XVI I 



lntr<xlm-tin 



ci 



TABLE XVI. 



To determine the Probable Error of the Biserial Expression for the Correlation 

 Coefficient. (H. E. Sopcr, Biometrika, Vol. x. pp. 384390.) 



If one variate y be given quantitatively and the other x by alternative cate- 

 gories thus : 



Frequencies of y. 



then, if the distribution be normal, the correlation is given by 



r = x 



where y and a y are the mean and standard deviation of the y-variate, y z is the 

 mean of the second or lesser series w 2 . (=%N(1 a)) and z is the reduced ordinate 

 corresponding to the |(1 + a) of Table II of Part I. 



If r be the correlation in the parent population, Soper shows that the mean 

 r (= r), and the standard deviation of r (o>), in samples of size n, are given approxi- 

 mately by 



(i), 



<r,-~ 



Table XVI gives \ lt \ 2 2 and X 3 - 



Now we do not know r in the population sampled, nor the distribution of the 

 correlation coefficient in samples, but when the size of the sample n is reasonably 

 large, say n ^ 50, and r is not extremely high, then r will not differ very widely 

 from the modal value, i.e. the value we are more likely to get than any other 

 single value. Hence 



= r/\l+ - (\i + \ r 2 ) >, approximately (iii), 



i 

 <r r = - {\ 2 2 AS r 2 + **}*, approximately (iv), 



where for r we insert the value observed in the sample. 



