ON CERTAIN PROBABLE ERRORS AND CORRELATION 
COEFFICIENTS OF MULTIPLE FREQUENCY DISTRI- 
BUTIONS WITH SKEW REGRESSION. 
By L. ISSERLIS, D.Sc. 
(1) In the systematic investigation of the statistical constants of multiple 
correlation and of their probable errors, it is important to have to hand the probable 
errors and the mutual correlations of the more fundamental constants — the means, 
the standard deviations and the correlation coefficients. For the case in which 
the frequency distribution follows the normal law this need is supplied in the 
memoir by Pearson and Filon entitled " On the Probable Errors of Frequency 
Constants and on the Influence of Random Selection on Variation and Correlation*." 
As regards the more general case in which the regression is skew, the probable 
error of a correlation coefficient was first given by Sheppard {Phil. Trans. Vol. 192, 
A, p. 128). 
The probable error of a mean and the correlation between deviations in the 
value of the mean and that of a standard deviation, or of a correlation coefficient, 
and the correlation between two standard deviations, are given by Pearson 
{Biometrika, Vol. ix. 1913, pp. 1-10). 
For reference we give here the results for the case of normal distributions 
obtained by Pearson and Filon in the memoir referred to above : 
Y.,^ = ii,lV2n : (1), 
2, =(1-A2)/Vn (2), 
"<7i<r2 • 12 (3), 
Vi2 = *'i2/V2 (4), 
J? — ^12 ^^13 ~ ^12^23) + ^13 ^^12 " ^'13^23) 
T> _ ^ _ ^ 12^13 (1 ~ ^"^23 ~ ^'^13 ~ ^'^12 + ^^V2Jn'j.z) , a\ 
^r,,rj3-»-23 2 (1 - r\2) (1 - r\,) ^ ^' 
'']3 - ^12^23) (^24 - ^'23 ''34) + ('"14 - »*34>*13) ('■23 " >"l2>'l3) 
+ (^13 - ^•14^34) (^ 2 4 - ^12^1 4) + ('^U - ^12 ^^24) ('■23 - »"24»34) 
(1 _ r\,) (1 - r^^) 
Jl ^ ( r VIS '14-34/ V' 2 4 '] 2'14; "T V U — ' 12' 2i) V 23 ~ '2 i'aill /rrvj. 
9 CI — r2 Wl — r2 \ y'lJ- 
* Phil, trans. Vol. 191, A (1898), pp. 229-311. 
t Phil. Trans. Vol. 191, A, Equations (xv)-{xviii), (xxxvi), (xxxvii) and (xl). 
