XXXI 1 ", LI-*] //i//W//r//o,, rlxxi 



(viii). Let us apply these results t<> h'ml more accurately the 

 of tin- r-distribution in our previous ,.\:un|l.-, p='6, n=16. 



\\V h.-ivosccn that formula (xxx) and Table LI" gave us the result r= '000,1 SO, 

 and therefore p^ fp -= '4015,0810 for a first, approximation, wlien<-<- 



p 4 = '1012,087.5, 1 -po = -8387,91 25. 

 Accordingly r<|ii:ilion.s (xxxii) and (xxxiii) become 



= 15 + 12-4467.5110ff' - 13-4206.6000 A" A'" 

 13'4206 7 ,6000 (A" + E")- 12-4467,5 1 10 ' 



12-5818,6875 A 1 ' 2 - 11-6437,3490 #' -14 = 0, 

 1 3-4206,6000 E" 2 - 12-4467,5 110 E" - 15=0. 



Solving these two quadratics for the positive roots, we find 

 E' = 1-6145,9598, E" = 1 '6 18 1,4766. 



Then substituting in the expression for e we find 



Accordingly for our next approximation 



p2 = p z + = -4015,0810 + -0010,6416 



= 4025,7226, 

 ?=p*/p = -67 0,954, 



a change from '669,180 deduced from formula (xxx) of only 0'27 / . 

 A second approximation reduces the value to "6709. 



Pearson's formula (see equation (ii)) gives f ='6708, or is correct to a unit in 

 the fourth figure, i.e. '013 / , but it involves a prior knowledge of <r r , /8 t and j. 

 Any of the above values are quite adequate for plotting the curve of r-frequency. 



(v) On the Determination of the "most likely" and "most reasonable " Values of 

 the Correlation in the Parental Population. 



We now turn to a still more difficult determination, that of p, the correlation 

 coefficient in the parent population from a knowledge of r in a small sample. 



Here we must consider the character of the parent population itself. Do we 

 know nothing whatever about it ? Or, does it belong to a certain class of populations 

 of which we know more or less accurately the range within which p is likely to lie? 

 For example, suppose we are dealing with a small sample of fathers and sons, and 

 the coefficient of correlation for some character came out for the sample > "7 ; in the 

 minds of those who have studied the subject of parental inheritance it would be un- 

 reasonable to suppose that the p of the sampled population could be as large as this. 

 Without knowing the particular value of p we can in many cases, from previous 

 experience, assert that it most probably lies within a certain range of values, and 

 that these cluster with a mqre or less definite variation round a central value. We 



