THEORY OF STATISTICS. 
9. We have now from §§ 7 and 8— 
0=3(2yy . . .. ne ios... nm) 
Fe 25,34 IIL {=} i Prost Le Zz, — terms in Zg to z,) 
= 3(2; . Zyg i ») — ba5 tL CR) SE > 
= S23... . pep. wm S@s....n) 
That is 
S(x nh ) 
b 24 1.34 ....mn 2.3. 
12.340 0. n TI ed 5 . (7) 
But this is the value that would have been obtained by taking a 
regression-equation of the form 
I a co ar 
and determining 4,5. , by the method of least-squares, <.e. 
b1234 . . .. nis the regression of z,, =, ona, a uiollows 
at once from (2) that r,, ___ , is the correlation between 
Tig. ...pnand Zag, and from (4) that we may write 
a n 
I . . (8) 
Eom 2 oh 
an equation identical with the familiar relation &;,=7,,0,/0,, 
with the secondary suffixes 34 . . . . n added throughout. 
To illustrate the meaning of the equation by the simplest case, 
if we had three variables only, z,, ,, and x,, the value of byy5 OF 
7193 could be determined (1) by finding the correlations 7, and 
755 and the corresponding regressions b;, and b,,; (2) working out 
the residuals #, — 65.2, and w, — b,,.2;, for all associated deviations ; 
(3) working out the correlation between the residuals associated 
with the same values of #,, The method would not, however, be 
a practical one, as the arithmetic would be extremely lengthy, 
much more lengthy than the method given below for expressing 
a correlation of order p in terms of correlations of order p — 1. 
- 10. Any standard-deviation of order p may be expressed in terms 
of a standard-deviation of order p — 1 and a correlation of order p — 1. 
For, 
3210s... n) = 2(210s. .. (n-1)+ £1.35... n) 
= Si. ol n=0)® = b1nss. .. (n—1y%n — terms in x, to Za) 
wa S(af as. 2) (n—-1)) = bins... (n-1) (Tras... (n=1)* Tn23. .. (n-1)) 
or, dividing through by the number of observations, 
Oo RL Li is m-1{1 EF b1n2s waits te (R=T)\® On1.23 Sie o=1)) 
=olas ... tll ss we £q) 
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