MISCELLANEA. 
I. On the Probable Error of a Coefficient of Mean Square 
Contingency. 
By KARL PEARSON, F.R.S. 
Let the sampled population be considei'ed as to two variates and be represented by the total 
M and the cell-frequency nipq for the pih. row and 5th column cell. Further let the vertical 
marginal frequencies be given by m.^ and the horizontal marginal frequencies by nip,, so that 
+ m.^q + . . . + m,,, + . . . = ?/i , J , 
Mipi + »»po + . . . + wip, + ... = inp.,. 
Let the corresponding quantities for the sample be N, Wp,, n.q and Wp.. 
Then we know that the mean square contingency c^^ given by 
the most probable values known to us*, namely, and . Doing this we obtain the usual 
value for the mean square contingency 
Starting from (ii) Blakeman and Pearson have found t the probable error of the mean square 
contingency. The process is admittedly very laborious and although it has now been used fairly 
often, it nnist be confessed that its chief value is to obtain appreciation of the probable errors of 
contingency coefficients in general, rather than in any usefulness in recording significant differences 
between long sei'ies of individual coefficients. 
But it has not been sufficiently recognised that the probable error thus found is that of the 
approximate value of the mean square contingency (ii) and not that of its true value (i). It is 
indeed the probable error of the expression actually used, but it is not the j^robable error of the 
true value as given by (i). The latter is easy to find and deserves consideration. Let us write 
(ii). 
JVm.,q7np.,IM^ = fj.pq, 
then 
* Pearson, Pliilosophical Magazine, Vol. l. 1900, pp. 164 et seq. 
t Biometrika, Vol. v. pp. 191 et seq. 
