45 



a critical importance. Unfortunately, as early as 1900, a mathe- 

 matical error was introduced into the statistical theory of good- 

 ness of fit, which has led to many inconsistencies. This error, 

 in its application to contingency tables, was pointed out by 

 Fisher (1922), and the method of correction was at the same time 

 indicated. In the present paper the disputed case of the four- 

 fold table is treated in detail. A mathematical proof of the 

 corrected formula is given, and the experiments of Yule, designed 

 to test this specific point, are shown to agree well with the 

 corrected formula, while they are wholly inconsistent with the 

 formula previously in use. 



ERRORS OF OBSERVATION. 



VII. R. A. Fisher. " Note on Dr. Burnside's recent 

 Paper on Errors of Observation." Proceedings of the 

 Cambridge Philosophical Society, 1923. Vol. XXL, 

 pp. 655-658. 



In small sample work, such as prevails in agricultural experi- 

 mentation, the traditional methods standardised in biometry and 

 in the theory of errors break down, so that more precise methods 

 must be used. The first of these was developed by " Student " 

 in 1908. In 1923 Burnside independently arrived at formula? 

 similar to, but not identical with, those of " Student." In the 

 present note attention is drawn to " Student's " paper, and an 

 exact proof is given of the accuracy of his formulae. 



THE PARTIAL CORRELATION COEFFICIENT. 



VIII. R. A. Fisher. " The Distribution of the Partial Cor- 

 relation Coefficient." Metron., 1924. Vol. III., pp. 

 329-332. 



In 1915 Fisher gave the exact sampling distribution of the 

 correlation coefficient, and showed that the current formula for 

 its probable error was inadequate when applied to small samples. 

 In the present paper it is shown that the same formula, with a 

 simple modification, is applicable to the distribution of the partial 

 correlation coefficient. The theoretical result so obtained is shown 

 to be in agreement with the experimental data hitherto available. 



STATISTICAL REQUIREMENTS OF ACCURATE TESTS. 



IX. R. A. Fisher. '* The Conditions under which y 2 



measures the Discrepancy between Observation and 



Hypothesis." Journal of the Royal Statistical Society, 



1924. Vol. LXXXVIL, pp. 442-450. 



In making tests of goodness of fit the expectations have 



often, or indeed usually, to be reconstructed from the actual data 



with which they are to be compared. In such cases it had not been 



observed that it is necessary that the methods used in this 



reconstruction should not involve errors of fitting comparable to 



the errors of random sampling. In the present paper it is 



demonstrated that this requirement can only be fulfilled if the 



statistics used in the reconstruction, are not only consistent, but 



efficient statistics. When all statistics so employed satisfy the 



criterion of efficiency, it is demonstrated that the measure of 



discrepancy, ^ 2 , may, in large samples, be used with precision. 



