214 THEORY OF STATISTICS. 
whole, to be safer, for it eliminates the assumption that the errors 
in # and in y, in the same series of observations, are uncorrelated. 
An insufficient though partial test of the correctness of the 
assumptions may be made by correlating #, — 2, with ¥1—¥,: this 
correlation should vanish. Evidently, however, it may vanish 
from symmetry without thereby implying that all the correlations 
of the errors are zero. 
8. Mean and Standard-deviation of an Index.—(Ref.11.) The 
means and standard-deviations of non-linear functions of two or 
more variables can in general only be expressed in terms of the 
means and standard-deviations of the original variables to a first 
approximation, on the assumption that deviations are small 
compared with the mean values of the variables. Thus let it be 
required to find the mean and standard-deviation of a ratio or 
index Z = X,/X,, in terms of the constants for X, and X,. Let [7 
be the mean of Z, M, and J, the means of X; and X,. “Then 
lin 2) zy ¥ 
7-533) rarx(+ 3) 
Expand the second bracket by the binomial theorem, assuming 
that »,/M, is so small that powers higher than the second can 
be neglected. Then to this approximation 
1M, 1 1 3 | 
I== 77 - I) == 7 ) . 
That is, if r be the correlation between x; and #,, and if v, = o,/M,, 
vy =0o/M,, . 
Y 
I= a! — 70,0, + Vy?) 9) 
If s be the standard-deviation of Z we have 
1_/7X\2 
CL Joe Nl] 
S47 7% 7) 
1 M2 ( xy \? z Ne 
“7A 3 03 
Expanding the second bracket again by the binomial theorem, 
and neglecting terms of all orders above the second, 
1.0.2 Z\2 7 7,2 
2 mae a iNT Sr 
=i -¥ 2A] +5) (1 I, +378) 
M2 
= 7 1 + 2,2 — drow, + 30,2)