MISCELLANEA. 
I. On a General Method of determining the successive terms in a 
Skew Regression Line*. 
By KARL PEARSON, F.R.S. 
Let X and y be the two variates. Suppose the r-range divided up into any not necessarily 
equal intervals giving arrays oi y of which the mean of those which centre at .v (lying some- 
where in h^) is and the array total Let ./', y, Ux, o-j, lie the means and standard deviations 
of the total population of the two variates. Assume the form of the regression line of y on x 
to be : 
= «o ^/'o + «i V'l + • • • + «»^>» (i), 
where a,,, ai...a„ are aV>solute constants to be determined and -^g is an orthogonal function of 
- — - , i.e. is suliject to the condition that : 
S {)ij.\lrg\p-s') = 0, s and s' unequal (ii), 
if the summation S be taken for all values of x corresponding to the arbitrary system of arrays. 
Further let us suppose t that : 
„ S {n^ {yr - y) - -y)'} _ s {n^,, {y - y) (x - xy} 
Na^^J =W('^.'^.) 
in the usual product-moment notation. 
Clearly if the yj/^gS can be determined we must have by virtue of (ii) from (i) 
Now if \p-n Vie determined as an integral function of (x — .v)j(rx of the wth degree, the left-hand 
side of (iii) is expressible in terms of the qn's, or the product-moments of the correlation distri- 
bution. Thus a„ will be determined. Let us write : 
\ (iv). 
Then a,j=K„/X,i (iii) bis 
is known, if the ^g's have once been determined, from the product-moments of the system of the 
correlated variates, and the moment coefficients of the a:-variate. 
Square (i), multiply by Wj./i\^and sum for all arrays, we have : 
S (^''[^;~/^'^ = n%..=<^,>'K + a,n, + ...+ (v). 
Here r;„ is the well-known correlation ratio of y-variate on A--variate, and must always lie 
between 0 and 1. The series on the right-hand side of (v) consists of a system of squares, and 
accordingly every additional term we take in our series for the regression line must carry us 
nearer this value of i;''„,,r. Unless X„ tends to zero the a„'s must grow smaller and smaller, or we 
have considerable anticipation of the convergency of the series, but this does not amount of course 
to definite proof. 
* Reproduced from lecture notes. 
t S denotes summation of all arrays, S denotes summation for each individual point. 
