250 
PROFESSOR K. PEARSf)K ON THE MATHEMATICAL THEORY 
containing all deviations falling within a certain small range of quantity, and the 
means, standard deviations, and correlations are deduced from these grouped observa¬ 
tions. If the means, standard deviations, and correlations he calculated from the 
grouped frequencies, as if these frequencies were actually the frequency of deviations 
coinciding with the midpoints of the small ranges which serve for the basis of the 
grouping, we do not obtain the same values as in the case of the ungrouped observa¬ 
tions. It becomes of some importance to determine what corrective terms ought to 
be applied to make the grouped and ungrouped results accord. This point has been 
considered by Mr. W. F. Sheppard,* who has shown that from the square of the 
standard deviation we ought to subtract of the square of the base element of 
grouping, but that the mean and product of the grouped deviations should be left 
uncorrected. Thus corrected the values of the constants of the distribution as found 
from the ungrouped and grouped deviations will nearljq but not of course absolutely, 
coincide. In })articular while the personal e(piation relation 
F21 == Fo 2 Foi 
will be absolutely satisfied for the ungrouped material, it will generally not be 
satisfied exactly for the grou23ed results. A test, however, of the practical justifica¬ 
tion for groui^ing is that the divergencies between the two methods ought to be of 
the order of tlie jjrobable errors of the results. If this be so, then we may safely 
groujD. The tact that my grouj^ed observations did not satisfy the relation cited 
above, led me to thiid^ it worth while that a comjiarison should at any rate be once 
made between ungrouj^ed and groujDed results on a large series of actual errors of 
observation. At the same time it gave me a means of verifying the accuracy of our 
very long arithmetical reductions by an indej^endent investigation. The ungroiqDed 
observations were dealt with in the case of ifine series involving 500 or 519 observa¬ 
tions each. The lal^our of scpiaring so many individual deviations each read to four 
figures was lessened by using Barlow’s Tables, and the series were added iqj by aid 
of an American Comjfiometer, which for some years jjast we have found of great aid 
in statistical investigations. 
[a.) Bisection of Line Series. 
In Table I. will be found a conqmrison of the ungrou})ed and grou23ed results so far 
as the means and S.D.’s are concerned for our first series. X' has been defined as the 
ratio of the error made in bisection to the leno’th of the line bisected. 
Here mean X/ denotes that Dr. Lee made an average error of about 12/1000 of 
the length of the line in Ijisecting it, and that this error was to the right of the true 
midjioint. Mr. Yule and I made average errors of 4 to 5/1000 of a line in bisecting 
* ‘ London Math. Soc. Proc.,’ vol. 29, pp. 3G8, 375. 
