xlii Tables for Statisticians and Biometricians [IV 



normal, then PI = '01 and P 2 = '02 would correspond to values of X 1 = 2'33 and 

 X 2 = 2'05 or to deviations greater than 3'45 and 3 '04 times the probable error. It 

 is not unusual to consider values greater than 2'5 or 3'0 times the probable error 

 as significant and values less than 2'5 as of doubtful significance. 



We have no need to consider values of if or R z , which are less than rf or R 2 , 

 i.e. cases in which X is negative, for such values being less than the mean values 

 for no correlation cannot indicate significance on the basis of a sample of the 

 given size. 



Illustrations. We shall throughout use the observed value of rj 2 , uncorrected 

 for number of arrays. We shall use \ d to denote the value of X deduced from the 

 observations. 



(i) Association of Crowding with General Astigmatism. We require to find if p . A , 

 the square of the correlation ratio of p the number of persons per room in the 

 home on general astigmatism. We have the data for 716 schoolboys divided into 

 eight arrays*, i.e. J!\T=716, n = 8. 



We have from the data y z p . A = '022,821, and from the table 

 $72= -009,790, c^ = -005,200. 



022,821 - -009,790 

 Hence X d = . 005>200 



P 2 = '02 corresponds to Xa = 2-54:. 



Accordingly, by rough extrapolation, we find that about once in 45 or 46 trials 

 such a value of tf would occur if there were no association between general 

 astigmatism and overcrowding. It is possible therefore that there may be some 

 relationship, but no great stress can be laid on the result. We need more extended 

 data to be certain of such a slight association. 



(N.B. Since we are assuming normality in our distribution of variates, it is 

 immaterial whether we test for if p . A or r) 2 A , p .) 



(ii) Influence of Familial Income on Gorneal Astigmatism. We wish to consider 

 the influence of poverty on corneal astigmatism. Our dataf consist of 228 boys 

 arranged in nine arrays. 



Thus N = 228, n = 9, and our table shows us that 



if = -035,242, 0^ = -017,332, 

 while the observations give ^ 2 c ^. r = '139,397. 



139,397 --035,242 



ThuS Xd= -017,332 



\ d is so much greater than X x = 2*96 that we conclude that the odds are far higher 

 than 99 to 1 against this correlation ratio arising from uncorrelated material. 

 We conclude therefore that corneal astigmatism is influenced by poverty. 



* Annals of Eugenics, Vol. in. pp. 29 et seq. f Ibid. p. 46. 



