IX.—CORRELATION. 17 
The regression of daughter-frond on mother-frond is 0:69 (a 
value which will not be altered by altering the units of measure- 
ment for both mother- and daughter-fronds, as such an alteration 
will affect both standard deviations equally). Hence the re- 
gression equation giving the average actual length (in millimetres) 
of daughter-fronds for mother-fronds of actual length X is 
Y=148+069X. 
We again leave it to the student to work out the second 
regression equation giving the average length of mother-fronds 
for daughter-fronds of length ¥, and to check the whole work 
by a diagram showing the lines of regression and the means of 
arrays for the central portion of the table. 
17. The student should be careful to remember the following 
points in working: — 
(1) To give p" and & their correct signs in finding the true 
mean deviation-product p. 
(2) To express o, and 0, in terms of the class-interval as a 
unit, in the value of »=p/o, o,, for these are the units in terms 
of which p has been calculated. 
(3) To use the proper units for the standard deviations (not 
class-intervals in general) in calculating the coefficients of 
regression : in forming the regression equation in terms of the 
absolute values of the variables, for example, as above, the work 
will be wrong unless means and standard deviations are ex- 
pressed in the same units. 
Further, it must always be remembered that correlation 
coefficients, like all other statistical measures, are subject to 
fuctuations of sampling (¢f. Chap. IIL § 7, 8). If we write 
on cards a series of pairs of strictly independent values of z and 
y and then work out the correlation coefficient for samples of, 
say, 40 or 50 cards taken at random, we are very unlikely ever 
to find r=0 absolutely, but will find a series of positive and 
negative values centring round 0. No great stress can therefore 
be laid on small, or even on moderately large, values of » as 
indicating a true correlation if the numbers of observations be 
small. For instance, if ¥=236, a value of r= +05 may be 
merely a chance result (though a very infrequent one); if 
N=100, r= +03 may similarly be a mere fluctuation of 
sampling, though again an infrequent one. If NN =900, a value 
of 7= #01 might occur as a fluctuation of sampling of the same 
degree of infrequency. The student must therefore be careful in 
interpreting his coefficients. (See Chap. XVII. § 15.) 
Finally, it should be borne in mind that any coefficient, e.g. the 
coefficient of correlation or the coefficient of contingency, gives 
S-