Student 
277 
I attribute this almost entirely to the grouping which was unfortunately 
rather coarse and which cannot be corrected by Sheppard's corrections in small 
samples. The slight divergency of the population from normal correlation may 
have helped to a very small extent but for the most part the excess in the lower 
values of r which cause the mean to be low and the S.D. to be high is due to 
those samples which have low s.D.'s and I incline to believe that if the grouping 
can be chosen so that the S.D.'s are not less than 3 the actual distribution of r 
will be found to be very close to the calculated for samples drawn from normally 
correlated material. 
In Table IX on the other hand the actual has a higher mean and lower S.D. 
than the calculated, but as the differences are in each case less than twice the 
probable error, I think we may put them down to the error of random sampling 
which is of course large in such a small sample as 100. 
Next we may note that Prof. Pearson's formulae, no doubt because they are 
correct for grades, do not enable us to correct rank correlations for small samples. 
The means of both rp and r^j are too low for samples of eight, and for samples of 
30 probably so. 
As for the s.D.'s of p and ?-p the values found are in the case of samples of eight 
much higher than those calculated from equations (v) and (vi) which are '258 
and '243 respectively. The samples of 30 however give values which agree 
sufficiently well, for the calculated s.D. is in each case "114, well within the p.e. 
Line 5 in Table IX shows that as determined in this investigation Sheppard's 
median division formula gives a mean value of /• well below the population value 
and a very high standard deviation *. While this is not unlikely to be the case 
for small samples the arbitrary division of the central groups makes it impossible 
to say that this is not due to the fact that we have only used an approximation to 
median division in this case. 
The chief point of interest however in Tables VIII and IX lies in column 4 
showing the number of samples which we must have to get the same accuracy by 
the various methods as that given by 100 samples in which r is determined with 
sufficiently fine grouping by the product moment method. 
Column 5 is put in in case there are any who do not accept ray explanation of 
the difference between the calculated and actual distribution of r namely that 
it is due to the coarse grouping. I have not been able to estimate the p.e.'s 
of the figures in column 5 as they are complicated by correlation between the 
numerator and denominator of the fractions from which the figures are calcu- 
lated. They must however be larger than the P.E.'s in column 4. 
\^ 
* The s.D. calculated from the formula (j,.= ^ — r — — l-^i ^ is however rather higher, being -lOl if 
r be taken as -Gfi and "207 if r be taken as -609. Miss Elderton kindly looked up this formula for me, 
but I cannot find that it has been published. [See, however, Biometrika, Vol. ix. p. 23. It is also involved 
in the early paper by Sheppard, Phil. Trans. Vol, 192, A, p. 147 etc. Ed.] 
Biometrika xin 18 
