Page:Proceedings of the Royal Society of London Vol 60.djvu/522

Rh that is, or (?*ia“ 7*12^29) > 0.
 * •»+ru*—2r12r13r23 > fi»*—rMVw8,

But (ria—^ 23) is the numerator of the net coefficient of correlation between xx and x3. Hence the standard error in the second case will be always less than in the first, so long as px3 is not zero. The condition is somewhat interesting.

To take an arithmetical example, suppose one had in some actual case ri2 = + 0*8 7*23— + 0 * 5 7*13 + 0 4.

One might very naturally imagine that the introduction of the third variable with a fairly high correlation coefficient (0*4) would considerably lessen the standard deviation of the avarray; but this is not so, for 0 * 4 — (0 * 5 X 0 * 8 ) P n ~ v ^ 0 ’7 5 X 0 * 3 6 °* so the third variable would be of no assistance. for Regrsion, #c., t» the case o f Skew 487

This case is, perhaps, of sufficient practical importance to warrant our developing the results at length as in the last.

If xx, Xt, %s, *4, be the associated deviations of the four variables from their respective means, the characteristic equation will be of the form

= 612*2+ 613*3 + buXi........................ (14). The normal equations for the 6’s are, in our previous notation, 7*1201 = ^ 1 2 0 * + ^ 1 8 7 * 2 3 0 3 + ^147*2104 ~j 7*1301 — &127*2302 + hl303 + ^147*3404 V 7*1401 — frl27*2402 + &137*340S + frl404*'

Hence 612 »*12 7*23 7*24 7*13 1 7*3i 7*14 7*34 1
 * 1 7*23 7j4

7*23 1 7*34 (TZ (15),
 * 24 *34 1 |

and so on for the others, bn, bn, &c., we may call the net regressions of *i on a?2, xx on xz, &c., as before. By parity of notation, we have