CORRELATION AS APPLIED TO FARM-SURVEY DATA. 5 
is the standard deviation of the first variable; and a y the standard 
deviation of the second. The value of " r " will always be between 
+1 and —1, +1 indicating perfect positive correlation, and — 1 
perfect negative correlation; and to be significant, the value should 
be appreciably greater than its probable error, 
^.±•6745(1-^), (II) 
■y/n 
In the example, r=-\-.277, and its probable error is ±.076, so there 
was a tendency for the heavier calves to return a greater profit, but 
the correlation is by no means perfect. 
PARTIAL CORRELATION. 
A study in which many factors are concerned is not complete 
until it is determined whether or not an apparent correlation, meas- 
ured in the manner explained above, is due to the fact that each 
of the two variables (or factors) under consideration is correlated 
with another or even several other variables. For instance, in the 
data under consideration there is apparently a high correlation be- 
tween the weight of the calves and the value per hundredweight 
received for them (r— +.56), and the question now arises if heavier 
calves really do demand a higher price on the market. This corre- 
lation might be due entirely or in part to the fact that the heavier 
calves in the records obtained were sold at a later date, and that 
the price of cattle in general was higher later in the season; that 
Is, the correlation exhibited here might be due to the fact that both 
weight and price are correlated with date of sale. 
In a problem of this type, where it is necessary to consider simul- 
taneously the relation between three variables and to determine the 
correlation between any two, a coefficient of net or partial correlation 1 
can be determined by the formula — 
r r ab — r ac' r bc /T"m 
Calling the three variables #, 5, and c, the terms of the formula are : 
Tab-c is the coefficient of net correlation between a and h, when the 
effect of c is considered ; r ah is the ordinary coefficient of gross corre- 
lation between a and b and is obtained as explained above ; r ac and r bG 
are the coefficients of gross correlation between a and c and b and <?, 
respectively. Continuing with the example above, let us endeavor to 
determine the degree of correlation between weight and value per 
hundredweight, after taking into account any effect that date of sale 
might have had. In other words, we seek an answer to the question : 
1 Yule, G. U. : " Introduction to the Theory of Statistics," p. 229 et seq. 
