18 



KAKL PEARSON 



Interpolating from Table lY we have, for „o-,= -0349, r=-05, logx' = 07303 

 Thus the correlation is less than -05, and an inspection of the abac and rough extra- 

 polation indicates that it must be about -03. To test this, remember that for such 

 low values of r, the Gaussian curve gives the area closely. Now log x' = 0;4942 

 corresponds to x' = 3-12, and this from Table II to - log P = 0-428, or logP = r572, 

 i.e. P = -3733, which gives a single tail of -1866, or we must enter the probability 

 integral table with i(l +a) = -8134. This gives a;= "89 =r/o(r, = r/(-0349),^ or r= -031, 

 agreeing excellently with the value read from the abac by extrapolation. Using 

 Everitt's Tables we have for a Gaussian distribution 



•009,631 = -156,650r+-000,621r-'-|--025,271r'+-000,46K 



which gives r= -061 ±"024, a result within the limits of random sampling of the r 

 obtained by the previous method, i.e. '031. 



Illustration V. I take for this illustration absolutely Gaussian material for 

 a population of 1000 destined to give '80 correlation. 



Such a table is easUy constructed from Everitt's Supplementary Tables of the 

 Tetrachoric Functions (Biometrika, Vol. vill. p. 385). 



We find log x' = 2-4013, 



i(l + a,) = -726, i(l + a,) = -864. 



Hence, by Table V, Xa, = 1*3391, Xa2=l"5713, 



°"'=7ik'^-^^-=''''''- 



and 



Interpolating from Table IV, we have 



r = -8, log x' = 2-3828, r = -9, log x' = 2-5873, 



whence we find for log x' = 2-4013, r=-809. From the abac r=-81, as against the 

 r = -800 + -022 actual value. 



Illustration VI. I take another illustration of truly Gaussian material, namely 

 1000 cases distributed so as to give r= -50. 



Here log x'= 1-9452. 



We have ^(1 + a,) = -709, ^(1 +a,) = -813, 



giving Xa,= 1-3248, x«,= 1-4512, 



and „o-,.= -06080. 



