•Student 
279 
In the case of the samples of eight x may be taken as r without Sheppard's 
corrections and y as Vp, when we have 
o-/ = (-271)= = -073,441, a^(Ty = -078,861, 
0-,/ = (-291)^= -084,681, 
r^y=-885, 
and hence from (xiii) 
z = •804r+ -196rp, 
and = '270, 
i.e. there is no appreciable gain in our case since a,, is '271. It may be that with 
a lower value of the population correlation the gain would be greater, but on the 
other hand if r had been determined for very fine grouping a/ would have been 
■062.5, the contribution of rp to z would have been practically negligible, and the 
gain in accuracy by the use of z less than that found. There is however another 
case where the above formulae might be applied, namely to the values of p obtained 
from the original grouping and those from coarse grouping. 
These are given in Table II from the first and third lines of which it appears 
that cTp^ and o-pg may both be taken as '288. 
In this case cr/ reduces to (1 + > pips) ^ 
and as ?-pjP3 = -903, o-., = '281. 
This is somewhat more encouraging, but the process is rather troublesome and 
could only be applied to cases where there is a proper scale. If however there is a 
proper scale greater accuracy could be obtained by the product moment method 
with very little more trouble (since we have now to make two calculations to 
find p). 
We may therefore conclude that as fixr as this sampling expei-iment may be 
taken as typical : 
(1) Where the unit of grouping is small (say < \ the standard deviation) the 
product moment method should be used if the most is to be made of the time and 
statistics at our disposal, however small the sample. 
(2) Where a coarse grouping has to be used, the mean value of v will fall 
below that calculated from the cooperative paper {Biometriht, xi. p. 328 et seq.) 
and the s.D. will rise. For small samples Sheppard's corrections will approxi- 
mately correct the former but will increase the latter still further. Indeed it is 
possible that for very coarse grouping p might vary less than r. 
(3) For this, or any other, purpose ties should be dealt with by one or other of 
the formulae in equations (x) and (xi) of this paper. 
(4) Where one or both variables can be ranked but not scaled, as frequently 
happens in some kinds of work, or for what Professor Pearson has called " purposes 
of assay," p can be determined with advantage and may be considered the natural 
method to adopt. 
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