222 On the Probable Error of a Coefficient of Contuigenci/ 
This leads to 
iv%V = x,x.s(f,) + x.(x,-x, + »){s(g 
+ X,(2x,+ 4,,-|-')s(y) 
..(xii). 
(6) Standard Deviation of cj)'^. Approximate Formulae. 
The result of the preceding section is an exact one since we have neglected no 
terms in arriving at it, but as mentioned before we can usually take M to be very 
large compared with N and make Xi = A!2 = Xa = = 1- With this simplification 
equation (xii) becomes 
+ 
+ 
2V3 
4^ I \ s 
s 
+ \s 
2S 
.(xiii). 
In the great majority of cases it will be impossible to make rigorous use of this 
formula since we have no other knowledge of the whole population than what 
is given by the sample. In particular the h's are usually unknown and we must 
simply make use of the approximations at our disposal, namely the w's of the 
observed sample. 
Again, it will usually happen that while ?), may be fairly large — will be 
small and it will give formulae which are much more convenient for computation 
if Ave write ijj^ = fig ~ K and substitute ijj^ + A^. for Ti^ in equation (xiii), remembering 
that S (A,) = N. 
After some reduction the formula becomes 
N 
+ 
+ 
6>S (^4) + (8 - S (t;) - 12>S (^^Q + (2 - icf>^) c - 22 - 56</,^ - 10</,* 
( t ) + © - \' iW - (g) - - (t) - 
(xiv), " 
where c is the number of classes or categories in the population in question. 
