302 
Miscellanea 
III. Second Note on the Coefficient of Correlation as determined from 
the Quantitative Measurement of one Variate and the Ranking of a 
Second Variate (see Bioinetrika.y Vol. x, p. 416). 
By KARL PEARSON, F.R.S. 
Ill a previous note I pointed out that if pg ^ be the correlation of the grade of a variate x with 
the quantitative value of a second variate y, then the variate correlation between x and y 
would be 
If we use ranks instead of grades, i.e. o-^^ instead of o-„p we have 
which gives a slight further correcting factor de2:)ending on but as a rule of very little 
importance. 
It follows from the above that on the assumption of a normal distribution the correlation of 
variate and grade for the same character will be 'OTTS. Accordingly when we are testing corre- 
lations of rank and variate we are actually sampling a population of which the correlation of the 
two characters is "OTTS. But we know from a paper in Biometrtl-u, Vol. xi, p. 401, that even for 
p='9 in the sampled population the curve of distribution of r in samples is markedly skew and 
accordingly deviations in defect can occur, which are not possible in excess. 
If we take samples of 25 the mean value of r found from (xxv), p. 336 of the above mentioned 
memoir is 
r=-9763, 
and for samples of 35 : 
r=-9766. 
We should thus anticipate that the mean of small samples of this order would come out slightly 
less than 9773. 
Accordingly with the help of my colleagues Miss Elderton, Miss Karn and Miss Moul, the 
correlations between rank and marks were worked out for 14 Civil Service Examinations and for 
16 examinations in a Technical School most kindly provided by my friend Dr Ritchie Scott. 
These examinations furnished the correlation coefficients given in the following table. 
It will be seen that the English results of the Technical School are somewhat erratic. This is 
an experience similar to that which I had some years ago, when drawing up a report on the 
Matriculation Examination of the University of London, the language marking in that case being 
peculiarly subject to personal ec[uation of an erratic kind. 
Taking the 14 Civil Service Examinations we find for the weighted mean, 
Correlation =- -9666 ± -0086. 
The result to be anticipated, using (xx)'''* to obtain the probable error of the mean is 
Correlation = - -9760 ± -0093. 
The difference is therefore slightly less than once the probable error of the last result and on 
the basis of this alone could not be considered significant. 
If we take the eight Second Year Examinations of the Technical Schools including English 
the result is 
Correlation = - -9001 + -0086, 
while the theoretical result is Correlation = - '9766+ "0123. 
The difference is about r3 times the probable error and is not in itself significant. 
* Actually the sign may be positive or negative according as the variate iu one case rises or falls with 
the rank of the other variate. 
