frequency carves. 



(1) Preliminary Proposition. Let ae u x 2 . . . x n be a system 

 of deviations from the means of n variables with standard 

 deviations er 1; a 2 . . . o~ n and with correlations r 12 , r 13 , r 2Z . . . 



**» 1, U' 



Then the frequency surface is given by 

 where R is the determinant 



(i.) 



1 



?'l2 



rn • 



. • r hl 



r 2\ 



1 



r-23 . 



• • r 2n 



r%i 



»'82 



1 . 



. . r Sn 



r n \ 



»'n2 



TvZ 



. 1 



and Rpp, R^g the minors obtained by striking out the pth row 

 and pth column, and the pth row and ^th column. Si is the 

 sum for every value of p, and S 2 for every pair of values of p 

 and q. 

 Now let 



Then : % 2 = constant, is the equation to a generalized " ellip- 

 soid," all over the surface of which the frequency of the 

 system of errors or deviations a? l5 a* 2 . . . x n is constant. The 

 values which % must be given to cover the whole of space 

 are from to go . Now suppose the " ellipsoid " referred to 

 its principal axes, and then by squeezing reduced to a sphere, 

 X,, X 2 , ... X being now the coordinates ; then the chances 

 of a system of errors with as great or greater frequency than 



* Communicated by the Author. 



