6 KAEL PEAESON 



Now let us determine the probable error of r^, for uncorrelated material, 



^hlc~ 



-1 



J{h + d){a + c){c + d){a + h) ^' 

 where y is zero for uncorrelated material. We have, using diiferentials, 



But for uncorrelated material we may put y = after the variation due to 

 random sampling has been allowed for, i.e. ■y = 0, but not Sy = 0. Hence 



or, summing and dividing by the number of random samples, 



1 



To find (Tr therefore for uncorrelated material, we require to find cr^. 

 Now y = ad — hc; 



.'. Sy = aSd + dSa — bSc — cSb. 

 Square, sum for all random samples and remember that 



, ab 



and oTaO'b^ab ~ ~W' 



we find 



j^j^-'-'y iv/-"^\^ ivy^^'^V N, 



2aH' 2&'c' 2abcd 2abcd 2abcd 2abcd 



= ad{a + d) + be [b + c) ^^ =r= '- 



— ad{a + d) + be {b + e) ^^ — ^^ — '- 



= {ad - bc){a + d) + Nbe ^" ~ ^^ 



= Nbc, for truly uncorrelated material. 



But if ad = be, 



,_ b{a + b + c + d) _ {b + d){a + b) 



N ~ N 



_ c{a + b + e + d) _ {a + e){c + d) 

 and c- -^ ^ 



