XIL—CORRELATION : MISCELLANEOUS THEOREMS, 211
Squaring both sides of the equation and summing,
3(2) = 3(ry?) + (2) £ 25(oy2,).
That is, if » be the correlation between z; and xy, and o, oy, oy
the respective standard-deviations,
o?=0.2+ 0,2 + 2r.0y0, +: (1)
If 2; and x, are uncorrelated, we have the important special case
o?=02+ 0,2 : (2)
The student should notice that in this case the standard-
deviation of the sum of corresponding values of the two variables
is the same as the standard-deviation of their difference.
The same process will evidently give the standard-deviation of a
linear function of an number of variables. For the sum of a
series of variables XV, |. | | X, we must have
ol=0 +0 : +0 + +024 20.000, + 275.000,
BNE hE are
7, being the correlation beween X, and X,, r,, the correlation
between X, and X,, and so on.
3. Influence of Errors of Observation on the Standard-deviation.
—The results of § 2 may be applied to the theory of errors of
observation. Let us suppose that, if any value of X be observed
a large number of times, the arithmetic mean of the observations
is approximately the true value, the arithmetic mean error being
zero. Then, the arithmetic mean error being zero for all values
of X, the error, say §, is uncorrelated with X. In this case if x, be
an observed deviation from the arithmetic mean, « the true devia-
tion, we have from the preceding
0,2=0.2+057 . . . (3)
The effect of errors of observation is, consequently, to increase the
standard-deviation above its true value. The student should
notice that the assumption made does not imply the complete in-
dependence of X and 8: he is quite at liberty to suppose that
errors fluctuate more, for example, with large than with small
values of X, as might very probably happen. In that case the
contingency-coefficient between X and J would not be zero,
although the correlation-coefficient might still vanish as supposed.
4. Influence of Grouping om the Standard-deviation.—The
consequence of grouping observations to form the frequency
distribution is to introduce errors that are, in effect, errors of
