438 
Miscellanea 
TABLE I. 
Constants of z=f{xi, x^. 
Mean of z 
Standard Deviation of « 
Z=.Vi-.V2 
Z = Xi. .V2 
Z = .Ti. Xo . .I':! 
mi + iiu 
— 111-2 
nil >;i2 + ''i20"i <^2 
'/Hl)no?)l., [1 +?'l2 2'l ''2 + ''l3fl '3 
+ '"23 "2 ''3] 
— (l+ V-'"l2'fl i'2) 
7/2/2 
v^((Ti2 + o-g^ + 2»-i2 0-1 0-2) 
\'{<ri^+ (T2^ + <T3- + 2;-i2 (Ti 0-2 + 2ri3 0-1 0-3 + 2»-23 (TaCTs) 
\'(o-i^ + 0-2^ - 2ri2 cTi (T2) 
»)l »l2 [I'l"^ + v.? + 2ri2 Vi + Vi^l'2^{l + ?-i2)]^ * 
or approximately 
»ii WI3 [ i',- + yo^ + 2ri2 Vi Vi]^ 
nil )»2?»3 [v{^ + V2' +v-^^ + 2ri2ViV2 + 27-i3Vi V3 
+ 2;'23 7^2 v^]^ approximately 
Vli 
~y[^^{fi'^ + ^>'/-^ri2t'iV2) 
III. The Correlation between a Variable and the Deviation of a 
Dependent Variable from its Probable Value. 
By J. ARTHUR HARRIS, Ph.D., Carnegie Institute, Station for Experimental Evolution, 
Long Island, U.S., and Biometric Laboratory, University College, London. 
The degree of interdependence of two quantitatively measurable variables is conveniently 
expressed by the correlation coefficient, r, or the correlation ratio, ij. These constants will both 
be used with increasing frequency to describe many relationships in evolutionary, morphological 
and physiological work. 
There appear to be, however, certain problems in which the correlation coefficient, while 
indispensable, does not yield all the information which we require. 
Take such a case as the relationship between the number of eggs per clutch and the number 
of birds hatching. The limits of the number of birds hatching per nest are fixed by the 
number of eggs laid, and it may be designated as the dependent variable. It is quite clear 
that if a fixed proportion of the eggs hatched the correlation between the number of eggs and 
the number of birds would be perfect. 
The biologist can suggest many analogous cases ; for instance number of flowers per 
inflorescence and number of fruits developing, number of ovules formed and number of seeds 
maturing. 
[* This formula, given by me to Mr J. F. Tocher, depends on the assumption of normal correlation, see 
Biometrika, Vol. iv. p. 320. The approximate value depends on neglecting higher powers of the coefficients 
of variation. The formula for the mean of the double product (Tocher, loc. cit.) is exact. The formula 
for the mean of the triple product is not exact, any more than the formula for the s.d. of the triple 
product (see Tocher, loc. cit. p. 321). The formulae for the mean and s.d. of an index are only true to the 
lowest powers in Vi and vo, and must not be applied if Vi and sire large. The formulae for« = a;i±.i;2) 
the sums or differences of any number of variables, are of course exact for both mean and s.d. Ed.] 
