Miscellanea 
215 
In applying this difference method to series of material already worked out by the product 
moment method I have found considerable variation in the weight to be given to the diagonal 
cell, but many more series should be examined before suggesting any other value than the 
one-sixth proposed by Professor Pearson. While engaged in this work some modifications 
of method which, I think, will lighten considerably the calculation of r in some cases came to 
my attention. 
It is well known that the formula 
= (<r,2 + rr,,,3-,r.2)/(2,r^,T,) (i), 
where (t^ and a-y are the standard deviations of the two characters and fr„ is the standard 
deviation of their difference, gives r with the same accuracy as the product moment method 
irrespective of the nature of the distributions. 
In symmetrical tables, in which each individual is used once as a first and once as a second 
member of a pair, (Tx = o'y and the above formula may be written 
2a/ =^-W 
From this fornuila the correlation coefficient may be calculated with great ease and rapidity 
as follows : 
First, determine the positive differences between the first and second member of the pairs 
from the table, as suggested by Professor Pearson* and as illustrated in the example given 
below. 
After the totals of the columns have been found they are multiplied by the squares of the 
differences as given at the heads of the several colunnis. Twice the sum of these products 
divided by iV gives (r„^, for the origin is at 0, the plus and minus deviations are equally great 
and the first moment necessarily 0. 
The standard deviation of the character will usually have been obtained for other purposes 
but if it is not wanted no roots need be extracted to obtain by this method. The multi- 
plication of the totals of the several columns by the squares of the differences which they 
represent requires only a little moi-e work than their multiplication by the numbers themselves, 
and if a Bruns\'iga or Comptometer be used all of the arithmetical work can be done in a 
few minutes, even when the tables are rather large. Thus the whole work can be completed 
almost if not quite as quickly as that for Pearson's formula (vii) t and with the same accuracy as 
the product moment method. 
Since all may not have access to Pearson's memoir referred to, I give the following illus- 
tration of the arithmetic of the method. 
llhistration I. Calculation of correlation from symmetrical table. Dr Fernando De 
Helguero's symmetrical table for number of flowers per inflorescence in Cicormm lntyhm% 
serves as an example. Beginning at the head of each column in the correlation table we copy 
down the frequencies to and including the diagonal cell — where the difference between the two 
variates is 0 — under appropriate headings in a series of difference columns. The work can be 
done systematically and rapidly by copying the first number of the first column in the 0 column. 
In beginning the copying of each succeeding column from the correlation table a new column of 
differences, one higher than the last, will be begun and be represented by a single entry for 
that column while the remaining entries will be placed successively one place to the left. The 
whole process will be clear at once from a comparison of the work of the illustration with the 
table of data. The numbers in clarendon type show the order in which the entries from the 
12th column were written down. 
* loc. cit. p. 7. 
t loc. cit. p. 6. 
t Diometrika, Vol. v. p. 188. 
