190 Probable Errors in Multiple Skew Regression 
and if we remember that for normal distributions 
"r^y = (1 - '•'--')/VF, a,^^ = (1 - r^,)/ VF, 
this result agrees with Pearson and Filon's value quoted above as equation (7). 
(4) To find the probable error of a standard deviation. 
.(40). 
Hence 
Px^ - P^x^ 
ip'x'N 
4iV ' 
This result is well known, and for normal distributions, i.e. when /Sj = 3, becomes 
^"^^•V^' ^^^^^^^^ ^^^^ 
To find the correlation between a standard deviation o-^, and a correlation 
coefficient fj,,, we multiply (40) by the equation 
dvy^ _ d'py^ _ dpjfi _ dp^2 
'Tyz Pvz '^Pv^ "^Pz^ 
and sum for all samples and divide by their number in the usual way, obtaining 
= {Px^vz - Px'Pyz)lpx'-Pvz - I ipx-u"' " Px-P,A/px"-py- - * iPx^z^ - Px^Pz")IVx^Pz^ 
<Ix^vz 9x'y' + (Ix'z' 
•(42), 
' yz " 
a result which as before can be approximated to in the case of linear regression, and 
which for normal distributions becomes* 
^ " 2 W T" -J' xy "T ~'i xz) 
"(^x^fiiz a n — 
^ ^ ' UZ 
V2N ' VN ' CTx^yz 
2'^xy'^xz xy ~^ ^^xz) yz 
V2 . (1 - r\,) 
agreeing with equation (5). 
.(43), 
* For the case z-x, i.e. R„ „ , cf. Biometrika, Vol. ix. p. 8. 
