Egon S. Pearson 
99 
In the Experiments B, G and D it has been found that the correlation between 
judgments at intervals of k, made in the same session, can be expressed approxi- 
mately in the form 
"Rk =qr^ (Ixv), 
while for Experiment A, owing to the large sessional change, the expression was 
Rfc' =P ff'^ (Ixvi). 
The decrease of correlation in geometrical progression expressed by (Ixv) 
follows precisely the law of ancestral heredity, for which the multiple regression 
equations required for prediction have already been worked out*. It is not 
therefore proposed to go further into the problem in the present Paper, nor to 
inquire whether the general multiple regression equations would reduce to as 
simple a form when the correlation is expressed by equation (Ixvi) rather 
than (Ixv). 
XIII. Summary and Conclusions. 
The secular change in personal equation is shown by the variation in the series' 
means, but it is only in Experiment A and perhaps Experiment G, where the general 
trend of the variations is markedly in one direction, that we find that type of 
change which is usually understood when a secular change is referred to. In the 
Bisection Experiment B the linear secular change is very small and its existence 
might well not be recognized, and yet the series' means are subject to fluctuations 
fiir exceeding those of random sampling. For the probable error of the mean of a 
series (or of the obsei'vations in Group 1) is 
+ -67449 X -fi= = + -00416, 
VoO ~ 
but if we take the distribution consisting of the 20 series means, cZi, we find that 
the standard deviation is -037875, giving for the probable error of a mean r/j 
± -02521, 
which is more than six times as large as the probable error wo have calculated by 
considering the variations within a series. It is therefore clear that the 50 
observations in a series are not random samples of the whole " universe " of 
observations, as they should be on the Gaussian hypothesis of normal errors. 
It is again only in Experiment A that there is a fairly consistent sessional 
change from series to series which an observer might easily recognize and possibly 
allow for, and yet if we turn to any of the graphs for the Bisection or tSeconds- 
counting which show the variations of judgment within a series (Figures 11 and 15), 
it will be seen how very often the mean of ten consecutive judgments will give but 
a pool- approximation to the mean oi the series; we cannot take the judgments 
within one series as scattered at random. When dealing with a sample of m 
* The Galton-Peaison Law of Ancestral Heredity; the offspring and the mean of the Ath grand- 
parents have qi* for their correlation. 
7—2 
