100 
Oh the Variations in Personal Equation 
correlated variates, the usual expression for the probable error of the mean is 
(1) ± -07449 cr,„ , as compared with (2) + -67449 , when the variates 
V 711 V m 
are not correlated, but owing to the sessional variations to which a large part of 
the correlation is due, the expression (1) being the smaller, is in the present case 
a worse measure than (2), of the probable limits of divergence of the mean of the 
sample from the mean of the series. The graphs of Figures 6, 11 and 15 show 
that there is a tendency for the judgments to vary in waves, to be first on one 
side of the mean for the series, and then to change to the other, but with no 
definite period of variation. It is owing to these large correlated variations which 
cannot be expressed in any simple sessional term, that the coefficients of corre- 
lation, ?pj,„, _ between o-j and have been found to have positive values ranging 
from + -.52 + "11 in Experiment A to + -18 + -15 in D, showing that greater 
variation is associated with higher correlation of successive judgments. 
An analysis has suggested that the coefficients of correlation of the crude values 
of the observations at intervals of k can be expressed in the generalized form 
,S/',?,+/'R," + i^, + - 2 (A - d,) ( A+. - 4+:) 
R,= - -- ^ - 
J\sr' + Ch- + ^ 2 { i). - rf, )4 + + 2 ( A+, - f4+o4 
V ( III III ) ( TO ) 
(Ixvii), 
where 
2 (-D] — di) (A+i ~ dic+i), 2 (A- — dk)" etc. are terms representing the secular change, 
TO m 
Fk and Gk are functions of the sessional change, and 
Ri" and »Sy are the correlation coefficients and standard deviations of the residuals 
left after secular and sessional changes have been removed. 
In two experiments it has been found that Rj is greater than + '80, which shows 
clearly that the estimates have not been distributed randomly in time. 
The coefficients R/' appear to fall off in geometrical progression, and to be 
closely represented hy expressions of the form in which q and r are constant 
for any experiment ; it has been found that the introduction of the quantities F 
and 0 in equation (Ixvii) in addition to the secular terms, is only necessary if there 
is a significant sessional change which repeats itself in series after series. Thus in 
Experiment G, where there was no such change, Rj,- could be expressed by the 
relation 
qr'S/S',+, + - 2 ( A - d,) (A« - 
R,= . ...(Ixviii). 
\^{'^" + i ? - f'"^-- + 5 - '^-4 
A tentative interpretation has been given to the results of this analysis. The 
observations in Experiment A suggested that there was some physiological signi- 
ficance in the distinction between the secular and sessional changes, and this was 
