34 On the Variations in Personal Equation 
and finally the coefficient of correlation is given by 
Rfc = (viii). 
and are the mean and standard deviation of the combined observations — 
1000 in all — of the 20 Groups k, while is the correlation between the 1000 
observations in the 20 Groups 1 and the corresponding 1000 observations in 
the 20 Groups A-'+l, where it must be remembered that owing to the break 
between each series the 50th observation in Series I is correlated with the 
(50 + /i;)th observation in that series, and not with the ^th in Series II, etc. 
It will be seen from the equations (vi) and (viii) that it is possible for R^ 
to have a large value even though the coefficients of correlation of successive 
judgments for the separate series are negligible. For though 2 (pi cr, o-jt^i) may be 
m 
zero for /I'^^j, let us say, where p may perhaps be 3 or 4, it is clear that the co- 
efficients for the combined series, Rjt, will not vanish as h increases unless 
S(A-c?i)(A-+:-f4+i) 
Lk = n-^ * 0. 
In fact if Lk (and therefore R^) does not vanish for values of k for which the 
pt's of the individual series vanish, this is a sign of the existence of a secular 
change running through the series ; the means of the separate series differ 
significantly from the moan of the combined 1000 observations, that is to say they 
differ significantly from each other. Now it is important to obtain a measure of 
the correlation of successive judgments, when freed from this secular term. First 
I define Sk by the relation 
("<)• 
(??i = 20, S indicating summation for the 20 series) ; it is the standard deviation 
m 
of the 1000 observations in the combined Groups k after the secular change has 
been removed. Then R^' is given by 
— 1 (pko-io-k+i) 
'Jl *J k+1 
this is the correlation of successive judgments freed from secular change; before 
correlating the observations we are in fact fitting the series means together, by 
subtracting i^i — A from the observations of the 1st Group of Series I, ]C?2 — A 
from the 2nd Group and so on, and again subtracting idt+i — from the obser- 
vations of the (k + l)th Group of Series I, etc. 
Again it may be desirable to examine the residuals after a sessional change 
has been removed from the observations of each series, in addition to the general 
secular term. Suppose that an observation in the ^th Series can be expressed in 
the form introduced on page 25 
pyt = (j> (Tp) +fp(t) + ,,rt (i) bis, 
