IX.—CORRELATION.
RR and CC given by the regression equations (a) and (3). In
doing this it is as well to determine a point at each end of both
lines, and then to check the work by seeing that they meet in the
mean of the whole distribution. Thus RR is determined from (a)
by the points ¥Y=0, X=19-13 and Y=6, X=1391: CC is
determined from (%) by the points X=12, ¥=564 and X=21,
¥ =114. Marking in these points, and drawing the lines, they
will be found to meet in the mean, X=15'94, Y=367. The
diagram gives a very clear idea of the distribution ; clearly the
regression is as nearly linear as may be with so very scattered a
distribution, and there are no very exceptional observations. The
most exceptional districts are Brixworth and St Neots with rather
low earnings but very low pauperism, and Glendale and Wigton
with the highest earnings but a pauperism well above the lowest—
over 2 per cent.
16. When a classified correlation-table is to be dealt with, the
procedure is of precisely the same kind as was used in the calcula-
tion of a standard deviation, the same artifices being used to shorten
the work. That is to say, (1) the product-sum is calculated in the
first instance with respect to an arbitrary origin, and is afterwards
reduced to the value it would have with respect to the mean; (2
the arbitrary origin is taken at the centre of a class-interval ; (3
the class-interval is treated as the unit of measurement throughout
the arithmetic.
Let deviations from the arbitrary origin be denoted by £7, and
let £7 be the co-ordinates of the mean. Then
¢=z +E g=y+7
- En=xy+ Ey +e + Gi.
Therefore, summing, since the second and third sums on the
right vanish, being the sums of deviations from the mean,
(én) = Z(xy) + NE7,
or bringing 2(zy) to the left,
(wy) = 3(&) - Ne3.
That is, in terms of mean-products, using »’ to denote the mean-
product for the arbitrary origin,
r=p -5&.
In any case where the origin from which deviations have been
measured is not the mean, this correction must be used. It will
sometimes give a sensible correction even for work in the form of
18,
