SELECTION ON THE VARIABILITY AND CORRELATION OF ORGANS. 
27 
Of course to bring them within any reasonable compass we have had to limit the 
values taken. In the first place we have considered only eleven grades of selecti^'e 
stringency given l)y 
5j/o-i = 0, l/IO, 2/10, 3/10, 4/10, 5/10, 6/10, 7/10, 8/10, 9/10, 1, 
the corresponding values of ro 3 in tlie tables are entered as 
TIO) Tto, Pv. 3 , 11^, I1-, ll(j, Ry, Rg, R;„ Ri^q. 
The tables are calculated for both positive and negative, hut and are 
always supposed positive. If and ^’^g ])e both nerjative, then r^g will 1)e the same 
as if they were both positive. If 'y’jo and he of opposite signs, then all we have 
to do is to look out r^g in the table in which ng lias a sign the reverse of its actual 
value, and having found the corresponding value of v.^g, then change its sign to obtain 
the actual coefficient of correlation after selection. This follows, if ^ ^g he the 
negative coefficient, by writing : 
1*23 — 
^’23 “t ^’13^42 sill" 
\/ {1 - siiP } {1 - rj sin2 {1 - giiP ^} {1 _ (_ giiP ’ 
Lastly, it would clearly he very laborious to tabulate 1*03 for a very great series of 
values of r, 2 , I’jg, 'i\^. Accordingly a selection had to be made of these coefficients of 
correlation. They were given the values 0 , ’25, ‘5, ‘75, and 1 . These may he spoken 
of as zero, small, medium, large, and perfect correlations, and the ranges 0 to '25, 
•25 to ‘5, ‘5 to '75, and '75 to 1, as the ranges of little, moderate, considerable, and 
high correlation respectively. There would thus appear to he 15 combinations of values 
for rj2, I’jg; these are given in the key to the tables as (a), [h), (c), {d) ... (to), (n), (p), 
see p. 63. If these 15 values had to be combined with the 10 values (5 positive 
and 5 negative) of rog and the 11 values of we should have 1650 entries in our 
tables. But this number is much reduced by the consideration that the expression 
1 — + 2r23rjg7’^3 has for the real correlation of three characters to lie 
always positive, v^g can also never he greater than unity. Accordingly all values of 
■J'gg, ?’i 3 , which do not satisfy these conditions, have been excluded from the tables ; 
they cannot arise in nature. A few impossilde values of Gg have been included in 
the tables, but these are placed there solely for the purpose of finding by interpolation 
values of igg, which are less than unity. The following pwrc/p hypothetical illustrations 
of formula (Ivi.) and the tables will serve to indicate their use. 
Illustration I .—Suppose the correlation of tiliia and femur with each other to he 
• 8 , and of both with the stature to be ‘ 6 . How would their correlation l)e altered if 
the variation in stature were reduced l)y selection to half its present value ? 
Let Si/o-j = /x^ as before, and suppose Vog = R ; then let /x^, r^g, r^g l)e the values 
of the constants next helom the required values occurring in the tables, and giving 
E 2 
