CHAPTER XI.
MISCELLANEOUS THEOREMS INVOLVING THE USE OF
THE CORRELATION-COEFFICIENT.
1. Introductory—2. Standard-deviation of a sum or diflerence—3-5. Influence
 of errors of observation and of grouping on the standarddeviation—6-7.
 Influence of errors of observation on the correlations
coefficient (Spearman’s theorems)—8. Mean and standard-deviation
of an index —9. Correlation between indices — 10. Correlationcoefficient
 for a two- x two-fold table—11. Correlation-coefficient
for all possible pairs of IV values of a variable—12. Correlation due
to heterogeneity of material —18, Reduction of correlation due to
mingling of uncorrelated with correlated material — 14-17. The
weighted mean—18-19. Application of weighting to the correction
of death-rates, ete., for varying sex and age-distributions—20. The
weighting of forms of average other than the arithmetic mean.
L. Ir has already been pointed out that a statistical measure, if
it is to be widely useful, should lend itself readily to algebraical
treatment. The arithmetic mean and the standard-deviation
derive their importance largely from the fact that they fulfil this
requirement better than any other averages or measures of dispersion
 ; and the following illustrations, while giving a number of
results that are of value in one branch or another of statistical
work, suffice to show that the correlation-coefficient can be treated
with the same facility. This might indeed be expected, seeing
that the coefficient is derived, like the mean and standard-deviation,
 by a straightforward process of summation.
2. To find the Standard-deviation of the sum or difference Z of
corresponding values of two variables X; and X,.
Let 2, x), #, denote deviations of the several variables from
their arithmetic means. Then if
Z=X +X,
evidently
f=; + x,

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