V.—MANIFOLD CLASSIFICATION. 73 
(2) Lrers, G. F., ‘““Die Bestimmung der Abhingigkeit zwischen den 
Merkmalen eines Gegenstandes,” Berichte der math.-phys. Klasse der 
kgl. Sachsischen Gesellschaft der Wissenschaften ; Leipzig, 1905. (A 
general discussion of the problems of association and contingency.) 
(3) PEARSON, KARL, “On a Coefficient of Class Heterogeneity or Divergence,” 
Biometrika, vol. v. p. 198, 1906. (An application of the contingency 
coefficient to the measurement of heterogeneity, e.g. in different 
districts of a country, by treating the observed frequencies of some 
quality A,, A, . ... A, in the different districts as rows of a con- 
tingency table and working out the coefficient: the same principle is 
also applicable to the comparison of a single district with the rest of 
the country.) 
Isotropy. 
(4) Youre, G. U., “On a Property which holds good for all Groupings of a 
Normal Distribution of Frequency for Two Variables, with applications 
to the Study of Contingency Tables for the Inheritance of Unmeasured 
Qualities,” Proc. Roy. Soc., Series A, vol. lxxvii., 1906, p. 324. (On 
the property of isotropy and some applications.) 
(6) YuLe, G. U., “On the Influence of Bias and of Personal Equation in 
Statistics of Ill-defined Qualities,” Jour. Anthrop. Inst., vol. Xxxvi., 
1906, p. 325. (Includes an investigation as to the influence of bias 
and of personal equation in creating divergences from isotropy in 
contingency tables.) 
Contingency Tables of two Rows only. 
(6) PEARSON, KARL, “On a New Method of Determining Correlation between 
a Measured Character 4 and a Character B of which only the Percentage 
of Cases wherein B exceeds (or falls short of) a given Intensity is recorded 
for each Grade of 4,” Biometrika, vol. vii., 1909, p. 96. (Deals with a 
measure of dependence for a common type of table, e.g. a table showing 
the numbers of candidates who passed or failed at an examination, for 
each year of age. The table of such a type stands between the con- 
tingency tables for unmeasured characters and the correlation table 
(chap. 1x.) for variables. Pearson’s method is based on that adopted 
for the correlation table, and assumes a normal distribution of fre- 
quency (chap. xv.) for B.) 
(7) PearsoN, KARL, “On a New Method of Determining Correlation, when 
one Variable is given by Alternative and the other by Multiple 
Categories,” Biometrika, vol. vii., 1910, p. 248. (The similar 
problem for the case in which the variable is replaced by an un- 
measured quality.) 
EXERCISES. 
(1) (Data from Karl Pearson, ‘“ On the Inheritance of the Mental and Moral 
Characters in Man,” Jour. of the Anthrop. Inst., vol. xxxiil., and Biometrika, 
vol. iii.) Find the coefficient of contingency (coefficient of mean square 
contingency) for the two tables below, showing the resemblance between 
brothers for athletic capacity and between sisters for temper. Show that 
neither table is even remotely isotropic. {As stated in § 7, the coefficient of 
contingency should not as a rule be used for tables smaller than 5 x 5-fold : 
these small tables are given to illustrate the method, while avoiding lengthy 
arithmetic.) 
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