209 THEORY OF STATISTICS.
rise from one year to the next when the other rises, and a fall
when the other falls. The movement of both variables is, how-
ever, of a much more regular kind than that of mortality,
resembling a series of “waves” superposed on a steady general
trend, and it is the “waves” in the two variables—the short-period
movements, not the slower trends—which are so clearly related.
16. It is not difficult, moreover, to separate the short-period
oscillations, more or less approximately, from the slower movement.
Suppose the marriage-rate for each year replaced by the average
of an odd number of years of which it is the centre, the number
being as near as may be the same as the period of the “waves ”—
e.g. nine years. If these short-period averages were plotted on
the diagram instead of the rates of the individual years, we should
evidently obtain a smoother curve which would clearly exhibit
the trend and be practically free from the conspicuous waves.
The excess or defect of each annual rate above or below the
trend, if plotted separately, would therefore give the * waves?”
apart from the slower changes. The figures for foreign trade
may be treated in the same way as the marriagerate, and we
can accordingly work out the correlation between the waves or
rapid fluctuations, undisturbed by the movements of longer period,
however great they may be. The arithmetic may be carried out
in the form of the following table, and the correlation worked out
in the ordinary way between the figures of columns 4 and 7.
L 2. 3. 4. 5. 6. .
Marriage-rate Nine Differ. Exports+Im- Nine Differ-
. d Years’ ts, £ Years’
Zan lf Uridine Rei snr Ye GEES
1855 16-2 — - 9:36 - —
1856 167 — - 11-14 — -_
1857 16' = 4 1185 — Z
1858 160 » 10-73 — LY
1859 17:0 16'6 +0'5 11-72 12°15 -043
1860 1:10 1 116 +05 13:03 12°94 +009
1861 16:3 167 “ut 13:01 13-52 -051
1862 1610 l163 — 1340 14°17 -077
1863 16:8 169 : 1513 14:31 40°32
1864 172 = 1643 g
1865 175 8 16°37 -
1866 176 - 1772 -
1867 165 : 16°47 -
17. Fig. 43 is drawn from the figures of columns 4 and 7, and
shows very well how closely the oscillations of the marriage-rate
are related to those of trade. For the period 1861-95 the
correlation between the two oscillations (Hooker, ref. 5) is 0-86.
The method may obviously be extended by correlating the devia
