448 
Intra-Class and Inter-Class Correlations 
formation of tables, for the calculation of the means of classes, the standard 
deviation of classes, or the correlations between them, thus reducing the labour 
involved in the determination of intra-class or inter-class correlations to a small 
fraction of that hitherto required. While these short methods when properly 
applied give coefficients agreeing exactly with those deduced from tables, they 
cannot be recommended without two special warnings, (a) In the hands of 
inexperienced calculators abstract formulae are likely to result in blunders 
which would not occur, or at least could be easily detected, if the computations 
were made from tabled data. (b) The tables are, in some of the cases at 
least, desirable for other purposes, e.g., the testing of linearity of regression. 
The underlying principle is very simple. The familiar product moment 
formula for r xy requires merely the first two moments of x and y around the 
means and the product moment 2 (oa'y'). Of course, the rough moments may 
be taken around any origin whatever and referred to the mean by a proper 
formula. In many cases, there are material advantages* in taking 0 as the 
arbitrary origin. 2 (x') } 2(3/') are then merely summations of the values of 
the individual variates ; 2 (#' 2 ), 2 (y'-) are summations of their squares; %{x')jN 
is the mean, m and a x = V2 (%'-) N — m x ~. Then 
_ Z(x'y')/N-m x m y 
Now the calculation of the rough moments and the product moments about 
0 as origin for individual classes is a very easy task. Indeed, with the aid of 
a proper machine they may be rapidly obtained without seriating the measure- 
ments at all f. Since these rough moments for classes are taken about 0, they 
can be summed, weighted and summed, or subtracted in any way we please to 
obtain the rough moments for a correlation surface for the whole population. 
All of this is rapid machine work requiring but a moiety of time needed in 
the formation of tables of thousands of entries. Formula (i) is so well known 
that I shall do nothing further in the way of proof of the validity of the 
formulae suggested here than to show that the data from the first two moments 
and the numbers of individuals in the classes can be thrown into an equivalent 
form. 
A word may be said concerning the illustrations given. All are, I believe, 
of intrinsic biological interest, but they have been chosen primarily for their 
convenient shortness, which facilitates not only the publication of the raw data, 
but the checking through of the arithmetical work by the reader. Thus they 
do not always represent the biologically best series which might have been 
selected. 
* Harris, J. Arthur, " The Arithmetic of the Product Moment of Calculating the Coefficient of 
Correlation,'' American Naturalist, Vol. xliv. pp. 693 — 099, 1910. 
t After a little practice with the machine, one can sum the squares almost as readily as the first 
powers entered on the record sheets, providing the grades are not too high. 
