436 
Association of Finger-Prints 
tables and in obtaining the standard deviations, instead of the observed totals 
lis, n's', etc. 
However, as we do not generally know M, m^, m's', etc., we are obliged to use 
the observed marginal totals as the nearest approximation we can get to the 
N . . 
correct values, although iig is not, in general, equal to vig A similar assumption 
is of course always made in the formulae for the probable errors of samples, where 
the sample value is put ultimately for the population value. 
8 h. Correlation Ratio of Restricted Tables. It is obvious that the ordinary 
method of calculating the correlation ratio also requires modification with Tables 
of this type ; for this method is based on the differences between the means of 
the marginal totals and the means of the arrays. Now, in restricted Tables it 
would be impossible for the means of about half the arrays to approximate to the 
means of the marginal totals and it would be fixUacious to base any conclusion on 
the deviations of the observed means from impossible values. 
A nearer approximation would be to take the pseudo 77 from the formula 
o _ 8 {nx (Va - aW] 
~ Nay' 
where is the mean of an array of the independent probability numbers; but 
the denominator of this formula must be modified in such a way that in a case 
of perfect association, tj = unity. The desired result is obtained if we put 2'^ 
instead of o-/, where 
We may write 
V2 _ 
N 
_ iy - Haf S {n^iya -ayiY} 2SS (y - ya) (ya - ayd 
N N ^ N ^ 
_ S (n^ o-/) S {n^ (ya - „yif] 
N ^ N ' 
since the third term vanishes ; hence 
S\n^(ya-ayiy]/N 
7}- = 
S {Tlx <Ta')IN + ;S [n^ (fu - ai/iT]!^ ' 
But " ~ - - ^(ri^^J) 
Na 
= 1 
and 
^ Wx (ya - ayd' 
where rj,, is the crude found by the ordinary method. 
