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)