160 Prof. Karl Pearson on Deviations from the 



have n correlated variables and we desire to ascertain whether 

 an outlying observed set is really anomalous. Then we 

 calculate % 2 from (ii.) ; next we take n' = n+i to enter our 

 table, i. e. if we have 7 correlated quantities we should look in 

 the column marked 8. The row ^ 2 and the column n + 1 

 will give the value of P, the probability of a system of 

 deviations as great or greater than the outlier in question. 

 For many practical purposes, the rough interpolation which 

 this table affords will enable us to ascertain the general order 

 of probability or improbability of the observed result, and 

 this is usually what we want. 



If n be very large, we have for the series in (v.) the value 



eix 2 1 e~ix 2 d% *, and accordingly 



Jo 



Again, the series in (vi.) for n very large becomes ei* 2 , 

 and thus again P = l. These results show that if we have 

 only an indefinite number of groups, each of indefinitely 

 small range, it is practically certain that a system of errors 

 as large or larger than that defined by any value of % will 

 appear. 



Thus, if we take a very great number of groups our test 

 becomes illusory. We must confine our attention in calcu- 

 lating P to a finite number of groups, and this is undoubtedly 

 what happens in actual statistics, n will rarely exceed 30, 

 often not be greater than 12. 



(3) Now let us apply the above results to the problem of 

 the fit of an observed to a theoretical frequency distribution. 

 Let there be an (n + l)-fold grouping, and let the observed 

 frequencies of the groups be 



m'x, m f 2 , m' z . . . m' n) m r H+h 

 and the theoretical frequencies supposed known a priori be 

 Wi, m 2 , m 3 . . . m H , m n+ y ; 



then S(m) = S(m')=N— total frequency. 



Further, if e = m' — m give the error, we have 



ei + e % + e z + • • • -he ii+ i = 0. 

 Hence only n of the n + 1 errors are variables; the n+ 1th is 



* Write the series as F, then we easily find dF/dx = l+xF, whence by 

 inte°ration the above result follows. Geometrically, P=l means that if 

 n be indefinitely large, the nth. moment of the tail of the normal curve 

 is equal to the nth moment of the whole curve, however much or 

 however little we cut off as " tail." 



