Godfrey H. Thomson - 357 
Then for two columns a and h, the true columnar correlation which we desire 
to know is 
/?' _ Sip'xnp'xb) 
'''"*-V|^(p'U^(p'^.)} 
by the Bravais-Pearson product-moment fonnula, S indicating siuumation over the 
various values of .r, i.e. summation up the column. 
This can be written 
V}*^' ipxaPxa) — (e<-«f.ia) - '-S {p'x.a6x„)\ \/ {^-' ( PxhPxb) - (e,bexh) - ~S {p'.^i,€,rb)' 
(2). 
In this expression, the three quantities of the form S (pp) are known. The three 
quantities of the form S(ee) are not known, but an attempt can be made to 
estimate their probable values from the known standaid deviations of the correla- 
tion coefficients. The four quantities of the form S(p'e) are treated by Dr Hart 
and Professor Spearman, in their paper, as negligible, on the ground that p' will 
not in general be correlated with e. It is the object of the next section of this 
paper to examine the nature of the correlation of these two quantities. 
(3) The Relationship betiveen the GoiTelation Coefficients and tJieir Sampling Errors, 
in the Case of Correlation between a N amber of Varlates taken in Pairs. 
Consider the formula for the standard deviation of a correlation coefficient, viz. 
1 - r' 
"'■"wr 
where N is the number in the sample. It follows from this that the larger 
correlation coefficients will probably have the smaller sampling errors e, disregarding 
the sign of e for the moment. 
But these signs of the quantities e are not likely to be indiscriminately positive 
and negative. On the contrary, they will have a tendency to be either all positive 
or all negative, if, as is the case in most of the columns of coefficients considered 
by Professor Spearman, the correlations in the square table are mainly positive. 
The errors in the correlation of a variate with a variate a are themselves 
correlated with the errors in the correlation of the variate a with another variate 
x^, according to the formula 
That is, the correlation of the sampling errors of with the sampling eri'ors of 
''"x.^a depends chiefly upon r^.^ ^,. To illustrate, let us take three correlations from 
an experiment in psychology, carried out by Mr Wyattf. 
* Karl Pearson and L. N. G. Filon, " On the Probable Errors of Frequency Constants," Phil. Trans, 
of the Rnijal Soc. 1898, cxci. A. p. 259. 
t Stanley Wyatt, "The Quantitative Investigation of Higher Mental Processes," Brit. Journ. 
Psychol. 1913, vi. p. 131. 
