VIII XI 6 ] /ntrodnrttttit Ixxix 



Here we find for 



r=-50, d/ N = -049,092; r = -55, d/N = -053,263, 

 and by interpolation for r = *5189 we have 



djN = '05066 and d = 507. 



Next we take "Tall" Fathers with "Tall" and "Tallish" Sons and find in the 

 same way d= 73'3; the difference of this and the former d gives 22'6 a the cell 

 frequency of "Tall" Fathers and "Tallish" Sons. Thus gradually the individual cell 

 frequency is built up from combinations of cell frequencies. 



It seemed worth while applying the "goodness of fit" test. Here, as we have 

 applied the normal surface assuming the marginal frequencies to be tfie same far 

 the surface and the sample, we have introduced nine restrictions instead of the 

 single usual one, accordingly we must look up P for n 25 8 = 17. The ^ 2 = 17*37 

 and we have accordingly P = '363, a reasonable fit. Thus the distribution of Stature 

 in Fathers and Sons may be described legitimately by a normal surface. 



TABLES X, XI XI 6 . 



For computing the Frequency Distribution of the First Product Moment Coefficient, 

 Pxy, in samples from an indefinitely large Normal Population. (Pearson, Jeffery and 

 Elderton, Biometrika, Vol. xxi. pp. 164 201.) 



(a) The aim of this table is to render it relatively easy to obtain a curve of fre- 

 quency for the distribution of p^y =pu in samples of size n drawn from a normal 

 bivariate population characterised by standard deviations <TI, 0-3 and correlation 

 coefficient p. 



We shall find it convenient to introduce a new variate v which is only p u 

 multiplied by a constant, a function of the characters of the sampled population, i.e. 



1 P 



Here p^ is the variable value of the product moment coefficient changing generally 

 with each sample of size n. The following are the constants for the distribution of v 

 and pu : 



_ a , u ia ............ (in), 



prf 



(v). 



The actual curve of frequency of v (or pu) depends upon a function T m (v), where 

 m = \n \, related to the Bessel function of the second kind with imaginary 

 argument. But beyond the value n = 25, or m= 11*5, Pearson curves of Types IV 

 and VI with their constants determined by (ii) (v) above, give excellent fits. Thus 



