Page:Proceedings of the Royal Society of London Vol 60.djvu/527

492 V » Vi, v 3, Vi their coefficients of variation, <r2/m2, <r3/m3, Wm4 respectively; r12, r23, r34, r41, r24, r13, the six coefficients of correlation ; cx, e2, 63, e4 the deviations of the four organs from their means, i.e, Xi = Wi + ej, ir2 = m2-\-e2, x 3 — ;r4 = -m/4 -|- <?4; the mean value of the index xx/x3, and iu the mean value of x2jxx; 2 b S2 the standard deviations of the indices xxjx3 and £r2/®4 respectively ; and n the total number of groups of organs.

We shall suppose the ratios of the deviations to the mean absolute values of the organs are so small that their cubes may be neglected. Then 1 mi r / \ / -- \ _i «i3 = “ S i —') = S | (1 + —V l + —) '} n \ x J nm 3 L\ m J \ _ 1mi (n | ^(e0 S(e3) >S(e163) S(g32)\ ^ nm3\ mx m3 m4m3 m3 ) ’

if we neglect quantities of the third order in But S (c 4) S(e3) = 0, ~ ' ‘ Hence: Similarly S ( ci£3) = n r13(ri<r3, and S (e 32) mx wg3 13 — (l + ^32— m3 0 ). m2 m4 (I + V42— ruv2V i)............................. (ii).

Thus we see that the mean of an index is not the ratio of the means of the corresponding absolate measurements, but differs by a quantity depending on the correlation and variation coefficients of the absolute measurements.

(3) Proposition I.—To find the standard deviation of an index in terms of the coefficients of variation, and coefficient of correlation of the two absolute measurements.

w2 132 s ( —- \«3 - i f mfi g rnf {(1+^) m x f S — m321 \ m x m 3 \ 21 quare terms j. i32(nvfi+ n v f—2 n, if we neglect cubic terms. . *. 2 13 = i13*/ (vfi