Raymond Pearl 
245 
was calculated for a particular group (the conjugants of Series A). This charac- 
teristic equation is the equation to the regression line which one would actually 
observe if one made a correlation table of index and length. In calculating it from 
the usual expression bxy = rxy~, Vj^y was put equal to the observed p, or gross 
index correlation ; ax was the observed standard deviation of index and ay the 
observed standard deviation of length. Then by the usual method a characteristic 
equation in terms of the units of measurement was formed. In the case of the 
conjugants of Series A this characteristic equation took the form 
/= 42-9018 --09854 
in which / denotes the probable mean index of an array of type L in leugth. This 
line was then plotted on decimal paper. The next step was to calculate for the 
same group what may be called the spurious regression of index on length, on the 
assumption that there is no correlation between length and breadth. The equation 
for this spurious regression coefficient we may write as bxy^ = 7\y — , in which 
ay 
rxy^ = Po, the "spurious" coefficient, and a^^ is the standard deviation of the 
"spurious" index distribution. This standard deviation is calculated from the 
usual formula for the standard deviation of an index* 
2i3 = (Vi- + Vs" - 2ri3 V;), 
by putting the organic correlation between length and breadth equal to 0. For 
the case in hand r^y,, = — "loTl. Forming the regression equation and remembering 
that it will pass through a mean of its own given by the equation 
"h 
in which nii and are the observed mean breadth and length respectively and 
is the coefficient of variation for length we get 
/„ = 52-9236 - ISTli. 
This is the equation of the regression line for index on length when there is no 
organic correlation of length with breadth. This was plotted to the same scale as 
the gross regression line, and the two lines are exhibited in Diagram IV. We see 
at once that, on account of the organic correlation between index and length, apart 
from the correlation between length and breadth, the regression line AB is pulled 
around through the shaded area to the position A'B' in the direction of the arrows. 
The amount and direction of this change is always given by the expression 
bxy — bxy„, so that we may say that the shaded area in the diagram is in each case 
the graph of what has taken place owing to /a — differing from zero. 
Diagram V. was prepared in exactly the same way and plotted to the same 
scale but represents the facts for the index-length correlation in the non- conjugants 
of Series A. Comparing the two diagrams we see that the effect of organic 
relationship between index and length is much greater in amount in the non- 
conjugants than in the conjugants. 
* Pearson, loc. cit. 
Biometrika v " 32 
