Beatrice M. Cave and Karl Pearson 
345 
probable errors as found by Dr Anderson's formulae. Tliese probable errors are 
of course those of the correlations of A™Z and A™F, and will not be the correct 
values for the probable errors of the correlations of A'"a; and A'^y, until A'",i' = A"'A' 
and A™2/ = A™F, i.e. until m is sufficiently great for t to have been eliminated. 
Further their accuracy depends on the vanishing of the means of the differences 
or on the equalities of the sums like 
1 n-l 1 n 
-^ 8{X,\ -^>S(Z,), etc. 
n — \ I ?i — i 2 
which, while true on the average, will only be approximately true in the actual 
instance if n be large. We give in Table II, the Mean Values of Index 
Differences. 
TABLE II. 
Mean Values of Indices and their Differences. 
Railways 
Shipping 
Revenue 
International 
Commerce 
Post and 
Telegraphs 
Stamp 
Duties 
.« c 
> 03 
o! CQ 
CC ^ 
Coal 
Tobacco 
Coffee 
Synthetical 
or 
Arithmetic 
Mean Index 
Quantity 
94-8 
116-3 
97-6 
96-4 
82-3 
103^3 
9.5-9 
99-0 
100^6 
98-6 
96-5 
1st difference 
-3-85 
- 3-93 
-2-22 
-3-96 
-4-44 
-2^19 
-5^37 
-4-67 
-2^56 
-2-30 
-3-55 
2nd 
- -35 
- '50 
- •08 
- -38 
- -27 
•00 
+ -04 
- -35 
- ^12 
- -42 
- -24 
3rd 
+ -16 
+ -28 
+ -20 
- -08 
•00 
+ •la 
- -08 
+ -40 
- -12 
-00 
+ -09 
4th 
- -33 
- -88 
- -13 
-1-17 
•00 
- ^28 
+ -04 
- -79 
- -16 
- 42 
- -40 
5th 
+ -57 
+ 2-26 
+ -26 
+ 2-52 
+ -30 
•00 
- -.52 
+ 1-87 
+ -22 
+ -87 
+ -83 
6th 
- -82 
-2-00 
- -5.5 
-6-41 
+ -18 
- -73 
- -05 
+ 2-00 
- -95 
-1-68 
-1-10 
It will be seen from this table that the means of the differences are far from 
zero even when we have reached a difference for which we may suppose the time 
to have been eliminated. This arises from the smallness of the series dealt with 
and shows us that we ought not to anticipate more than a rough accordance with 
theory, or only an approximate steadiness, for sums like 
1 n — m 
n — m I 
may grow less and less steady as m increases. 
Similar considerations apply to the standard deviations of the differences. 
These will not at first obey Dr Anderson's formulae because they are the values 
for o"^„j^ and cr^,,,^, and when we have taken m sufficiently high for 
°'A'»r ^^"'^ ^A>ny to be theoretically equal to cr^,„,^, and o-^,,,^,. 
