Miscellanea 
195 
appreciation of "negligible terms" when the contingency is not very small. The whole formula 
may be written : 
If we endeavour to get some idea of the general magnitude of this expression, by evaluating 
it for a normal correlation surface we find, for infinitesimal groupings, that the last two terms 
become infinite if r does not lie between — 1/\''3 to +1/n/3 and the first term becomes infinite 
when r does not lie between - 1/2 to +1/2. In fact in such cases we seem to reach indefinitely 
large probable errors. We doubt, however, the justice of this view and believe it merely signifies 
that with indefinitely fine grouping beyond a certain range of values of the assumption that 
the errors of random sampling may be treated as differentials is incorrect, and thus our process 
of reaching (xxxvi) is no longer legitimate when applied to such normal distributions. The 
whole matter, however, deserves careful investigation from the theoretical standpoint. Even 
from the practical side the error in any constituent due to random sampling must be at least 
unity, and accordingly if the theoretical value of the constituent be only a few units or a 
fraction even of a unit Biipjupg is not necessarily a small quantity. We ought accordingly to 
provide in practice for a contingency grouping which leaves no constituent to consist of but 
a few units, if we wish to justify our fundamental assumption in determining the probable error. 
In actual practice with fairly coarse grouping and not replacing the summations by inte- 
grations, the value for o-^ will always be finite, for we make no summation where either 
Jip or are zero, i.e. we do not as in using the normal surface extend our distribution all over 
space. For the cases in which we have tried it (xxxvi) then gives reasonable results, and we will 
now indicate how the calculations can be made fairly briefly. 
In the accompanying table we have the contingency between Intelligence and Handwriting 
in schoolgirls. The columns correspond to grades of intelligence, the rows to grades of hand- 
writing. The first number of each constituent group is the actual frequency in the total of 
1801 girls with the characteristics of that group. The reciprocal of 1801 is 555,247/10^. This 
is put on the calculator and the column of row totals multiplied by it, with the result 
put under each row total ; each one of these is now jjut on the machine in succession and 
multiplied by the series of column totals w,, ; we thus obtain UpnJN, which is registered as 
the second number in each constituent. The difference of the first and second number of each 
constituent with due regard to sign is N^pq the constituent contingency. This is registered as 
the third number in the constituent. The square of this — taken from Barlow's tables — and 
divided by the second number is ]V({>\q, or iV times the mean square contingency contribution 
of each constituent. This is the fourth number registered in the constituents. The sum 
of these fourth numbers for each row gives J^(f)q^, and for each column jV(f)ji^. These are 
registered in the column and row beyond " totals." Adding up this column or row, we have 
Sg{iV(t>,j^) = Sp{iV(f)/) = n2-62 = iV(j)'^, hence <^2 = .o9580 and ^7=^(^7(1+^) = -2957. 
This is the coefficient of mean squared contingency between handwriting and intelligence, and 
is our standard method of finding C. So far all the work is usual and necessary. Now square 
from Barlow the column of JV(f)g^ and the row of i\^<^,/ ; we obtain the column and row of iV'c^,/ 
and N'^<py^. Divide these by their respective column and row total frequencies and we have the 
numbers given underneath N'^<j)g^ and N'^<pjj% or N'^cf),*/ng and N'^(j>p*/njj respectively. Adding up 
these column numbers and row numbers we find on division by 
IV 
values registered on the table. These are two of the sums needed for (xxxvi). If the distri- 
bution were normal and the group ranges infinitesimal these should be equal. They clearly 
differ widely. Next divide iV^^- by 1801, i.e. multiply these quantities by the reciprocal, placed 
25—2 
