48 
On the Variations in Personal Equation 
Figures 8 and 4 together show very cleai'ly the marked sessional change ; while 
the former shows that except in a few series, notably Series I, IV and X, the 
regression is remarkably constant in its value, the latter indicates that the sessional 
change is better represented by a parabola than by a straight line. 
The sessional change can also be represented numerically with the help of the 
correlation ratio of yt upon t. If we are dealing with the observations freed from 
the secular change, that is after the removal of the means pd from the 63 observa- 
tions of the jjth series we have r/,,^ given by 
V,:- = ^"g3,v'- ' = 1260 :,r/^'^' -^'^^ ' 
or S' is the standard deviation of the whole 1260 observations after the removal 
of the secular term*. Then the ratio of the mean square distance of every observa- 
tion fiom the regression line or line of means J/t, to the standard deviation of the 
observations is 
yl ^'■^ 
1260 
= ^/l - w (xxxi), 
where S indicates summation for the 20 series. 
Ill 
This is a. measure of the closeness of fit of the observations in a series to the 
mean sessional change as represented by the values Ijt ; the larger rjy^ and therefore 
the smaller VI — 77,^^" is, the more nearly does a sessional change of the same form 
recur in series after series. A comparison of the values of Vl — rjy^^ for the different 
experiments will show the relative significance of their mean sessional changes. 
In the present case the value of 7;,,^ is found to be "579 + '013, while 
\^T^Vvf = '815. 
It would be an interesting problem to obtain the correlation of the successive 
residuals left after the ordinates of the " best " fitting parabola for each series had 
been subtracted from the observations of that series ; but although this has not 
been done, a fair idea of the degree to which the correlation of the successive 
judgments in the individual series is due to the sessional change can be obtained 
by removing the " best " fitting straight lines from each series. The values calcu- 
lated for the pb's have been referred to above, and using these and the equations 
(xxii) — (xxiv) of pp. 41 — 43, the values of cr/ and p/, or the standard deviations 
and correlations of successive observations freed from the linear sessional changes, 
have been calculated and are given in the 4th and 6th columns of Table IV. The 
p/'s are all less than the corresponding p/s, except in Series X where they are 
* Actually it is only the values of the Group Standard Deviations Sj', S2' ... Sn' which have been 
calculated ; they are not all equal (as shown in Table V) owing to the sessional change in standard 
deviation, but an approximation to S' sufficiently accurate for the purpose will be given by taking 
