278 On the Prohahle Error of Dr Spearman's Correlation Coefficients 
In any case there is a strong indication that with samples of eight the loss of 
accuracy due to the use of rp instead of r will practically always more than 
counterbalance the gain of time in calculation. Either method is however so 
little to be depended upon for a single sample of very small size, except as the 
merest indication, that very little is lost by the use of rp. If however a number of 
small samples can be averaged so as to obtain a coefficient of some value, the 
product moment method should be used when possible. 
With samples of 30 the 8 more samples required compares fairly with 
Prof. Pearson's 10°/^ more for large samples, but seeing that the particular sample 
of 100 gave too low a value for a^, the value of a-y^ which must be correlated with 
it is likely to be low also and the 8 "/^ may easily be 18 or more. 
In any case it would very seldom pay to have to collect 8 more samples of 
30 even if one could save 8 "/^ of the time on samples of that size. 
In both tables there is a considerable loss from the use of instead of r^, since 
from 13 ° I ^ to 16 7o more samples would be required of the former to give the same 
accuracy as the latter. The gain in calculation is not very appreciable since most 
of the time is spent in ranking the samples. Dr Spearman prefers i2 to p at times 
because less importance attaches to outlying samples, but as the extremes of 
small samples tend to be outliers even in normally correlated material owing 
to the phenomenon to which attention was drawn in Galton's Difference problem*, 
it seems to me that as much weight as possible should be given to them. 
At an early stage in the investigation I hoped to be able to combine r and 
to get a value less subject to error than either. Curiously enough Prof Pearson in 
his editorial in the last number of this Journal gives the equations which I proposed 
to use for the purpose (p. 7 (29)). 
As they are perfectly general I will state them in a slightly more general 
form. 
If X and y be two estimates of any quantity obtained in different ways then a 
quantity z can always be found which will have a lower error than either of them, 
unless X and y are perfectly correlated. 
To Combine tivo Methods of Determination. 
Thus 
' xy ^y 
. X + 
a. 
■y 
(xiii). 
and 
CX^ O-y- (1 — ry;y") 
O'x' + 
(xiv). 
<^x — 2? X,/ (Jx <^ y 
* Biometrika, Vol. i. pp. 385 — 399. In this connection it is of interest to note that the correlation 
surface of ranks is not an elliptical hill as is the normal correlation surface but two comparatively 
steep ridges joined by a saddle, the ridges having a skew section. 
