CHAPTER XI. 
MISCELLANEOUS THEOREMS INVOLVING THE USE OF 
THE CORRELATION-COEFFICIENT. 
1. Introductory—2. Standard-deviation of a sum or diflerence—3-5. In- 
fluence of errors of observation and of grouping on the standard- 
deviation—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. Correlation- 
coefficient 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 dis- 
persion ; 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-devia- 
tion, 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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