THE MEAN AND VARIANCE OF χ2, WHEN USED AS A TEST OF HOMOGENEITY, WHEN EXPECTATIONS ARE SMALL
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=N(E i) N. ij (8s tj) Fisher (1922) showed that when every si and tj was sufficiently large, x2 has the usual distribution, with (m 1) (n 1) degrees of freedom. Thus its mean is (mr-1) (n-I ), its variance 2(m1) (n-1). Haldane (1937, 1938, 1939) investigated the exact values of the moments of x2 when expectations are small, in (m x n)-fold tables with men or m(n 1) degrees of freedom, but did not attempt the present problem. This had previously been done by Cochran (1936), in the
[1] J. Haldane. THE EXACT VALUE OF THE MOMENTS OF THE DISTRIBUTION OF x2 USED AS A TEST OF GOODNESS OF FIT, WHEN EXPECTATIONS ARE SMALL , 1937 .
[2] R. Fisher. 019: On the Interpretation of x2 from Contingency Tables, and the Calculation of P. , 1922 .
[3] Harold Jeffreys,et al. Addenda: The Law of Error and the Combination of Observations , 1938 .
[4] W. G. Cochran. THE X2 DISTRIBUTION FOR THE BINOMIAL AND POISSON SERIES, WITH SMALL EXPECTATIONS , 1936 .