378 Testing the Goodness of Fit of Frequency Distributions. 



In the Eye-colour case the determination of P 2 (11*54671) 

 must be a little less accurate, for the ordinary tables do not 

 go to such high values of the argument. Interpolating 

 between the values of — log F for 11 and 12 in Table IV. of 

 the ' Tables for Statisticians/ we find 



-logF(ll\54671) = 30-46966, 

 or log F = 31-53034. 



Hence i(l~«) = 3-3911/10 31 



and P 2 ( % ) = 6-7822/10 31 = '007/10 28 . 



But* P 4 (%) = 1-035/10 23 ; 



tlms <r p = i 2-4548(l-035/10 28 - -007/10 28 ) 



|2S 



= P2618/10 2 



Thus the probable error of P is '851 /10 28 , 

 The numerical results for the two cases are : 



(i.) P = -8713 + -4049, 



(ii.) P = l-035/10 23 + -851/10 28 . 



Now the whole range of P is from to 1, and the 

 probable error of P shows that within the limits of random 

 variation P for the Datura table may be anything from 

 to 1. The data are accordingly absolutely inadequate for 

 the purpose of indicating whether or no colour of flower 

 and prickliness of fruit are associated. In the second case 

 the probable error is such that P might really be zero, 

 but it is not such that P could ever be replaced by anything 

 but another indefinitely small probability. In other words, 

 the data are amply sufficient to demonstrate that eye-colours 

 in parent and child are definitely associated. On the other 

 hand, the large value of the probable error of P, relative to 

 the size of P itself ) justifies a further emphasis of the caution 

 given above to compare indefinitely small magnitudes of P 

 of different orders with considerable reservation. 



* • Tables for Statisticians,' p. xxxv. 



