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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 measurement for both mother- and daughter-fronds, as such an alteration will affect both standard deviations equally). Hence the regression 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 expressed 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-