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REPORTS ON THE STATE OF SCIENCE, ETC. 



class w 6 , in the largest number of cases (101*5) the ' second egg ' is also in this class, 

 and the next largest numbers are in the neighbouring classes w 5 (95-5) and w 7 (75-75). 



The mere arrangement therefore in the table brings out the similarity in colour- 

 value between eggs in the same clutch. 



With pairs of measurements the same considerations as to class interval should 

 be applied, as with single variates, save that with close associations a finer grouping 

 may be required to make the class interval as small as a quarter of the average 

 standard deviation of the arrays (separate rows or columns). 



5. Summary of a Two-way Table. 



In all cases where two characters are considered the results can be displayed most 

 compactly in a two-way table, and this forms a convenient basis for the calculation 

 required if it is wished to study the relationship between them. Where the data 

 are in the form of numerical measurements, as in the first example, and if they con- 

 form to what is termed a Normal Correlation distribution, the contents of the table 

 may be described by five quantities. These, in addition to being useful in them- 

 selves, will serve as a statistical summary of the data when the two-way table cannot 

 be presented in full. These are (i) and (ii) the means and (iii) and (iv) the 

 variances of the two marginal distributions, and (v) the ' product-moment coefficient ' 

 which is calculated like the variance from the deviations from the means, using the 

 products of the deviations of the two variates (having regard to the positive and 

 negative sign of these deviations) instead of the squares of the deviations of a single 

 variate. From the three latter statistics any of the following may be at once obtained, 

 one or other of which will in almost all cases be of importance in the interpretation 

 of the data. Denoting the two variates by x and y, 



(a) the regression coefficient of y on x is the ratio of the product moment coeffi- 

 cient to the variance of x ; here x is regarded as the independent variate, and y 

 as dependent upon it ; 



(b) the regression coefficient of x on y is the ratio of the product moment coefficient 

 to the variance of y ; 



(c) the coefficient of correlation is the geometric mean of the two regressions, and 

 may be found either from them, or by dividing the product moment by the 

 geometric mean of the two variances, or by the product of the standard deviations. 

 (Section D, 2.) 



This method of description becomes inadequate if the material differs markedly 

 from Normal in the form of its distribution ; in such cases the two-way distribution 

 table should not be replaced by a summary. 



6. More than Two Variates. 



There is no compact form for the complete publication of sets of three or more 

 measurements. When the number of individuals is not too great, these may be set 

 out seriatim, each occupying a line of the table. If the number of class combinations 

 is sufficiently small, which can only occur if very broad classes are employed, it may 

 happen that the class combinations and the corresponding frequencies of occurrence 

 can be compactly listed. For storage in a form ready for immediate use, cards are 

 recommended, each card representing an individual with its numerical measures 



