IX. —CORRELATION. 123 
+034 x 645/298 = 0-74, and the regression equation accordingly 
z=0'74y, or 
X=139+0747Y, 
the standard error made in using the equation for estimating X 
from Y being 0, o/T = 72= 6-07. 
This is the equation of greatest practical interest, telling us 
that, as we pass from one district to another, a rise of 1 in the 
ratio of the numbers relieved in their own homes to the numbers 
relieved in the workhouse corresponds on an average to a rise of 
0-74 in the percentage in receipt of relief. The result is such as 
to create a presumption in favour of the view that the giving of 
out-relief tends to increase the numbers relieved, and this can be 
taken as a working hypothesis for further investigation. 
The student should work out the second regression equation, 
and check both by calculating the means of the principal rows 
and columns, and drawing a diagram like figs. 36, 37, and 38. 
Example iii., Table IX.—(Unpublished data ; measurements by 
G. U. Yule.) The two variables are (1) X, the length of a mother- 
frond of duckweed (Lemna minor); (2) Y, the length of the 
daughter-frond. The motherfrond was measured when the 
daughter-frond separated from it, and the daughter-frond when 
its first daughter-frond separated. Measures were taken from 
camera drawings made with the Zeiss-Abbé camera under a low 
power, the actual magnification being 24 : 1. The units of length 
in the tabulated measurements are millimetres on the drawings. 
The arbitrary origin for both X and ¥ was taken at 105 mm. 
The following are the values found for the constants of the single 
distributions :— 
§= -1'058 intervals= — 63 mm. M,= 98'7 mm. on drawing. 
= 4°11 mm. actual. 
oz= 2°'828 intervals= 17'0 mm. on drawing= 0°707 mm. actual. 
#=-0208 , =- 12mm. My=103"8 mm. on drawing. 
= 4°32 mm. actual. 
oy= 3°08¢ , = 185mm. ondrawing= 0771 mm. actual, 
The values of & are entered in every compartment. of the 
table as before, and the frequencies then collected, according to 
the magnitude and sign of &n, in columns 2 and 3 of Table IXa. 
The entries in these two columns are next checked by adding to 
the totals the frequency in the row and column for which &, is 
zero, and seeing that it gives the total number of observations 
(266). The numbers in column 4 are given by deducting the 
entries in column 3 from those in column 2. The totals so 
obtained are multiplied by & (column 1) and the products entered 
RF