BIOMETRIC NOTES. 
I. An Abac for determining the Probable Errors of 
Correlation Coefficients. 
By DAVID HERON, M.A. 
The probable errors of correlation coefficients are so often required that the accompanying 
abac which enables them to be determined at once will save considerable labour. From this 
abac the probable errors can be read off correct to at least two decimal places ; greater 
accuracy is seldom required. 
The principle on which the abac is constructed is simple. If we take the well-known 
formula for the probable error of a correlation coefficient 
„ -67449(1-^2) 
^'•= J^—' 
where r is the correlation and n the number of olxservations, and express it in logarithmic 
form, we get 
log (E,) = 1 -8289755 + log ( 1 - r^) - i log n. 
For any constant value of r there is thus a linear relationship between the logarithms 
of Er and ?i. Hence by plotting E^ and n on logarithmic scales and ruling lines for a 
sufficient number of values of r, we can find E,. from r and n without difficulty. 
The use of the abac can best be illustrated by an example. Let the correlation between 
two variables be -55 and the number of observations 160; then to find the probable error 
of r, we must read along the perpendicular line from the number 160 on the scale of frequency 
until it crosses the diagonal line representing a correlation of -55. The position of the 
point of intersection of these two lines is then i-ead, by aid of the vertical lines, on the scale 
of )u-obable errors and the value -037 so obtained. 
After the diagram had been drawn in pencil, the whole of the laborious work of ruling 
in the lines in ink and preparing the diagram for reproduction was undertaken by Miss H. 
G. Jones, and I have to thank her most heartily for her careful work. 
II. On the Probable Error of a Partial Correlation Coefficient. 
By DAVID HERON, M.A. 
In a paper " On the Theory of Correlation for any Number of Variables, treated by a New 
System of Notation,"* G. Udny Yule has suggested that the probable error of a partial correla- 
tion coefficient, giving the correlation between two variables for a constant value of a third, is of 
the same form as the probable error of the direct correlation between any two variables, but his 
proof by reason of its very general nature is by no means easy to follow and it seems desirable 
* Proc. U.S., Vul. 79, pp. 182 et seq. 
