228 0)1 the Probahle Error of a Coefficient of Contlngencu 
as is obtained by substitution of A, = in (viii), or, if X\ = 1j 
In the same way Ave derive from equation (xiii) 
-V = |i {x4 + ' + - 1) - Xi - 1) j (^vii), 
where H is the harmonic mean of the mean cell contents, or for the usual particular 
case when M is very large as compared with N 
''-^.{^"-'^^^^(-1)} 
an expression which is very easily calculated especially as ^ will usually be small 
coinpared with c and hence a good rough approximation for a fairly large table 
will be f'ot from 
9 
2 
.(xix). 
Thus if we take twice the standard deviation as a limit to the probability of a 
deviation being that of a random sample, we have as a rough upper limit to the 
value, which j^^gy expected to take in any sample, 
(10) Numerical Illustration. 
In the example of the Contingency-table for Handwriting and Intelligence 
in Girls 
^2 = ^(c- 1)= -01943, 
and when calculated from the more exact formula (xviii) 
= -004879, 
the approximation given by (xix) being -0046. 
Hence in accordance with our assertion above, we should regard any observed 
value of ^2 ^vhicii exceeds -01943 + 2 x -00488, i.e. -02919 or, say, -03, as being 
incompatible with zero contingency. The observed value of </)2 = -0958. 
The corresponding mean value of C — the coefficient of contingency — is 
1-01943=^^^'^'^' 
and the upper limit for C according to our assertion is -1684 or, say, -17. The 
observed value of C is -2957. Clearly there is definite association between 
intelligence and handwriting. 
