Volume XI 
MAY, 1916 
No. 3 
\ ^'Ul 8 1916 
ON THE GENERAL THEORY OF MULTIPLE 
CONTINGENCY WITH SPECIAL REFERENCE 
TO PARTIAL CONTINGENCY. 
By KARL PEARSON, F.R.S. 
(1) Let there be I variates or characteristics A, B,C, ... L, each of these variates 
or characteristics being subdivided into categories A-^, A^, ... Aa, B^, B^, ... -B^, 
Cj, C2, ... Cy, ... Li, L.^, ... L/,, where a, ^, y, ... X are arbitrary numbers. Then if 
N be the total population, and na^, na^, ... naa the number of individuals in the 
^-categories ; m^^, ... n^^ those in the jB-categories and so on, we have relations 
(%J = {nj,) = S,y {nj =... = N. 
Further, if there be no relationship whatever between the variates or characteristics, 
we should anticipate that the frequency of the group ... in a sample 
of M would on the average be 
^ % ^ 
' N ' N ' N "' N ' 
Actually we find in the sample M the number ^, and the problem arises 
whether the system represented by m„^^..,^ is so improbable that in the selected 
population M the characteristics A, B, C, ... L cannot be considered independent, 
i.e. M is really not a random sample from the supposed population N. Clearly 
the answer to this problem has already been given. We have to find the value 
of x': 
M "St % 
' N ' N "' N 
and apply the tables for "goodness of fit." Of course in many cases the sampled 
population is not known and accordingly we can only put for , ~jfy-'J^ 
values given by the sample itself, i.e. , ... and test from this substitu- 
tion the degree of divergence from independence. If we take the mean value of 
X^, I.e. (f)^ = x^l^> is termed the mean square contingency, and C2 = V (f>^l{l + 
Biometrika xi 10 
