188 
Miscellanea 
the oell contingency and after playing about with such cell contingencies for a time succeeded in 
finding a function of them which for indefinitely fine grouping for a bi-variate normal frequency 
distribution gave the correlation r as : 
where <^' = i/S 
M 
•(«)• 
M ■ 
I see no reason for confusing this 0^ as a measure of correlation with the x'' which is a measure 
of variability in the samples of constant size drawn from an indefinitely large poi)ulation. It was 
different in its origin, as far as I am concerned, and different in its use. It is only when we come 
to consider the probable error of </)-' that we have to distinguish between (n) the actual marginal 
totals of the sample and {h) the probable constitution of the marginal totals as deduced from an 
indefinitely large sampled population. 
There are, as those who have read Biometrika* will I'ecognise, considerable difficulties about 
determining the probable error of (^", where 
l+rf)'' = *S' 
m 
m 
and the determination of the mean (f,- and of the standard deviation of involves very trouble- 
some analysis. 
So lal)orions is the arithmetic involved that for t)rdinary statistical use it became doubtful 
whether it would not be better to define as the mean squared contingency measured not from 
the marginal totals of the sample, but from the " probable constitution " of the marginal totals 
of the sample as deduced from the sampled population. In this case if 
M , _M , _M 
M 
tr 111 si.m, . 
•(/3) 
with this change of definition the probable error and mean of are more easily obtainable, and 
in this case for the first time, i/i^^ can be looked upon as equivalent to a x^. 
The form (n) from my standpoint cannot be treated as a x^, because it is not the deviation- 
measure of a given sample from the sampled population. Nor again is (/3) the deviation-measure 
of the sample from the sampled population, iniless we assume that population to have zero 
contingency, i.e. m'gsf = m'g,m',g>/M. 
But x'^ may in the form (jS) be treated as a deviation-measure of the actual samjile from an 
artificial sam^jled population, which differs from the actual 2)02Julation in having no correlation 
or contingency, but having the same marginal distributions of the two characters. 
The moment, however, we assume form (/3) for our contingency we are giving, what we clearly 
must give, absoKite freedom to the marginal totals of-our samples. The sole limit on our sample 
is its total size 31. But when we come to actually calculating cj)'^ for the individual sample, or the 
mean value or the standard deviation (i.e. probable error) of cfy^ for a series of samples, we have 
only one course open to us, if we do not know the constants of the sampled population, we must 
insert the marginal totals of the individiial sample of which we have cognizance in place of the 
* Vol. V. p. 191, Vol. X. p. 570, Vol. xi. p. 570, and Vol. xii. p. 259. 
