Page:Proceedings of the Royal Society of London Vol 60.djvu/514

Rh X — a-b Y

be the equation to RR. Then for any one array S{*—(a-bfy/) }3 == —(a-b&Y)}2 = n<P-\-nd?. Hence, extending the meaning of S to summation over the whole surface

S(wd2) = S{*—(o + &y)}2—Sw«r2. But in this expression S(wo2) is independent of a and 6, it is, in fact,, a characteristic of the surface. Therefore, making S(»d2) a minimum is equivalent to making

S {x—(a + by)Y a minimum. That is to say, we may regard our method in another light. We may say that we form a single-valued relation x — a + by

between a pair of associated deviations, such that the sum of thesquares of our errors in estimating any one x from its y by the relation is a minimum. This single-valued relation, which we may call the characteristic relation, is simply the equation to the line of regression RR. There will be two such equations to be formed corresponding to the two lines of regression.

The idea of the method may at once be extended to the case of correlation between several variables x%, x3, &c. Let n be thenumber of observations in an array of *i’s associated with fixed values X2, X3, X4, &c., of the remaining variables, let be the standard deviation of this array, and let d be the difference of itsmean from the value given by a regression equation 2Cj — cli‘iX.2 "1” O13X3 -b 014X4 -b ••«•«>

Then, as before, we shall determine the coefficients Oi2, ax3 &c., so as to make S nd?a minimum. But this is again equivalent tomaking

S{*i—(anas2 -b anx3+<*14*4+••••) }2

a minimum for S{*i—(<*12^2 + OiS*3-b axixx + .. ►•)}*= S(Wi2) + S(wd2).

Hence, we may say that we solve for- a single-valued relation xx — anx2 + 013*3+ 014*4 *b.. • •

between our variables; the relation being such that the sum of thesquares of the errors made in estimating xx from its associated values *2, *3, &c., is the least possible. In the case of normal correla*