346 Illudratioiis of the Vmv'ate Difference Correlation Method 
and accordingly the correlation should have begun to be steady, then some failure 
to obey Dr Anderson's formulae will arise, because the means of the differences 
are not truly zero and equalities of the tj^pe 
1 - 1 1 m - 1 
j S (X^) = -~ S (X'W,), etc. 
will not be satisfied when 7i is relatively small. 
Dr Anderson gives the value of cr''^,,,-^ in terms of a-^, but we do not of course 
know a"^, which will be very different from a\., and can only practically be found 
from the value of o--^,,,^. itself, after that value has become equal to o-^^,^^, i-e. after 
steadiness has set in. In order therefore to test the formulae we have formed the 
ratio of 
2 . 
This equals 4 — - in Dr Anderson's formulae for a-- .,„^rla- and therefore 
' Qyi A'"A' A'" 'A 
we have a good measure of the approach of A'"*' to A™X, or of the growth of 
steadiness as apart from the correlations. The following Table III gives the 
values of the ratios of the squares of the standard deviations, theoretical values, 
actual values and the mean value for each of the differences of the ten 
individual indices. 
TABLE III. 
Values of a'^^m.^l o'^^m^-if cind their approach to 4< — 
in 
Theoretical 
Series 
Synthetic 
Index 
Kail 
Shipping 
Revenue 
International 
Commerce 
Post 
Stamp 
Duties 
Savings 
Coal 
Tobacco 
Coffee 
Mean of 10 
Index 
Difference 
Standard 
Deviation 
Ratios 
1 
2 
-012 
-012 
-031 
•019 
-038 
-009 
•040 
-010 
•035 
-022 
-036 
•025 
2 
3 
-705 
-708 
1-834 
-763 
1 -720 
-799 
-585 
-350 
2^074 
-352 
•843 
1-003 
3 
3-333 
3-107 
2-816 
3-093 
2-124 
3 032 
1 -959 
1 -660 
2-214 
3-075 
2-213 
3-307 
2-549 
4 
3-5 
3-167 
3-128 
3-174 
2-747 
3-213 
2-597 
2-008 
3-106 
3-379 
3-025 
3-619 
3-000 
5 
3-6 
3-143 
3-449 
3-189 
3-020 
3-104 
3-010 
2-328 
3-275 
3-580 
3-117 
3-701 
3-177 
6 
3-667 
3-149 
3-711 
3-195 
3-164 
2-881 
3-208 
2-499 
3-455 
3-682 
3-101 
3-791 
3-269 
It will be clear that until we reach the ratio of the square standard deviations 
of the third and second differences, there is no general approach to steadiness. 
After m = 3, however, for m = 4, 5 and G, the ratio of the values for the mean of the 
series of individual indices to the theoretical value is '86, "88 and '89, respectively. 
Thus, there is increasing approach to agreement in the observed and theoretical 
values, but the approach is slow, and we believe that there is greater steadiness than 
is really indicated by this test. The source of this apparent unsteadiness lies we 
