392 Oti the Partial Correlation Ratio 
Here we are dealing with N pairs of two characters A and B. of these have 
the character x of A. yx '^s the mean of this a-'-array of B's. a^^ is the standard 
deviation of this array, cr,„^ is the weighted standard deviation of the means of the 
arrays and Y is the value of y given by the regression straight line, i.e. 
Y:^y + r^{x-x) (5). 
fx 
This is the " best fitting " straight line (in the Gaussian sense) to the means 
of the arrays, and is the regression line when the regression is linear. 
§ 2. Consider now three correlated characters A, B, C. If N combinations 
of A, B, G are taken, we may denote by Ux the number of these which have the 
character A = x and by n^y the number in which A has the value x and B has 
the value y. Let x, y, z be the mean values of the total population, and let z^tj 
be the mean of z for a given x and y. The frequency of A=x, B = y, G = z 
is 
We define the correlation ratio of z on x and y, which may be denoted by 
x,/Hz, or if no confusion is likely to arise by 5^^, by the equation 
a/ [1 - = - ^^ ^""^^''^^ ' ^^y^'^ . (6). . 
The triple sum in the definition can be written 
SSS {llxyz (Z-Z + Z- ZxyY} 
= SS {7lxy (Z - Z^yY} + 2SS {{Z - Z,y)} X S [ll^y, {Z - Z)] + SSS {« {Z " Zf] 
= SS {n^y {z - z^yf] - 2SS [n^y {z - z^yf] + 
Hence .yH.^ = ^^^^^^^^£p^ (7) 
This is a generalisation of the property of a;?;,^ given by (3). 
Further, the " best fitting " plane to the means z„j is given by 
z — z x — x ,y-v 
<^Z CTx O-y 
where y,, =. ^> ~ ^'^^ (9), 
7/ = ''"!^:^^ 
-l xy 
Let xyRz denote as usual the maximum correlation of z with any linear function 
of X and y, then 
XyRz" = Jsr.x + Js 1\y (11), 
