Miscellanea 
where we shall use (^(^ and for the true and approximate values of the mean square con- 
tingency. Thus 
'+*''4K7^) ("«■ 
Now for a sample of constant size /Xp, is constant and therefore representnig small deviations 
by dififerentials 
Square, add for all samples and divide by the number of such samples and we have 
where o-„ is the standard deviation of and r„ „ , , is the correlation of deviations in n„„ and 
rip'q' ; S is & summation for every cell and 2 for e\'ery pair of cells. 
But it is well known* that 
"M M \ M 
'^»pq"'»p'q''npq''l,'q'- J^f Jf X> 
where x is the factor l-{iV - 1)/{M- 1), usually put unity, since # is as a rule large compared 
with iV, and which will be here put unity for the remainder of the work. 
Hence a ^^2_- ^ ^^-^^ - ^ i> [jP^) ' W ^ \,„q,.,^, M'^ 
= 4 -S {'tM^!hi\ _ A j ^ (" ^'pg'Ml (iv). 
This is the standard deviation of the true value of the mean square contingency, and in most 
cases will be of no service, for we do not know the true values of m^q and /ip,. 
If we put these equal to the values obtained from the actual sample under consideration we 
obtain the approximate value of the standard deviation of the true mean square contingency, 
which we may rejjresent by the symbol {^^f) *iid compare with what Blakeman and Pearson 
found, i.e. (o-^ 2)^. Thus our alternatives are 
<^,;-± -67449 (.^^.)^, 
and + . 67449(^0-^^2)^. 
The real thing is ^^±■'S1^\^(T^^. 
Shall we obtain a better insight into the variation of this by taking the approximate values of 
both and o-^^a, or by taking the probable error of <^„2 ? The problem is a subtle one, and, 
perhaps, only to be solved by experiment, not by theory. Of course when we take numerous 
samples and calculate then o-^^a will measure their variability. But this is not what we seek. 
We use as an approximation to (^^^^ r^^^ \^ jg i^^. variability of the true value that we want. 
Are we not right in choosing {^<f,^)^^ as its best value ? In short would not — on the average of a 
great number of samples— ( 0-0^2)^ give us a closer result to (t^^ than ? 
Returning now to equation (iv) and putting in the observed values for ?rtj,q, jup, we have 
('*A4[^!(IS^rM<==fe)F] 
* The values here given are the true values before we approximate by putting m^,qlM=npqlN, etc. 
Biometrika x 73 
